text:
### Meager Nowhere-Dense Games (IV): $n$-Tactics

Marion Scheepers
Source: J. Symbolic Logic Volume 59, Issue 2 (1994), 603-605.

#### Abstract

We consider the infinite game where player ONE chooses terms of a strictly increasing sequence of first category subsets of a space and TWO chooses nowhere dense sets. If after $\omega$ innings TWO's nowhere dense sets cover ONE's first category sets, then TWO wins. We prove a theorem which implies for the real line: If TWO has a winning strategy which depends on the most recent $n$ moves of ONE only, then TWO has a winning strategy depending on the most recent 3 moves of ONE (Corollary 3). Our results give some new information concerning Problem 1 of [S1] and clarifies some of the results in [B-J-S] and in [S1].

Primary Subjects: 03E99
Secondary Subjects: 04A99, 90D44
Full-text: Remote access
If you are a member of the ASL, log in to Euclid for access.
Full-text is available via JSTOR, for JSTOR subscribers. Go to this article in JSTOR.


text:
# Partial Sum Arithmetic Sequence Calculator

First Member (a1)
Common Difference (d)

# Partial Sum of Arithmetic Sequence Calculator

Free partial sum of arithmetic sequence calculator tool that helps you to display the partial sums of the arithmetic sequence very easily in a small number of seconds. You can use this calculator by giving the input parameters in the given fields and by tapping the calculate button that gives you the desired output.

Partial Sums of Arithmetic Sequence Calculator: Know the steps to calculate the partial sum of arithmetic sequence very easily? By using this calculator finding a partial sum is not a big deal. And not only with the calculator tool, but we can also calculate the partial sum with the formula of the partial sum of an arithmetic sequence. We are going to share the formulas, steps, and solved examples in further sections below. So, let's start...

## How to Find the Partial Sum of Arithmetic Sequence?

In mathematics, the arithmetic sequence is the set of numbers where the difference between the consecutive terms is different.

Now, if we see the formula for the partial sum of a sequence,

Sn = [(a1 + an )n ] / 2

where,

Sn = partial sum of a sequence.

a = a1 + (n-1)d

d = common difference.

a= first term of sequence.

Follow the guidelines carefully that are shown below to calculate the partial sum of arithmetic sequence easily.

• Initially, write the values that were given in the problem.
• Then, now you need to find the value of an by using the formula, a = a1 + (n-1)d.
• After that, we need to apply the formula to find the partial sum of the sequence, i.e., Sn = [(a1 + an )n ] / 2.
• Finally, after substituting the values and simplifying them you will get the output.

Example:

Question: Find the partial sum of the sequence S10, if a= 2, d = 2?

Solution:

Given, a= 2, d = 2, S10 = ?

To find the avalue we need to apply the formula,

a = a1 + (n-1)d

a10 = 2 + (9)(2) = 2 + 18 = 20.

Now to find the partial sum of S10

Sn = [(a1 + an )n ] / 2

As we found the ''an" value already, substitute that in the formula.

S10= [(a1 + an )n ] / 2

S10= [(2 + 20)10 ] / 2

S10= [(22)10 ] / 2

S10= [(220)] / 2

S10= 110.

Hence, the partial sum of the sequence S10 = 110.

For more calculators tools like this, you can go to sequencecalculators.com and make your work easier.

## FAQs on Partial Sum of Sequence Calculator

1. How do you find the partial sum of the sequence?

We find the partial sum of a sequence by using the formula, i.e., Sn = [(a1 + an )n ] / 2.

2. What are the steps to use this partial sum of sequence calculator?

• Firstly, give the input parameters in the fields that are given.
• Next, you need to click on the calculate button.
• Finally, you will see the desired result on the page in seconds.

3. What is the partial sum of sequence?

The partial sum of the sequence is nothing but the sum of the part of the sequence.

4. What do you mean by arithmetic sequence?

The sequence of numbers that have the difference of any two consecutive terms is constant then is called an arithmetic sequence.

format: plain


text:
# How to get from $2^{99} \cdot (\cos{(99\times \frac{5\pi}{6}) + i\cdot \sin{(99\times \frac{5\pi}{6})}})$ to $0+2^{99}i$?

How do I get from 2nd last to last step? How did they simplify cos & sin $99\times \frac{5\pi}{6}$?

-

## 3 Answers

Well, $99\cdot \frac{5\pi}{6}=33\cdot \frac{5\pi}{2}=\frac{165\pi}{2}$, so

$\sin\left(\frac{165\pi}{2}\right)=\sin\left(82\pi + \frac{\pi}{2}\right)=\sin(\frac{\pi}{2})=1$. Similarly, $\cos\left(\frac{165\pi}{2}\right)=0$.

-

$99\cdot \frac {5 \pi}{6}=82.5\pi$ and you can ignore the multiples of $2 \pi$

-

$\cos \left(99 \times \frac{5 \pi}{6} \right) = \cos \left(33 \times \frac{5 \pi}{2} \right) = \cos \left(\frac{165 \pi}{2} \right) = 0$ since $\cos(\frac{n \pi}{2}) = 0$ whenever $n$ is odd.

$\sin \left(99 \times \frac{5 \pi}{6} \right) = \sin \left(33 \times \frac{5 \pi}{2} \right) = \sin \left(\frac{165 \pi}{2} \right) = \sin \left(82 \pi + \frac{\pi}{2} \right) = 1$ since $\sin(2n \pi + \frac{\pi}{2}) = 1$.

-


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# SABR Calibration: Normal vs Log-Normal Market Data

This question is about getting some clarification as to how to understand market quotes for normal & log-normal vols together with certain model assumptions.

So let us define

1. $C_{BS}(F_0,K,T,\sigma,\beta)=\mathbb{E}[(F_T-K)^+]\quad \text{with}\quad dF_t=\sigma F_t^\beta dW_t$$2. C_{SABR}(F_0,K,T,\sigma_0,\beta,\nu,\rho)=\mathbb{E}[(F_T-K)^+]$$\text{with}\quad dF_t=\sigma_t F_t^\beta dW_t,\quad \sigma_t=\nu \sigma_t dZ_t,\quad dW_tdZ_t = \rho dt$$And for any given combination of F_0,K,T,\sigma_0,\beta,\nu,\rho the SABR-implied vol v_{SABR} is the quantity such that the following relationship holds$$C_{BS}(F_0,K,T,v_{SABR},1) = C_{SABR}(F_0,K,T,\sigma_0,\beta,\nu,\rho)$$See http://www.math.ku.dk/~rolf/SABR.pdf right-hand side of page 89. Now let us assume that for a fixed expiry/tenor we are given a set of volatility market quotes: Ideally, I want to calibrate the SABR model to it. So when I set \beta=1 and calibrate \sigma_0,\nu,\rho to the log-normal vols, I get a very nice fit: However, when I set \beta=0 and calibrate \sigma_0,\nu,\rho to the normal vols, I get a very poor fit: So I have two questions: 1. Is my definition of the SABR vol v_{SABR} correct? For example, would$$C_{BS}(F_0,K,T,v_{SABR},\beta) = C_{SABR}(F_0,K,T,\sigma_0,\beta,\nu,\rho)$$perhaps be more correct? Note that the difference here is the \beta in C_{BS} as opposed to having a 1 there. 2. Is maybe my normal vol market data of an atypical shape causing SABR to only find a poor fit? Or is my SABR implementation faulty? ## 2 Answers I think you did something wrong in translating the input to numerics. As pointed out by dm63 normal vols are quoted in basis points. Using equation A.67a) from the Hagan paper you linked we see (setting \beta = 0)$$\sigma_N(K) = \alpha\frac{\xi}{x(\xi)}\left[1+\frac{2-3\rho^2}{24}\nu^2\tau_{exp}\right]$$where \tau_{exp} = 0.25 in your example and$$\xi = \frac{\nu}{\alpha}(f-K)x(\xi) = \log{\left(\frac{\sqrt{1-2\rho\xi+\xi^2}-\rho+\xi}{1-\rho}\right)}$$I've implemented a very simple (not at all optimized) code just as an example:  #maket data iVol <- c(46.6,49.8,52.3,55.2,58.8,62.8,72,92.2)/10000 strike <- c(0.298,0.798,1.048,1.298,1.548,1.798,2.298,3.298)/100 exp <- 0.25 atmF <- 1.298/100 #object function f.obj <- function(x,strike,iVol,exp,atmF) { return(1/length(strike)*sum((iVol-sigma.hat(x,strike,atmF,exp))^2)) } # approximatino using formula A.67a) in Hagan paper sigma.hat <- function(x,strike,atmF,exp) { #x[1] = alpha #x[2] = nu #x[3] = rho xi <- x[2]/x[1]*(atmF-strike) x.xi <- log(((sqrt(1-2*x[3]*xi+xi^2)-x[3]+xi)/(1-x[3]))) ret <- ifelse(abs(strike-atmF)<10^(-4), x[1]*(1+((2-3*x[3]^2)/(24))*x[2]^2*exp),x[1]*((xi)/(x.xi))*(1+((2-3*x[3]^2)/(24))*x[2]^2*exp)) return(ret) } # fit the model sol1 <- nlm(f.obj, c(atmF, 0.1, 0.5), strike = strike, iVol = iVol, exp= exp, atmF = atmF)$estimate
sol2 <- nlm(f.obj, c(atmF,sol1[2],sol1[3]),strike = strike, iVol = iVol, exp = exp, atmF = atmF)$estimate sol1 sol2 x.seq <- seq(0.9*min(strike),1.1*max(strike),0.001) y.seq <- sigma.hat(sol2,x.seq,atmF,exp)*10000 plot(100*x.seq,y.seq,type="l",col="red",xlab = "Strike", ylab = "implied Vol", main = "Sabr Normal model") points(100*strike,iVol*10000,col="blue")  leading to the following fit: • Agree that the formula for$\sigma_N(K)$is way more accurate than the$\sigma_{BS}(K)$with$\beta=0\$. – Jaehyuk Choi Oct 13 '18 at 17:34
• @JaehyukChoi the main issue here is not about which formula is more accurate. This is about what the formula output is: normal (Bachelier) vols or Black-Scholes vols. If you try to fit normal vols with the formula that gives you Black vols, you will of course run into problems, unless you convert those Black vols back to normal vols. – jherek Mar 20 '19 at 21:34

I think (1) is the issue. You need to compare market normal vols to normal vols implied by the sabr model. (2) is not the issue - these vols look reasonable.

By the way we express normal vols in bp per annum, not percent!


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# zbMATH — the first resource for mathematics

## Maggioni, Mauro

Compute Distance To:
 Author ID: maggioni.mauro Published as: Maggioni, M.; Maggioni, Mauro External Links: MGP · Wikidata
 Documents Indexed: 31 Publications since 2000
all top 5

#### Co-Authors

 6 single-authored 4 Coifman, Ronald Raphaël 3 Chen, Guangliang 3 Little, Anna V. 2 Jones, Peter Wilcox 2 Rosasco, Lorenzo A. 2 Schul, Raanan 2 Szlam, Arthur D. 1 Allard, William K. 1 Bongini, Mattia 1 Bremer, James C. 1 Chui, Charles Kam-tai 1 Crosskey, Miles M. 1 Czaja, Wojciech 1 Drineas, Petros 1 Fornasier, Massimo 1 Gerber, Samuel 1 Guinney, Justin 1 Hansen, Markus 1 Iwen, Mark A. 1 Katz, Nets Hawk 1 Krop, Elliot 1 Lanteri, Alessandro 1 Liao, Wenjing 1 Lu, Fei 1 Mahadevan, Sridhar 1 Mahoney, Michael W. 1 Minsker, Stanislav 1 Mukherjee, Sayan 1 Murphy, James M. 1 Strawn, Nate 1 Tang, Sui 1 Vigogna, Stefano 1 Weiss, Guido L. 1 Wu, Qiang 1 Zhong, Ming
all top 5

#### Serials

 7 Journal of Machine Learning Research (JMLR) 5 Applied and Computational Harmonic Analysis 2 Monatshefte für Mathematik 2 Proceedings of the National Academy of Sciences of the United States of America 2 Notices of the American Mathematical Society 1 SIAM Journal on Matrix Analysis and Applications 1 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 1 Mathematical Research Letters 1 The Journal of Fourier Analysis and Applications 1 Annales Academiae Scientiarum Fennicae. Mathematica 1 Multiscale Modeling & Simulation 1 Information and Inference
all top 5

#### Fields

 11 Computer science (68-XX) 10 Statistics (62-XX) 9 Harmonic analysis on Euclidean spaces (42-XX) 6 Numerical analysis (65-XX) 6 Information and communication theory, circuits (94-XX) 4 Probability theory and stochastic processes (60-XX) 3 Global analysis, analysis on manifolds (58-XX) 2 Partial differential equations (35-XX) 2 Calculus of variations and optimal control; optimization (49-XX) 2 Biology and other natural sciences (92-XX) 1 Combinatorics (05-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Group theory and generalizations (20-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Abstract harmonic analysis (43-XX) 1 Operator theory (47-XX) 1 Differential geometry (53-XX) 1 Operations research, mathematical programming (90-XX) 1 Systems theory; control (93-XX) 1 Mathematics education (97-XX)


text:
# Newton-metre

(Redirected from Newton metre)
Newton-metre
One newton-metre is the torque resulting from a force of one newton applied perpendicularly to the end of a moment arm that is one metre long.
General information
Unit systemSI derived unit
Unit oftorque
SymbolN⋅m or N m
Conversions
1 N⋅m in ...... is equal to ...
FPS system   0.73756215 ft lbf
inch⋅pound-force   8.8507 in lbf
inch⋅ounce-force   141.6 in oz

The newton-metre (also newton metre; symbol N⋅m[1] or N m[1]) is a unit of torque (also called moment) in the SI system. One newton-metre is equal to the torque resulting from a force of one newton applied perpendicularly to the end of a moment arm that is one metre long. The nonstandard abbreviation Nm occurs in some fields.

It is also used less commonly as a unit of work, or energy, in which case it is equivalent to the more common and standard SI unit of energy, the joule.[2] In this usage the metre term represents the distance travelled or displacement in the direction of the force, and not the perpendicular distance from a fulcrum as it does when used to express torque. This usage is generally discouraged,[3] since it can lead to confusion as to whether a given quantity expressed in newton-metres is a torque or a quantity of energy.[4] However, since torque represents energy transferred or expended per angle of revolution, one newton-metre of torque is equivalent to one joule per radian.[4]

Newton-metres and joules are dimensionally equivalent in the sense that they have the same expression in SI base units:

${\displaystyle 1\,{\text{N}}{\cdot }\mathrm {m} =1{\frac {{\text{kg}}{\cdot }{\text{m}}^{2}}{{\text{s}}^{2}}}\quad ,\quad 1\,\mathrm {J} =1{\frac {\mathrm {kg} {\cdot }\mathrm {m} ^{2}}{\mathrm {s} ^{2}}}}$

Again, N⋅m and J are distinguished in order to avoid misunderstandings where a torque is mistaken for an energy or vice versa. Similar examples of dimensionally equivalent units include Pa versus J/m3, Bq versus Hz, and ohm versus ohm per square.


text:
• English only

# Satisfiability of Sets of Formulas

We next introduce sets of formulas as a way of talking about potentially infinite conjunctions of formulas. We will need this when reducing reasoning in first-order logic to reasoning in propositional logic.

We say that interpretation is a model for a set of formulas , written , iff for each , . In other words, we view a set of formulas as a (potentially infinite) conjunction; when is finite then is the same condition as .

Example: the set is an infinite satisfiable set. The example of one satisfying interpretation is the interpretation which evaluates all variables to .

Clearly, if and , then also .

We say that a set of formulas is satisfiable iff there exists an interpretation such that .


text:
In this tutorial, we’ll guide you through the essential steps of building a TensorFlow and a PyTorch working enrivonment on your local machine. For the course we'll be using TensorFlow, but PyTorch is also a very popular framework that you might encounter in the future.

System requirements

• Windows 7, 8 and 10, or Ubuntu Linux 14.04 and 16.04.
• NVIDIA graphics card with CUDA compatibility higher than version 3.0. Check out CUDA gpu for your card’s compatibility. Laptops are usually equipped with NVIDIA GeForce or Quadro graphics cards.
• Note: Because Mac doesn't have NVIDIA GPU, you can't install a GPU version of TensorFlow. However, a CPU version is available in case you want to debug your code and run it on the cloud. An official MAC OS X installation guide is proivded by TensorFlow official document: https://www.tensorflow.org/install/install_mac.

Installation Guide

Note: The process is time-consuming/complex for starters, so please prepare an external power supply if necessary. For reference, an official instruction is here: https://www.tensorflow.org/install/.

Step 1: Anaconda Installation

Anaconda is a data science platform using Python. It also provides a Python package manager that lets you install, update and remove packages.

• Install Anaconda to your local computer. If you have installed Python before, the Anaconda installer will prompt you whether you want to use Anaconda Python as your system default Python interpreter. It’s recommended that you use Anaconda Python as your system default Python.
• Check if Anaconda is correctly installed:
• Windows: open a Windows command prompt window by pressing win+R and typing cmd. In the window, type python and hit Enter. If you see the following python interpreter, which indicates it is an Anaconda interpreter, then it is finished. Type exit() to quit Python.
• Linux: open a console, and the rest is the same as in Windows.
• If python doesn't work (in which case the system doesn't find this command), or the interpreter is not the Anaconda iterpreter (in which case you've installed Python before and Anaconda is not specified as your default Python), you may choose one of the following three:
• Try reinstalling Anaconda again;
• Search "anaconda prompt" in your system and use it. This works as well.
• locate the path where the Anaconda python executable is installed, and include it into your system's PATH environment variable. Instructions on how to update PATH variable can be found here: Windows and Ubuntu. After that, type python again to test.

You can use conda list in the command window to see all the packages you’ve installed.

Step 2: Create Anaconda virtual environment

We are going to create a virtual environment on this Anaconda platform, and install necessary modules. A virtual environment is a named, isolated, working copy of Python that maintains its own files, directories and paths so that you can work with specific versions of libraries or Python itself without affecting other Python projects. Virtual environments make it easy to cleanly separate different projects and avoid problems with different dependencies and version requirements across components. The conda command is the preferred interface for managing installations and virtual environments with the Anaconda Python distribution. (managing conda environments.)

• In the command window, type conda create -n dlWorks python=3.5. If you wish, you can replace the name that we chose for the virtual environment “dlWorks” with the name of your choice. The reason we use python 3.5 version is because TensorFlow runs on python version 3.5.
• After installation, type activate dlWorks to activate the virtual enrivonment. (For Linux users, type source activate dlWorks) Your command prompt will contain the name of your environment, as in the cmd window below. (The name for this environment is ‘gpu’ rather than 'dlWorks' in this case)
• Install some necessary packages into your environment 'dlWorks' for future use. For example, to install multiple packages, type conda install pandas numpy scipy pillow matplotlib scikit-learn  in the command window.
• To list all of your virtual environments, activate/deactivate and switch between them, see managing conda environments.

Step 3: Install CUDA and GPU drivers

Compute Unified Device Architecture (CUDA) is a parallel computing platform and programming model created by NVIDIA. It harnesses the full power of graphics processing units (GPUs) for deep learning purposes.

• Install CUDA: install CUDA to your local machine. Remember the install path because it’ll be helpful during cuDNN installation. This installation can last for tens of minutes.

Step 4: Install cuDNN

cuDNN is a GPU-accelerated library for deep learning.

Note 1: This step requires you to create an NVIDIA account.

Note 2: For this step, if you're familiar with PATH environment variable setup, you can follow the official instructions provided after you login with your account. The following is a more understandable way.

• Extract the downloaded file. There would be 3 directories named ‘lib’,‘include’ and ‘bin’.
• Go to the CUDA install path. There you will also see 3 folders with the same names:

Put the contents of cuDNN into their corresponding CUDA path, i.e., move the files under cuDNN ‘bin’ directory to CUDA ‘bin’ directory, and do the same to ‘include’ and ‘bin’.

Note: If this step goes wrong, you can still install the rest, but when you initiate TensorFlow you will probably see a message: Couldn't open CUDA library cudnn.so.x. The computation speed might be compromised.

Step 5: Install TensorFlow

TensorFlow is an open source deep learning framework created and maintained by Google Brain Team. It is the most popular deep learning framework nowadays.

For windows users:

• Go to the command prompt (or anaconda command) window, and make sure that your virtual environment is active.
• If you simply want to use CPU version of TensorFlow, in the command window, type:
                pip install --ignore-installed --upgrade tensorflow

• If you want to enable GPU for TensorFlow, in the command window, type:
                pip install --ignore-installed --upgrade tensorflow-gpu

For Ubuntu users:

• Go to the command prompt (or anaconda command) window, and make sure that your virtual environment is active.
• If you simply want to use CPU version of TensorFlow, in the command window, type:
                pip install --ignore-installed --upgrade \
https://storage.googleapis.com/tensorflow/linux/cpu/tensorflow-1.3.0-cp35-cp35m-linux_x86_64.whl

• If you want to enable GPU for TensorFlow, in the command window, type:
                pip install --ignore-installed --upgrade \
https://storage.googleapis.com/tensorflow/linux/gpu/tensorflow_gpu-1.3.0-cp35-cp35m-linux_x86_64.whl

Note: Always try to activate your virtual environment.

• For Ubuntu users, building from source is a better way of installing TensorFlow, but requires very complicated steps and some time. If you’re interested, visit TensorFlow build from Sources.
• Verify that TensorFlow is correctly installed. In the command window, type python, and then type:
        >>>import tensorflow as tf
>>>a = tf.constant('Hello TensorFlow!')
>>>with tf.Session() as sess:
...    print(sess.run(a))
# If you see the following output, then you're all set!
b'Hello TensorFlow!'

Step 6: (Optional) Install PyTorch(Ubuntu and OSX users only)

PyTorch is another deep learning framework for Python. Its predecessor is Torch which relys on Lua. Compared to TensorFlow, Pytorch has its own advantages such as dynamic network design.

To install PyTorch, go to the official website http://pytorch.org/ and follow the install instructions. That’s it!

ECBM E4040 Neural Networks and Deep Learning, 2017.

Columbia University


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# Fight Finance

#### CoursesTagsRandomAllRecentScores

 Scores keithphw $6,011.61 Jade$1,815.80 Chu $789.98 royal ne...$750.00 Leehy $713.33 Visitor$650.00 JennyLI $625.61 Visitor$590.00 ZOE HY $570.00 Visitor$555.33 Visitor $550.00 Visitor$550.00 Visitor $540.00 Visitor$500.00 Yizhou $489.18 Visitor$480.00 Visitor $464.70 LWH$460.00 Jasper.sun $460.00 Visitor$460.00

For a price of $102, Andrea will sell you a share which just paid a dividend of$10 yesterday, and is expected to pay dividends every year forever, growing at a rate of 5% pa.

So the next dividend will be $10(1+0.05)^1=10.50$ in one year from now, and the year after it will be $10(1+0.05)^2=11.025$ and so on.

The required return of the stock is 15% pa.

Would you like to the share or politely ?

For a price of $129, Joanne will sell you a share which is expected to pay a$30 dividend in one year, and a $10 dividend every year after that forever. So the stock's dividends will be$30 at t=1, $10 at t=2,$10 at t=3, and $10 forever onwards. The required return of the stock is 10% pa. Would you like to the share or politely ? A person is thinking about borrowing$100 from the bank at 7% pa and investing it in shares with an expected return of 10% pa. One year later the person will sell the shares and pay back the loan in full. Both the loan and the shares are fairly priced.

What is the Net Present Value (NPV) of this one year investment? Note that you are asked to find the present value ($V_0$), not the value in one year ($V_1$).

Which of the following statements about cash in the form of notes and coins is NOT correct? Assume that inflation is positive.

Notes and coins:

Which of the following statements about book and market equity is NOT correct?

Who is most in danger of being personally bankrupt? Assume that all of their businesses' assets are highly liquid and can therefore be sold immediately.

Business people make lots of important decisions. Which of the following is the most important long term decision?

You expect a nominal payment of $100 in 5 years. The real discount rate is 10% pa and the inflation rate is 3% pa. Which of the following statements is NOT correct? A firm is considering a business project which costs$11m now and is expected to pay a constant $1m at the end of every year forever. Assume that the initial$11m cost is funded using the firm's existing cash so no new equity or debt will be raised. The cost of capital is 10% pa.

Which of the following statements about net present value (NPV), internal rate of return (IRR) and payback period is NOT correct?

A firm is considering a business project which costs $10m now and is expected to pay a single cash flow of$12.1m in two years.

Assume that the initial $10m cost is funded using the firm's existing cash so no new equity or debt will be raised. The cost of capital is 10% pa. Which of the following statements about net present value (NPV), internal rate of return (IRR) and payback period is NOT correct? You have$100,000 in the bank. The bank pays interest at 10% pa, given as an effective annual rate.

You wish to consume an equal amount now (t=0), in one year (t=1) and in two years (t=2), and still have $50,000 in the bank after that (t=2). How much can you consume at each time? Your neighbour asks you for a loan of$100 and offers to pay you back $120 in one year. You don't actually have any money right now, but you can borrow and lend from the bank at a rate of 10% pa. Rates are given as effective annual rates. Assume that your neighbour will definitely pay you back. Ignore interest tax shields and transaction costs. The Net Present Value (NPV) of lending to your neighbour is$9.09. Describe what you would do to actually receive a $9.09 cash flow right now with zero net cash flows in the future. A home loan company advertises an interest rate of 6% pa, payable monthly. Which of the following statements about the interest rate is NOT correct? All rates are given to four decimal places. A semi-annual coupon bond has a yield of 3% pa. Which of the following statements about the yield is NOT correct? All rates are given to four decimal places. On his 20th birthday, a man makes a resolution. He will deposit$30 into a bank account at the end of every month starting from now, which is the start of the month. So the first payment will be in one month. He will write in his will that when he dies the money in the account should be given to charity.

The bank account pays interest at 6% pa compounding monthly, which is not expected to change.

If the man lives for another 60 years, how much money will be in the bank account if he dies just after making his last (720th) payment?

You want to buy an apartment priced at $300,000. You have saved a deposit of$30,000. The bank has agreed to lend you the $270,000 as a fully amortising loan with a term of 25 years. The interest rate is 12% pa and is not expected to change. What will be your monthly payments? Remember that mortgage loan payments are paid in arrears (at the end of the month). You just borrowed$400,000 in the form of a 25 year interest-only mortgage with monthly payments of $3,000 per month. The interest rate is 9% pa which is not expected to change. You actually plan to pay more than the required interest payment. You plan to pay$3,300 in mortgage payments every month, which your mortgage lender allows. These extra payments will reduce the principal and the minimum interest payment required each month.

At the maturity of the mortgage, what will be the principal? That is, after the last (300th) interest payment of $3,300 in 25 years, how much will be owing on the mortgage? You're advising your superstar client 40-cent who is weighing up buying a private jet or a luxury yacht. 40-cent is just as happy with either, but he wants to go with the more cost-effective option. These are the cash flows of the two options: • The private jet can be bought for$6m now, which will cost $12,000 per month in fuel, piloting and airport costs, payable at the end of each month. The jet will last for 12 years. • Or the luxury yacht can be bought for$4m now, which will cost $20,000 per month in fuel, crew and berthing costs, payable at the end of each month. The yacht will last for 20 years. What's unusual about 40-cent is that he is so famous that he will actually be able to sell his jet or yacht for the same price as it was bought since the next generation of superstar musicians will buy it from him as a status symbol. Bank interest rates are 10% pa, given as an effective annual rate. You can assume that 40-cent will live for another 60 years and that when the jet or yacht's life is at an end, he will buy a new one with the same details as above. Would you advise 40-cent to buy the or the ? Note that the effective monthly rate is $r_\text{eff monthly}=(1+0.1)^{1/12}-1=0.00797414$ An industrial chicken farmer grows chickens for their meat. Chickens: 1. Cost$0.50 each to buy as chicks. They are bought on the day they’re born, at t=0.
2. Grow at a rate of $0.70 worth of meat per chicken per week for the first 6 weeks (t=0 to t=6). 3. Grow at a rate of$0.40 worth of meat per chicken per week for the next 4 weeks (t=6 to t=10) since they’re older and grow more slowly.
4. Feed costs are $0.30 per chicken per week for their whole life. Chicken feed is bought and fed to the chickens once per week at the beginning of the week. So the first amount of feed bought for a chicken at t=0 costs$0.30, and so on.
5. Can be slaughtered (killed for their meat) and sold at no cost at the end of the week. The price received for the chicken is their total value of meat (note that the chicken grows fast then slow, see above).

The required return of the chicken farm is 0.5% given as an effective weekly rate.

Ignore taxes and the fixed costs of the factory. Ignore the chicken’s welfare and other environmental and ethical concerns.

Find the equivalent weekly cash flow of slaughtering a chicken at 6 weeks and at 10 weeks so the farmer can figure out the best time to slaughter his chickens. The choices below are given in the same order, 6 and 10 weeks.

You own a nice suit which you wear once per week on nights out. You bought it one year ago for $600. In your experience, suits used once per week last for 6 years. So you expect yours to last for another 5 years. Your younger brother said that retro is back in style so he wants to wants to borrow your suit once a week when he goes out. With the increased use, your suit will only last for another 4 years rather than 5. What is the present value of the cost of letting your brother use your current suit for the next 4 years? Assume: that bank interest rates are 10% pa, given as an effective annual rate; you will buy a new suit when your current one wears out and your brother will not use the new one; your brother will only use your current suit so he will only use it for the next four years; and the price of a new suit never changes. You own some nice shoes which you use once per week on date nights. You bought them 2 years ago for$500. In your experience, shoes used once per week last for 6 years. So you expect yours to last for another 4 years.

Your younger sister said that she wants to borrow your shoes once per week. With the increased use, your shoes will only last for another 2 years rather than 4.

What is the present value of the cost of letting your sister use your current shoes for the next 2 years?

Assume: that bank interest rates are 10% pa, given as an effective annual rate; you will buy a new pair of shoes when your current pair wears out and your sister will not use the new ones; your sister will only use your current shoes so she will only use it for the next 2 years; and the price of new shoes never changes.

Bonds X and Y are issued by the same US company. Both bonds yield 10% pa, and they have the same face value ($100), maturity, seniority, and payment frequency. The only difference is that bond X and Y's coupon rates are 8 and 12% pa respectively. Which of the following statements is true? Calculate the price of a newly issued ten year bond with a face value of$100, a yield of 8% pa and a fixed coupon rate of 6% pa, paid annually. So there's only one coupon per year, paid in arrears every year.

You just signed up for a 30 year fully amortising mortgage loan with monthly payments of $1,500 per month. The interest rate is 9% pa which is not expected to change. To your surprise, you can actually afford to pay$2,000 per month and your mortgage allows early repayments without fees. If you maintain these higher monthly payments, how long will it take to pay off your mortgage?

A 'fully amortising' loan can also be called a:

Which of the following statements about the capital and income returns of a 25 year fully amortising loan asset is correct?

Assume that the yield curve (which shows total returns over different maturities) is flat and is not expected to change.

Over the 25 years from issuance to maturity, a fully amortising loan's expected annual effective:

An 'interest only' loan can also be called a:

Calculate the price of a newly issued ten year bond with a face value of $100, a yield of 8% pa and a fixed coupon rate of 6% pa, paid semi-annually. So there are two coupons per year, paid in arrears every six months. A young lady is trying to decide if she should attend university. Her friends say that she should go to university because she is more likely to meet a clever young man than if she begins full time work straight away. What's the correct way to classify this item from a capital budgeting perspective when trying to find the Net Present Value of going to university rather than working? The opportunity to meet a desirable future spouse should be classified as: A man has taken a day off from his casual painting job to relax. It's the end of the day and he's thinking about the hours that he could have spent working (in the past) which are now: Find Candys Corporation's Cash Flow From Assets (CFFA), also known as Free Cash Flow to the Firm (FCFF), over the year ending 30th June 2013.  Candys Corp Income Statement for year ending 30th June 2013$m Sales 200 COGS 50 Operating expense 10 Depreciation 20 Interest expense 10 Income before tax 110 Tax at 30% 33 Net income 77
 Candys Corp Balance Sheet as at 30th June 2013 2012 $m$m Assets Current assets 220 180 PPE Cost 300 340 Accumul. depr. 60 40 Carrying amount 240 300 Total assets 460 480 Liabilities Current liabilities 175 190 Non-current liabilities 135 130 Owners' equity Retained earnings 50 60 Contributed equity 100 100 Total L and OE 460 480

Note: all figures are given in millions of dollars ($m). Find Ching-A-Lings Corporation's Cash Flow From Assets (CFFA), also known as Free Cash Flow to the Firm (FCFF), over the year ending 30th June 2013.  Ching-A-Lings Corp Income Statement for year ending 30th June 2013$m Sales 100 COGS 20 Depreciation 20 Rent expense 11 Interest expense 19 Taxable Income 30 Taxes at 30% 9 Net income 21
 Ching-A-Lings Corp Balance Sheet as at 30th June 2013 2012 $m$m Inventory 49 38 Trade debtors 14 2 Rent paid in advance 5 5 PPE 400 400 Total assets 468 445 Trade creditors 4 10 Bond liabilities 200 190 Contributed equity 145 145 Retained profits 119 100 Total L and OE 468 445

Note: All figures are given in millions of dollars ($m). The cash flow from assets was: In Germany, nominal yields on semi-annual coupon paying Government Bonds with 2 years until maturity are currently 0.04% pa. The inflation rate is currently 1.4% pa, given as an APR compounding per quarter. The inflation rate is not expected to change over the next 2 years. What is the real yield on these bonds, given as an APR compounding every 6 months? An Australian company just issued two bonds: • A 1 year zero coupon bond at a yield of 10% pa, and • A 2 year zero coupon bond at a yield of 8% pa. What is the forward rate on the company's debt from years 1 to 2? Give your answer as an APR compounding every 6 months, which is how the above bond yields are quoted. You're trying to save enough money to buy your first car which costs$2,500. You can save $100 at the end of each month starting from now. You currently have no money at all. You just opened a bank account with an interest rate of 6% pa payable monthly. How many months will it take to save enough money to buy the car? Assume that the price of the car will stay the same over time. A student won$1m in a lottery. Currently the money is in a bank account which pays interest at 6% pa, given as an APR compounding per month.

She plans to spend $20,000 at the beginning of every month from now on (so the first withdrawal will be at t=0). After each withdrawal, she will check how much money is left in the account. When there is less than$500,000 left, she will donate that remaining amount to charity.

In how many months will she make her last withdrawal and donate the remainder to charity?

Which one of the following bonds is trading at a discount?

For a bond that pays fixed semi-annual coupons, how is the annual coupon rate defined, and how is the bond's annual income yield from time 0 to 1 defined mathematically?

Let: $P_0$ be the bond price now,

$F_T$ be the bond's face value,

$T$ be the bond's maturity in years,

$r_\text{total}$ be the bond's total yield,

$r_\text{income}$ be the bond's income yield,

$r_\text{capital}$ be the bond's capital yield, and

$C_t$ be the bond's coupon at time t in years. So $C_{0.5}$ is the coupon in 6 months, $C_1$ is the coupon in 1 year, and so on.

An investor bought two fixed-coupon bonds issued by the same company, a zero-coupon bond and a 7% pa semi-annual coupon bond. Both bonds have a face value of $1,000, mature in 10 years, and had a yield at the time of purchase of 8% pa. A few years later, yields fell to 6% pa. Which of the following statements is correct? Note that a capital gain is an increase in price. A 30 year Japanese government bond was just issued at par with a yield of 1.7% pa. The fixed coupon payments are semi-annual. The bond has a face value of$100.

Six months later, just after the first coupon is paid, the yield of the bond increases to 2% pa. What is the bond's new price?

Cash Flow From Assets (CFFA) can be defined as:

One year ago you bought $100,000 of shares partly funded using a margin loan. The margin loan size was$70,000 and the other $30,000 was your own wealth or 'equity' in the share assets. The interest rate on the margin loan was 7.84% pa. Over the year, the shares produced a dividend yield of 4% pa and a capital gain of 5% pa. What was the total return on your wealth? Ignore taxes, assume that all cash flows (interest payments and dividends) were paid and received at the end of the year, and all rates above are effective annual rates. Hint: Remember that wealth in this context is your equity (E) in the house asset (V = D+E) which is funded by the loan (D) and your deposit or equity (E). A company increases the proportion of debt funding it uses to finance its assets by issuing bonds and using the cash to repurchase stock, leaving assets unchanged. Ignoring the costs of financial distress, which of the following statements is NOT correct: Value the following business project to manufacture a new product.  Project Data Project life 2 yrs Initial investment in equipment$6m Depreciation of equipment per year $3m Expected sale price of equipment at end of project$0.6m Unit sales per year 4m Sale price per unit $8 Variable cost per unit$5 Fixed costs per year, paid at the end of each year $1m Interest expense per year 0 Tax rate 30% Weighted average cost of capital after tax per annum 10% Notes 1. The firm's current assets and current liabilities are$3m and $2m respectively right now. This net working capital will not be used in this project, it will be used in other unrelated projects. Due to the project, current assets (mostly inventory) will grow by$2m initially (at t = 0), and then by $0.2m at the end of the first year (t=1). Current liabilities (mostly trade creditors) will increase by$0.1m at the end of the first year (t=1).
At the end of the project, the net working capital accumulated due to the project can be sold for the same price that it was bought.
2. The project cost 0.5m to research which was incurred one year ago. Assumptions • All cash flows occur at the start or end of the year as appropriate, not in the middle or throughout the year. • All rates and cash flows are real. The inflation rate is 3% pa. • All rates are given as effective annual rates. • The business considering the project is run as a 'sole tradership' (run by an individual without a company) and is therefore eligible for a 50% capital gains tax discount when the equipment is sold, as permitted by the Australian Tax Office. What is the expected net present value (NPV) of the project? One method for calculating a firm's free cash flow (FFCF, or CFFA) is to ignore interest expense. That is, pretend that interest expense $(IntExp)$ is zero: \begin{aligned} FFCF &= (Rev - COGS - Depr - FC - IntExp)(1-t_c) + Depr - CapEx -\Delta NWC + IntExp \\ &= (Rev - COGS - Depr - FC - 0)(1-t_c) + Depr - CapEx -\Delta NWC - 0\\ \end{aligned} Does this annual FFCF with zero interest expense or the annual interest tax shield? There are many ways to calculate a firm's free cash flow (FFCF), also called cash flow from assets (CFFA). One method is to use the following formulas to transform net income (NI) into FFCF including interest and depreciation tax shields: $$FFCF=NI + Depr - CapEx -ΔNWC + IntExp$$ $$NI=(Rev - COGS - Depr - FC - IntExp).(1-t_c )$$ Another popular method is to use EBITDA rather than net income. EBITDA is defined as: $$EBITDA=Rev - COGS - FC$$ One of the below formulas correctly calculates FFCF from EBITDA, including interest and depreciation tax shields, giving an identical answer to that above. Which formula is correct? Find UniBar Corp's Cash Flow From Assets (CFFA), also known as Free Cash Flow to the Firm (FCFF), over the year ending 30th June 2013.  UniBar Corp Income Statement for year ending 30th June 2013m Sales 80 COGS 40 Operating expense 15 Depreciation 10 Interest expense 5 Income before tax 10 Tax at 30% 3 Net income 7
 UniBar Corp Balance Sheet as at 30th June 2013 2012 $m$m Assets Current assets 120 90 PPE Cost 360 320 Accumul. depr. 40 30 Carrying amount 320 290 Total assets 440 380 Liabilities Current liabilities 110 60 Non-current liabilities 190 180 Owners' equity Retained earnings 95 95 Contributed equity 45 45 Total L and OE 440 380

Note: all figures are given in millions of dollars ($m). Find Piano Bar's Cash Flow From Assets (CFFA), also known as Free Cash Flow to the Firm (FCFF), over the year ending 30th June 2013.  Piano Bar Income Statement for year ending 30th June 2013$m Sales 310 COGS 185 Operating expense 20 Depreciation 15 Interest expense 10 Income before tax 80 Tax at 30% 24 Net income 56
 Piano Bar Balance Sheet as at 30th June 2013 2012 $m$m Assets Current assets 240 230 PPE Cost 420 400 Accumul. depr. 50 35 Carrying amount 370 365 Total assets 610 595 Liabilities Current liabilities 180 190 Non-current liabilities 290 265 Owners' equity Retained earnings 90 90 Contributed equity 50 50 Total L and OE 610 595

Note: all figures are given in millions of dollars ($m). Find World Bar's Cash Flow From Assets (CFFA), also known as Free Cash Flow to the Firm (FCFF), over the year ending 30th June 2013.  World Bar Income Statement for year ending 30th June 2013$m Sales 300 COGS 150 Operating expense 50 Depreciation 40 Interest expense 10 Taxable income 50 Tax at 30% 15 Net income 35
 World Bar Balance Sheet as at 30th June 2013 2012 $m$m Assets Current assets 200 230 PPE Cost 400 400 Accumul. depr. 75 35 Carrying amount 325 365 Total assets 525 595 Liabilities Current liabilities 150 205 Non-current liabilities 235 250 Owners' equity Retained earnings 100 100 Contributed equity 40 40 Total L and OE 525 595

Note: all figures above and below are given in millions of dollars ($m). The equations for Net Income (NI, also known as Earnings or Net Profit After Tax) and Cash Flow From Assets (CFFA, also known as Free Cash Flow to the Firm) per year are: $$NI=(Rev-COGS-FC-Depr-IntExp).(1-t_c)$$ $$CFFA=NI+Depr-CapEx - \varDelta NWC+IntExp$$ For a firm with debt, what is the formula for the present value of interest tax shields if the tax shields occur in perpetuity? You may assume: • the value of debt (D) is constant through time, • The cost of debt and the yield on debt are equal and given by $r_D$. • the appropriate rate to discount interest tax shields is $r_D$. • $\text{IntExp}=D.r_D$ Question 121 capital structure, leverage, costs of financial distress, interest tax shield Fill in the missing words in the following sentence: All things remaining equal, as a firm's amount of debt funding falls, benefits of interest tax shields __________ and the costs of financial distress __________. A young lady is trying to decide if she should attend university or not. The young lady's parents say that she must attend university because otherwise all of her hard work studying and attending school during her childhood was a waste. What's the correct way to classify this item from a capital budgeting perspective when trying to decide whether to attend university? The hard work studying at school in her childhood should be classified as: When using the dividend discount model to price a stock: $$p_{0} = \frac{d_1}{r - g}$$ The growth rate of dividends (g): Which statement about risk, required return and capital structure is the most correct? Question 99 capital structure, interest tax shield, Miller and Modigliani, trade off theory of capital structure A firm changes its capital structure by issuing a large amount of debt and using the funds to repurchase shares. Its assets are unchanged. Assume that: • The firm and individual investors can borrow at the same rate and have the same tax rates. • The firm's debt and shares are fairly priced and the shares are repurchased at the market price, not at a premium. • There are no market frictions relating to debt such as asymmetric information or transaction costs. • Shareholders wealth is measured in terms of utiliity. Shareholders are wealth-maximising and risk-averse. They have a preferred level of overall leverage. Before the firm's capital restructure all shareholders were optimally levered. According to Miller and Modigliani's theory, which statement is correct?  Portfolio Details Stock Expected return Standard deviation Correlation $(\rho_{A,B})$ Dollars invested A 0.1 0.4 0.5 60 B 0.2 0.6 140 What is the standard deviation (not variance) of the above portfolio? Two risky stocks A and B comprise an equal-weighted portfolio. The correlation between the stocks' returns is 70%. If the variance of stock A increases but the: • Prices and expected returns of each stock stays the same, • Variance of stock B's returns stays the same, • Correlation of returns between the stocks stays the same. Which of the following statements is NOT correct? What is the correlation of a variable X with itself? The corr(X, X) or $\rho_{X,X}$ equals: What is the correlation of a variable X with a constant C? The corr(X, C) or $\rho_{X,C}$ equals: The covariance and correlation of two stocks X and Y's annual returns are calculated over a number of years. The units of the returns are in percent per annum $(\% pa)$. What are the units of the covariance $(\sigma_{X,Y})$ and correlation $(\rho_{X,Y})$ of returns respectively? Hint: Visit Wikipedia to understand the difference between percentage points $(\text{pp})$ and percent $(\%)$. Let the standard deviation of returns for a share per month be $\sigma_\text{monthly}$. What is the formula for the standard deviation of the share's returns per year $(\sigma_\text{yearly})$? Assume that returns are independently and identically distributed (iid) so they have zero auto correlation, meaning that if the return was higher than average today, it does not indicate that the return tomorrow will be higher or lower than average. According to the theory of the Capital Asset Pricing Model (CAPM), total risk can be broken into two components, systematic risk and idiosyncratic risk. Which of the following events would be considered a systematic, undiversifiable event according to the theory of the CAPM? Which statement is the most correct? A firm changes its capital structure by issuing a large amount of equity and using the funds to repay debt. Its assets are unchanged. Ignore interest tax shields. According to the Capital Asset Pricing Model (CAPM), which statement is correct? The CAPM can be used to find a business's expected opportunity cost of capital: $$r_i=r_f+β_i (r_m-r_f)$$ What should be used as the risk free rate $r_f$? Assume that there exists a perfect world with no transaction costs, no asymmetric information, no taxes, no agency costs, equal borrowing rates for corporations and individual investors, the ability to short the risk free asset, semi-strong form efficient markets, the CAPM holds, investors are rational and risk-averse and there are no other market frictions. For a firm operating in this perfect world, which statement(s) are correct? (i) When a firm changes its capital structure and/or payout policy, share holders' wealth is unaffected. (ii) When the idiosyncratic risk of a firm's assets increases, share holders do not expect higher returns. (iii) When the systematic risk of a firm's assets increases, share holders do not expect higher returns. Select the most correct response: All things remaining equal, according to the capital asset pricing model, if the systematic variance of an asset increases, its required return will increase and its price will decrease. If the idiosyncratic variance of an asset increases, its price will be unchanged. What is the relationship between the price of a call or put option and the total, systematic and idiosyncratic variance of the underlying asset that the option is based on? Select the most correct answer. Call and put option prices increase when the: Your friend claims that by reading 'The Economist' magazine's economic news articles, she can identify shares that will have positive abnormal expected returns over the next 2 years. Assuming that her claim is true, which statement(s) are correct? (i) Weak form market efficiency is broken. (ii) Semi-strong form market efficiency is broken. (iii) Strong form market efficiency is broken. (iv) The asset pricing model used to measure the abnormal returns (such as the CAPM) is either wrong (mis-specification error) or is measured using the wrong inputs (data errors) so the returns may not be abnormal but rather fair for the level of risk. Select the most correct response: Fundamentalists who analyse company financial reports and news announcements (but who don't have inside information) will make positive abnormal returns if: Australians usually quote the Australian dollar in USD per 1 AUD. For example, in October 2015 the Australian dollar was quoted as 0.72 USD per AUD. Is this an or terms quote? If the AUD appreciates against the USD, the American terms quote of the AUD will or ? One and a half years ago Frank bought a house for$600,000. Now it's worth only $500,000, based on recent similar sales in the area. The expected total return on Frank's residential property is 7% pa. He rents his house out for$1,600 per month, paid in advance. Every 12 months he plans to increase the rental payments.

The present value of 12 months of rental payments is $18,617.27. The future value of 12 months of rental payments one year in the future is$19,920.48.

What is the expected annual rental yield of the property? Ignore the costs of renting such as maintenance, real estate agent fees and so on.

This annuity formula $\dfrac{C_1}{r}\left(1-\dfrac{1}{(1+r)^3} \right)$ is equivalent to which of the following formulas? Note the 3.

In the below formulas, $C_t$ is a cash flow at time t. All of the cash flows are equal, but paid at different times.

Most listed Australian companies pay dividends twice per year, the 'interim' and 'final' dividends, which are roughly 6 months apart.

You are an equities analyst trying to value the company BHP. You decide to use the Dividend Discount Model (DDM) as a starting point, so you study BHP's dividend history and you find that BHP tends to pay the same interim and final dividend each year, and that both grow by the same rate.

You expect BHP will pay a $0.55 interim dividend in six months and a$0.55 final dividend in one year. You expect each to grow by 4% next year and forever, so the interim and final dividends next year will be $0.572 each, and so on in perpetuity. Assume BHP's cost of equity is 8% pa. All rates are quoted as nominal effective rates. The dividends are nominal cash flows and the inflation rate is 2.5% pa. What is the current price of a BHP share? You are an equities analyst trying to value the equity of the Australian telecoms company Telstra, with ticker TLS. In Australia, listed companies like Telstra tend to pay dividends every 6 months. The payment around August is called the final dividend and the payment around February is called the interim dividend. Both occur annually. • Today is mid-March 2015. • TLS's last interim dividend of$0.15 was one month ago in mid-February 2015.
• TLS's last final dividend of $0.15 was seven months ago in mid-August 2014. Judging by TLS's dividend history and prospects, you estimate that the nominal dividend growth rate will be 1% pa. Assume that TLS's total nominal cost of equity is 6% pa. The dividends are nominal cash flows and the inflation rate is 2.5% pa. All rates are quoted as nominal effective annual rates. Assume that each month is exactly one twelfth (1/12) of a year, so you can ignore the number of days in each month. Calculate the current TLS share price. You own an apartment which you rent out as an investment property. What is the price of the apartment using discounted cash flow (DCF, same as NPV) valuation? Assume that: • You just signed a contract to rent the apartment out to a tenant for the next 12 months at$2,000 per month, payable in advance (at the start of the month, t=0). The tenant is just about to pay you the first $2,000 payment. • The contract states that monthly rental payments are fixed for 12 months. After the contract ends, you plan to sign another contract but with rental payment increases of 3%. You intend to do this every year. So rental payments will increase at the start of the 13th month (t=12) to be$2,060 (=2,000(1+0.03)), and then they will be constant for the next 12 months.
Rental payments will increase again at the start of the 25th month (t=24) to be $2,121.80 (=2,000(1+0.03)2), and then they will be constant for the next 12 months until the next year, and so on. • The required return of the apartment is 8.732% pa, given as an effective annual rate. • Ignore all taxes, maintenance, real estate agent, council and strata fees, periods of vacancy and other costs. Assume that the apartment will last forever and so will the rental payments. Three years ago Frederika bought a house for$400,000.

Now it's worth $600,000, based on recent similar sales in the area. Frederika's residential property has an expected total return of 7% pa. She rents her house out for$2,500 per month, paid in advance. Every 12 months she plans to increase the rental payments.

The present value of 12 months of rental payments is $29,089.48. The future value of 12 months of rental payments one year ahead is$31,125.74.

What is the expected annual capital yield of the property?

Two years ago Fred bought a house for $300,000. Now it's worth$500,000, based on recent similar sales in the area.

Fred's residential property has an expected total return of 8% pa.

He rents his house out for $2,000 per month, paid in advance. Every 12 months he plans to increase the rental payments. The present value of 12 months of rental payments is$23,173.86.

The future value of 12 months of rental payments one year ahead is $25,027.77. What is the expected annual growth rate of the rental payments? In other words, by what percentage increase will Fred have to raise the monthly rent by each year to sustain the expected annual total return of 8%? You just bought a residential apartment as an investment property for$500,000.

You intend to rent it out to tenants. They are ready to move in, they would just like to know how much the monthly rental payments will be, then they will sign a twelve-month lease.

You require a total return of 8% pa and a rental yield of 5% pa.

What would the monthly paid-in-advance rental payments have to be this year to receive that 5% annual rental yield?

Also, if monthly rental payments can be increased each year when a new lease agreement is signed, by how much must you increase rents per year to realise the 8% pa total return on the property?

Ignore all taxes and the costs of renting such as maintenance costs, real estate agent fees, utilities and so on. Assume that there will be no periods of vacancy and that tenants will promptly pay the rental prices you charge.

Note that the first rental payment will be received at t=0. The first lease agreement specifies the first 12 equal payments from t=0 to 11. The next lease agreement can have a rental increase, so the next twelve equal payments from t=12 to 23 can be higher than previously, and so on forever.

If housing rents are constrained from growing more than the maximum target inflation rate, and houses can be priced as a perpetuity of growing net rental cash flows, then what is the implication for house prices, all things remaining equal? Select the most correct answer.

Background: Since 1990, many central banks across the world have become 'inflation targeters'. They have adopted a policy of trying to keep inflation in a predictable narrow range, with the hope of encouraging long-term lending to fund more investment and maintain higher GDP growth.

Australia's central bank, the Reserve Bank of Australia (RBA), has specifically stated their inflation target range is between 2 and 3% pa.

Some Australian residential property market commentators suggest that because rental costs comprise a large part of the Australian consumer price index (CPI), rent costs across the nation cannot significantly exceed the maximum inflation target range of 3% pa without the prices of other goods growing by less than the target range for long periods, which is unlikely.

A man inherits $500,000 worth of shares. He believes that by learning the secrets of trading, keeping up with the financial news and doing complex trend analysis with charts that he can quit his job and become a self-employed day trader in the equities markets. What is the expected gain from doing this over the first year? Measure the net gain in wealth received at the end of this first year due to the decision to become a day trader. Assume the following: • He earns$60,000 pa in his current job, paid in a lump sum at the end of each year.
• He enjoys examining share price graphs and day trading just as much as he enjoys his current job.
• Stock markets are weak form and semi-strong form efficient.
• He has no inside information.
• He makes 1 trade every day and there are 250 trading days in the year. Trading costs are $20 per trade. His broker invoices him for the trading costs at the end of the year. • The shares that he currently owns and the shares that he intends to trade have the same level of systematic risk as the market portfolio. • The market portfolio's expected return is 10% pa. Measure the net gain over the first year as an expected wealth increase at the end of the year. A managed fund charges fees based on the amount of money that you keep with them. The fee is 2% of the start-of-year amount, but it is paid at the end of every year. This fee is charged regardless of whether the fund makes gains or losses on your money. The fund offers to invest your money in shares which have an expected return of 10% pa before fees. You are thinking of investing$100,000 in the fund and keeping it there for 40 years when you plan to retire.

What is the Net Present Value (NPV) of investing your money in the fund? Note that the question is not asking how much money you will have in 40 years, it is asking: what is the NPV of investing in the fund? Assume that:

• The fund has no private information.
• Markets are weak and semi-strong form efficient.
• The fund's transaction costs are negligible.
• The cost and trouble of investing your money in shares by yourself, without the managed fund, is negligible.

A residential real estate investor believes that house prices will grow at a rate of 5% pa and that rents will grow by 2% pa forever.

All rates are given as nominal effective annual returns. Assume that:

• His forecast is true.
• Real estate is and always will be fairly priced and the capital asset pricing model (CAPM) is true.
• Ignore all costs such as taxes, agent fees, maintenance and so on.
• All rental income cash flow is paid out to the owner, so there is no re-investment and therefore no additions or improvements made to the property.
• The non-monetary benefits of owning real estate and renting remain constant.

Which one of the following statements is NOT correct? Over time:

A managed fund charges fees based on the amount of money that you keep with them. The fee is 2% of the end-of-year amount, paid at the end of every year.

This fee is charged regardless of whether the fund makes gains or losses on your money.

The fund offers to invest your money in shares which have an expected return of 10% pa before fees.

You are thinking of investing $100,000 in the fund and keeping it there for 40 years when you plan to retire. How much money do you expect to have in the fund in 40 years? Also, what is the future value of the fees that the fund expects to earn from you? Give both amounts as future values in 40 years. Assume that: • The fund has no private information. • Markets are weak and semi-strong form efficient. • The fund's transaction costs are negligible. • The cost and trouble of investing your money in shares by yourself, without the managed fund, is negligible. • The fund invests its fees in the same companies as it invests your funds in, but with no fees. The below answer choices list your expected wealth in 40 years and then the fund's expected wealth in 40 years. How is the AUD normally quoted in Australia? Using or terms? A company's share price fell by 20% and its number of shares rose by 25%. Assume that there are no taxes, no signalling effects and no transaction costs. Which one of the following corporate events may have happened? What is the covariance of a variable X with itself? The cov(X, X) or $\sigma_{X,X}$ equals: A stock is expected to pay the following dividends:  Cash Flows of a Stock Time (yrs) 0 1 2 3 4 ... Dividend ($) 8 8 8 20 8 ...

After year 4, the dividend will grow in perpetuity at 4% pa. The required return on the stock is 10% pa. Both the growth rate and required return are given as effective annual rates.

What is the current price of the stock?

A stock is expected to pay the following dividends:

 Cash Flows of a Stock Time (yrs) 0 1 2 3 4 ... Dividend ($) 8 8 8 20 8 ... After year 4, the dividend will grow in perpetuity at 4% pa. The required return on the stock is 10% pa. Both the growth rate and required return are given as effective annual rates. What will be the price of the stock in 5 years (t = 5), just after the dividend at that time has been paid? A stock is expected to pay the following dividends:  Cash Flows of a Stock Time (yrs) 0 1 2 3 4 ... Dividend ($) 2 2 2 10 3 ...

After year 4, the dividend will grow in perpetuity at 4% pa. The required return on the stock is 10% pa. Both the growth rate and required return are given as effective annual rates.

What will be the price of the stock in 5 years (t = 5), just after the dividend at that time has been paid?

The following is the Dividend Discount Model used to price stocks:

$$p_0=\frac{d_1}{r-g}$$

All rates are effective annual rates and the cash flows ($d_1$) are received every year. Note that the r and g terms in the above DDM could also be labelled as below: $$r = r_{\text{total, 0}\rightarrow\text{1yr, eff 1yr}}$$ $$g = r_{\text{capital, 0}\rightarrow\text{1yr, eff 1yr}}$$ Which of the following statements is NOT correct?

You deposit cash into your bank account. Have you or debt?

You deposit cash into your bank account. Have you or debt?

One of Miller and Modigliani's (M&M's) important insights is that a firm's managers should not try to achieve a particular level of leverage or interest tax shields under certain assumptions. So the firm's capital structure is irrelevant. This is because investors can make their own personal leverage and interest tax shields, so there's no need for managers to try to make corporate leverage and interest tax shields. This is true under the assumptions of equal tax rates, interest rates and debt availability for the person and the corporation, no transaction costs and symmetric information.

This principal of 'home-made' or 'do-it-yourself' leverage can also be applied to other topics. Read the following statements to decide which are true:

(I) Payout policy: a firm's managers should not try to achieve a particular pattern of equity payout.

(II) Agency costs: a firm's managers should not try to minimise agency costs.

(III) Diversification: a firm's managers should not try to diversify across industries.

(IV) Shareholder wealth: a firm's managers should not try to maximise shareholders' wealth.

Which of the above statement(s) are true?

A company advertises an investment costing $1,000 which they say is underpriced. They say that it has an expected total return of 15% pa, but a required return of only 10% pa. Of the 15% pa total expected return, the dividend yield is expected to always be 7% pa and rest is the capital yield. Assuming that the company's statements are correct, what is the NPV of buying the investment if the 15% total return lasts for the next 100 years (t=0 to 100), then reverts to 10% after that time? Also, what is the NPV of the investment if the 15% return lasts forever? In both cases, assume that the required return of 10% remains constant, the dividends can only be re-invested at 10% pa and all returns are given as effective annual rates. The answer choices below are given in the same order (15% for 100 years, and 15% forever): Suppose the Australian cash rate is expected to be 8.15% pa and the US federal funds rate is expected to be 3.00% pa over the next 2 years, both given as nominal effective annual rates. The current exchange rate is at parity, so 1 USD = 1 AUD. What is the implied 2 year forward foreign exchange rate? A bathroom and plumbing supplies shop offers credit to its customers. Customers are given 60 days to pay for their goods, but if they pay within 7 days they will get a 2% discount. What is the effective interest rate implicit in the discount being offered? Assume 365 days in a year and that all customers pay on either the 7th day or the 60th day. All rates given in this question are effective annual rates. In the below term structure of interest rates equation, all rates are effective annual yields and the numbers in subscript represent the years that the yields are measured over: $$(1+r_{0-3})^3 = (1+r_{0-1})(1+r_{1-2})(1+r_{2-3})$$ Which of the following statements is NOT correct? Which of the following is NOT a synonym of 'required return'? An economy has only two investable assets: stocks and cash. Stocks had a historical nominal average total return of negative two percent per annum (-2% pa) over the last 20 years. Stocks are liquid and actively traded. Stock returns are variable, they have risk. Cash is riskless and has a nominal constant return of zero percent per annum (0% pa), which it had in the past and will have in the future. Cash can be kept safely at zero cost. Cash can be converted into shares and vice versa at zero cost. The nominal total return of the shares over the next year is expected to be: Which of the following statements about yield curves is NOT correct? Which of the following statements about Australian franking credits is NOT correct? Franking credits: A levered company's required return on debt is always less than its required return on equity. or ? A levered firm has a market value of assets of$10m. Its debt is all comprised of zero-coupon bonds which mature in one year and have a combined face value of $9.9m. Investors are risk-neutral and therefore all debt and equity holders demand the same required return of 10% pa. Therefore the current market capitalisation of debt $(D_0)$ is$9m and equity $(E_0)$ is $1m. A new project presents itself which requires an investment of$2m and will provide a:

• $6.6m cash flow with probability 0.5 in the good state of the world, and a • -$4.4m (notice the negative sign) cash flow with probability 0.5 in the bad state of the world.

The project can be funded using the company's excess cash, no debt or equity raisings are required.

What would be the new market capitalisation of equity $(E_\text{0, with project})$ if shareholders vote to proceed with the project, and therefore should shareholders proceed with the project?

A levered firm has zero-coupon bonds which mature in one year and have a combined face value of $9.9m. Investors are risk-neutral and therefore all debt and equity holders demand the same required return of 10% pa. In one year the firm's assets will be worth: •$13.2m with probability 0.5 in the good state of the world, or
• $6.6m with probability 0.5 in the bad state of the world. A new project presents itself which requires an investment of$2m and will provide a certain cash flow of $3.3m in one year. The firm doesn't have any excess cash to make the initial$2m investment, but the funds can be raised from shareholders through a fairly priced rights issue. Ignore all transaction costs.

Should shareholders vote to proceed with the project and equity raising? What will be the gain in shareholder wealth if they decide to proceed?

Which of the following statements about the capital and income returns of an interest-only loan is correct?

Assume that the yield curve (which shows total returns over different maturities) is flat and is not expected to change.

An interest-only loan's expected:

Let the 'income return' of a bond be the coupon at the end of the period divided by the market price now at the start of the period $(C_1/P_0)$. The expected income return of a premium fixed coupon bond is:

Which of the following investable assets are NOT suitable for valuation using PE multiples techniques?

Which firms tend to have high forward-looking price-earnings (PE) ratios?

A project's Profitability Index (PI) is less than 1. Select the most correct statement:

You just started work at your new job which pays $48,000 per year. The human resources department have given you the option of being paid at the end of every week or every month. Assume that there are 4 weeks per month, 12 months per year and 48 weeks per year. Bank interest rates are 12% pa given as an APR compounding per month. What is the dollar gain over one year, as a net present value, of being paid every week rather than every month? What is the net present value (NPV) of undertaking a full-time Australian undergraduate business degree as an Australian citizen? Only include the cash flows over the duration of the degree, ignore any benefits or costs of the degree after it's completed. Assume the following: • The degree takes 3 years to complete and all students pass all subjects. • There are 2 semesters per year and 4 subjects per semester. • University fees per subject per semester are$1,277, paid at the start of each semester. Fees are expected to stay constant for the next 3 years.
• There are 52 weeks per year.
• The first semester is just about to start (t=0). The first semester lasts for 19 weeks (t=0 to 19).
• The second semester starts immediately afterwards (t=19) and lasts for another 19 weeks (t=19 to 38).
• The summer holidays begin after the second semester ends and last for 14 weeks (t=38 to 52). Then the first semester begins the next year, and so on.
• Working full time at the grocery store instead of studying full-time pays $20/hr and you can work 35 hours per week. Wages are paid at the end of each week. • Full-time students can work full-time during the summer holiday at the grocery store for the same rate of$20/hr for 35 hours per week. Wages are paid at the end of each week.
• The discount rate is 9.8% pa. All rates and cash flows are real. Inflation is expected to be 3% pa. All rates are effective annual.

The NPV of costs from undertaking the university degree is:

Your poor friend asks to borrow some money from you. He would like $1,000 now (t=0) and every year for the next 5 years, so there will be 6 payments of$1,000 from t=0 to t=5 inclusive. In return he will pay you $10,000 in seven years from now (t=7). What is the net present value (NPV) of lending to your friend? Assume that your friend will definitely pay you back so the loan is risk-free, and that the yield on risk-free government debt is 10% pa, given as an effective annual rate. Your friend is trying to find the net present value of a project. The project is expected to last for just one year with: • a negative cash flow of -$1 million initially (t=0), and
• a positive cash flow of $1.1 million in one year (t=1). The project has a total required return of 10% pa due to its moderate level of undiversifiable risk. Your friend is aware of the importance of opportunity costs and the time value of money, but he is unsure of how to find the NPV of the project. He knows that the opportunity cost of investing the$1m in the project is the expected gain from investing the money in shares instead. Like the project, shares also have an expected return of 10% since they have moderate undiversifiable risk. This opportunity cost is $0.1m $(=1m \times 10\%)$ which occurs in one year (t=1). He knows that the time value of money should be accounted for, and this can be done by finding the present value of the cash flows in one year. Your friend has listed a few different ways to find the NPV which are written down below. (I) $-1m + \dfrac{1.1m}{(1+0.1)^1}$ (II) $-1m + \dfrac{1.1m}{(1+0.1)^1} - \dfrac{1m}{(1+0.1)^1} \times 0.1$ (III) $-1m + \dfrac{1.1m}{(1+0.1)^1} - \dfrac{1.1m}{(1+0.1)^1} \times 0.1$ (IV) $-1m + 1.1m - \dfrac{1.1m}{(1+0.1)^1} \times 0.1$ (V) $-1m + 1.1m - 1.1m \times 0.1$ Which of the above calculations give the correct NPV? Select the most correct answer. A fairly priced unlevered firm plans to pay a dividend of$1 next year (t=1) which is expected to grow by 3% pa every year after that. The firm's required return on equity is 8% pa.

The firm is thinking about reducing its future dividend payments by 10% so that it can use the extra cash to invest in more projects which are expected to return 8% pa, and have the same risk as the existing projects. Therefore, next year's dividend will be $0.90. What will be the stock's new annual capital return (proportional increase in price per year) if the change in payout policy goes ahead? Assume that payout policy is irrelevant to firm value and that all rates are effective annual rates. Which of the following investable assets is the LEAST suitable for valuation using PE multiples techniques? A firm has 1 million shares which trade at a price of$30 each. The firm is expected to announce earnings of $3 million at the end of the year and pay an annual dividend of$1.50 per share.

What is the firm's (forward looking) price/earnings (PE) ratio?

The below screenshot of Microsoft's (MSFT) details were taken from the Google Finance website on 28 Nov 2014. Some information has been deliberately blanked out.

What was MSFT's backwards-looking price-earnings ratio?

A zero coupon bond that matures in 6 months has a face value of $1,000. The firm that issued this bond is trying to forecast its income statement for the year. It needs to calculate the interest expense of the bond this year. The bond is highly illiquid and hasn't traded on the market. But the finance department have assessed the bond's fair value to be$950 and this is its book value right now at the start of the year.

Assume that:

• the firm uses the 'effective interest method' to calculate interest expense.
• the market value of the bond is the same as the book value.
• the firm is only interested in this bond's interest expense. Do not include the interest expense for a new bond issued to refinance the current one, as would normally happen.

What will be the interest expense of the bond this year for the purpose of forecasting the income statement?

There are many ways to calculate a firm's free cash flow (FFCF), also called cash flow from assets (CFFA). Some include the annual interest tax shield in the cash flow and some do not.

Which of the below FFCF formulas include the interest tax shield in the cash flow?

$$(1) \quad FFCF=NI + Depr - CapEx -ΔNWC + IntExp$$ $$(2) \quad FFCF=NI + Depr - CapEx -ΔNWC + IntExp.(1-t_c)$$ $$(3) \quad FFCF=EBIT.(1-t_c )+ Depr- CapEx -ΔNWC+IntExp.t_c$$ $$(4) \quad FFCF=EBIT.(1-t_c) + Depr- CapEx -ΔNWC$$ $$(5) \quad FFCF=EBITDA.(1-t_c )+Depr.t_c- CapEx -ΔNWC+IntExp.t_c$$ $$(6) \quad FFCF=EBITDA.(1-t_c )+Depr.t_c- CapEx -ΔNWC$$ $$(7) \quad FFCF=EBIT-Tax + Depr - CapEx -ΔNWC$$ $$(8) \quad FFCF=EBIT-Tax + Depr - CapEx -ΔNWC-IntExp.t_c$$ $$(9) \quad FFCF=EBITDA-Tax - CapEx -ΔNWC$$ $$(10) \quad FFCF=EBITDA-Tax - CapEx -ΔNWC-IntExp.t_c$$

The formulas for net income (NI also called earnings), EBIT and EBITDA are given below. Assume that depreciation and amortisation are both represented by 'Depr' and that 'FC' represents fixed costs such as rent.

$$NI=(Rev - COGS - Depr - FC - IntExp).(1-t_c )$$ $$EBIT=Rev - COGS - FC-Depr$$ $$EBITDA=Rev - COGS - FC$$ $$Tax =(Rev - COGS - Depr - FC - IntExp).t_c= \dfrac{NI.t_c}{1-t_c}$$

A project has the following cash flows:

 Project Cash Flows Time (yrs) Cash flow ($) 0 -400 1 0 2 500 The required return on the project is 10%, given as an effective annual rate. What is the Internal Rate of Return (IRR) of this project? The following choices are effective annual rates. Assume that the cash flows shown in the table are paid all at once at the given point in time. Currently, a mining company has a share price of$6 and pays constant annual dividends of $0.50. The next dividend will be paid in 1 year. Suddenly and unexpectedly the mining company announces that due to higher than expected profits, all of these windfall profits will be paid as a special dividend of$0.30 in 1 year.

If investors believe that the windfall profits and dividend is a one-off event, what will be the new share price? If investors believe that the additional dividend is actually permanent and will continue to be paid, what will be the new share price? Assume that the required return on equity is unchanged. Choose from the following, where the first share price includes the one-off increase in earnings and dividends for the first year only $(P_\text{0 one-off})$ , and the second assumes that the increase is permanent $(P_\text{0 permanent})$:

Note: When a firm makes excess profits they sometimes pay them out as special dividends. Special dividends are just like ordinary dividends but they are one-off and investors do not expect them to continue, unlike ordinary dividends which are expected to persist.

 Portfolio Details Stock Expected return Standard deviation Covariance $(\sigma_{A,B})$ Beta Dollars invested A 0.2 0.4 0.12 0.5 40 B 0.3 0.8 1.5 80

What is the standard deviation (not variance) of the above portfolio? Note that the stocks' covariance is given, not correlation.

A stock's standard deviation of returns is expected to be:

• 0.09 per month for the first 5 months;
• 0.14 per month for the next 7 months.

What is the expected standard deviation of the stock per year $(\sigma_\text{annual})$?

Assume that returns are independently and identically distributed (iid) and therefore have zero auto-correlation.

Stocks in the United States usually pay quarterly dividends. For example, the retailer Wal-Mart Stores paid a $0.47 dividend every quarter over the 2013 calendar year and plans to pay a$0.48 dividend every quarter over the 2014 calendar year.

Using the dividend discount model and net present value techniques, calculate the stock price of Wal-Mart Stores assuming that:

• The time now is the beginning of January 2014. The next dividend of $0.48 will be received in 3 months (end of March 2014), with another 3 quarterly payments of$0.48 after this (end of June, September and December 2014).
• The quarterly dividend will increase by 2% every year, but each quarterly dividend over the year will be equal. So each quarterly dividend paid in 2015 will be $0.4896 ($=0.48×(1+0.02)^1$), with the first at the end of March 2015 and the last at the end of December 2015. In 2016 each quarterly dividend will be$0.499392 ($=0.48×(1+0.02)^2$), with the first at the end of March 2016 and the last at the end of December 2016, and so on forever.
• The total required return on equity is 6% pa.
• The required return and growth rate are given as effective annual rates.
• All cash flows and rates are nominal. Inflation is 3% pa.
• Dividend payment dates and ex-dividend dates are at the same time.
• Remember that there are 4 quarters in a year and 3 months in a quarter.

What is the current stock price?

You believe that the price of a share will fall significantly very soon, but the rest of the market does not. The market thinks that the share price will remain the same. Assuming that your prediction will soon be true, which of the following trades is a bad idea? In other words, which trade will NOT make money or prevent losses?

Question 589  future, contango, market efficiency

In general, stock prices tend to rise. What does this mean for futures on equity?

A project has the following cash flows. Normally cash flows are assumed to happen at the given time. But here, assume that the cash flows are received smoothly over the year. So the $250 at time 2 is actually earned smoothly from t=1 to t=2:  Project Cash Flows Time (yrs) Cash flow ($) 0 -400 1 200 2 250

What is the payback period of the project in years?

A project has the following cash flows. Normally cash flows are assumed to happen at the given time. But here, assume that the cash flows are received smoothly over the year. So the $105 at time 2 is actually earned smoothly from t=1 to t=2:  Project Cash Flows Time (yrs) Cash flow ($) 0 -90 1 30 2 105

What is the payback period of the project in years?

A firm pays a fully franked cash dividend of $100 to one of its Australian shareholders who has a personal marginal tax rate of 15%. The corporate tax rate is 30%. What will be the shareholder's personal tax payable due to the dividend payment? The "interest expense" on a company's annual income statement is equal to the cash interest payments (but not principal payments) made to debt holders during the year. or ? On 22-Mar-2013 the Australian Government issued series TB139 treasury bonds with a combined face value$23.4m, listed on the ASX with ticker code GSBG25.

The bonds mature on 21-Apr-2025, the fixed coupon rate is 3.25% pa and coupons are paid semi-annually on the 21st of April and October of each year. Each bond's face value is $1,000. At market close on Friday 11-Sep-2015 the bonds' yield was 2.736% pa. At market close on Monday 14-Sep-2015 the bonds' yield was 2.701% pa. Both yields are given as annualised percentage rates (APR's) compounding every 6 months. For convenience, assume 183 days in 6 months and 366 days in a year. What was the historical total return over those 3 calendar days between Friday 11-Sep-2015 and Monday 14-Sep-2015? There are 183 calendar days from market close on the last coupon 21-Apr-2015 to the market close of the next coupon date on 21-Oct-2015. Between the market close times from 21-Apr-2015 to 11-Sep-2015 there are 143 calendar days. From 21-Apr-2015 to 14-Sep-2015 there are 146 calendar days. From 14-Sep-2015 there were 20 coupons remaining to be paid including the next one on 21-Oct-2015. All of the below answers are given as effective 3 day rates. A Chinese man wishes to convert AUD 1 million into Chinese Renminbi (RMB, also called the Yuan (CNY)). The exchange rate is 6.35 RMB per USD, and 0.72 USD per AUD. How much is the AUD 1 million worth in RMB? The Chinese government attempts to fix its exchange rate against the US dollar and at the same time use monetary policy to fix its interest rate at a set level. To be able to fix its exchange rate and interest rate in this way, what does the Chinese government actually do? 1. Adopts capital controls to prevent financial arbitrage by private firms and individuals. 2. Adopts the same interest rate (monetary policy) as the United States. 3. Fixes inflation so that the domestic real interest rate is equal to the United States' real interest rate. Which of the above statements is or are true? The 'initial margin', also known as the performance bond in a futures contract, is paid at the start when the futures contract is agreed to. or ? Find Scubar Corporation's Cash Flow From Assets (CFFA), also known as Free Cash Flow to the Firm (FCFF), over the year ending 30th June 2013.  Scubar Corp Income Statement for year ending 30th June 2013$m Sales 200 COGS 60 Depreciation 20 Rent expense 11 Interest expense 19 Taxable Income 90 Taxes at 30% 27 Net income 63
 Scubar Corp Balance Sheet as at 30th June 2013 2012 $m$m Inventory 60 50 Trade debtors 19 6 Rent paid in advance 3 2 PPE 420 400 Total assets 502 458 Trade creditors 10 8 Bond liabilities 200 190 Contributed equity 130 130 Retained profits 162 130 Total L and OE 502 458

Note: All figures are given in millions of dollars ($m). The cash flow from assets was: Read the following financial statements and calculate the firm's free cash flow over the 2014 financial year.  UBar Corp Income Statement for year ending 30th June 2014$m Sales 293 COGS 200 Rent expense 15 Gas expense 8 Depreciation 10 EBIT 60 Interest expense 0 Taxable income 60 Taxes 18 Net income 42
 UBar Corp Balance Sheet as at 30th June 2014 2013 $m$m Assets Cash 30 29 Accounts receivable 5 7 Pre-paid rent expense 1 0 Inventory 50 46 PPE 290 300 Total assets 376 382 Liabilities Trade payables 20 18 Accrued gas expense 3 2 Non-current liabilities 0 0 Contributed equity 212 212 Retained profits 136 150 Asset revaluation reserve 5 0 Total L and OE 376 382

Note: all figures are given in millions of dollars ($m). The firm's free cash flow over the 2014 financial year was: The hardest and most important aspect of business project valuation is the estimation of the:  Project Data Project life 2 yrs Initial investment in equipment$600k Depreciation of equipment per year $250k Expected sale price of equipment at end of project$200k Revenue per job $12k Variable cost per job$4k Quantity of jobs per year 120 Fixed costs per year, paid at the end of each year $100k Interest expense in first year (at t=1)$16.091k Interest expense in second year (at t=2) $9.711k Tax rate 30% Government treasury bond yield 5% Bank loan debt yield 6% Levered cost of equity 12.5% Market portfolio return 10% Beta of assets 1.24 Beta of levered equity 1.5 Firm's and project's debt-to-equity ratio 25% Notes 1. The project will require an immediate purchase of$50k of inventory, which will all be sold at cost when the project ends. Current liabilities are negligible so they can be ignored.

Assumptions

• The debt-to-equity ratio will be kept constant throughout the life of the project. The amount of interest expense at the end of each period has been correctly calculated to maintain this constant debt-to-equity ratio. Note that interest expense is different in each year.
• Thousands are represented by 'k' (kilo).
• All cash flows occur at the start or end of the year as appropriate, not in the middle or throughout the year.
• All rates and cash flows are nominal. The inflation rate is 2% pa.
• All rates are given as effective annual rates.
• The 50% capital gains tax discount is not available since the project is undertaken by a firm, not an individual.

What is the net present value (NPV) of the project?

 Project Data Project life 1 year Initial investment in equipment $6m Depreciation of equipment per year$6m Expected sale price of equipment at end of project 0 Unit sales per year 9m Sale price per unit $8 Variable cost per unit$6 Fixed costs per year, paid at the end of each year $1m Interest expense in first year (at t=1)$0.53m Tax rate 30% Government treasury bond yield 5% Bank loan debt yield 6% Market portfolio return 10% Covariance of levered equity returns with market 0.08 Variance of market portfolio returns 0.16 Firm's and project's debt-to-assets ratio 50%

Notes

1. Due to the project, current assets will increase by $5m now (t=0) and fall by$5m at the end (t=1). Current liabilities will not be affected.

Assumptions

• The debt-to-assets ratio will be kept constant throughout the life of the project. The amount of interest expense at the end of each period has been correctly calculated to maintain this constant debt-to-equity ratio.
• Millions are represented by 'm'.
• All cash flows occur at the start or end of the year as appropriate, not in the middle or throughout the year.
• All rates and cash flows are real. The inflation rate is 2% pa.
• All rates are given as effective annual rates.
• The 50% capital gains tax discount is not available since the project is undertaken by a firm, not an individual.

What is the net present value (NPV) of the project?

 Project Data Project life 1 year Initial investment in equipment $8m Depreciation of equipment per year$8m Expected sale price of equipment at end of project 0 Unit sales per year 4m Sale price per unit $10 Variable cost per unit$5 Fixed costs per year, paid at the end of each year $2m Interest expense in first year (at t=1)$0.562m Corporate tax rate 30% Government treasury bond yield 5% Bank loan debt yield 9% Market portfolio return 10% Covariance of levered equity returns with market 0.32 Variance of market portfolio returns 0.16 Firm's and project's debt-to-equity ratio 50%

Notes

1. Due to the project, current assets will increase by $6m now (t=0) and fall by$6m at the end (t=1). Current liabilities will not be affected.

Assumptions

• The debt-to-equity ratio will be kept constant throughout the life of the project. The amount of interest expense at the end of each period has been correctly calculated to maintain this constant debt-to-equity ratio.
• Millions are represented by 'm'.
• All cash flows occur at the start or end of the year as appropriate, not in the middle or throughout the year.
• All rates and cash flows are real. The inflation rate is 2% pa. All rates are given as effective annual rates.
• The project is undertaken by a firm, not an individual.

What is the net present value (NPV) of the project?

Find the cash flow from assets (CFFA) of the following project.

 One Year Mining Project Data Project life 1 year Initial investment in building mine and equipment $9m Depreciation of mine and equipment over the year$8m Kilograms of gold mined at end of year 1,000 Sale price per kilogram $0.05m Variable cost per kilogram$0.03m Before-tax cost of closing mine at end of year $4m Tax rate 30% Note 1: Due to the project, the firm also anticipates finding some rare diamonds which will give before-tax revenues of$1m at the end of the year.

Note 2: The land that will be mined actually has thermal springs and a family of koalas that could be sold to an eco-tourist resort for an after-tax amount of $3m right now. However, if the mine goes ahead then this natural beauty will be destroyed. Note 3: The mining equipment will have a book value of$1m at the end of the year for tax purposes. However, the equipment is expected to fetch $2.5m when it is sold. Find the project's CFFA at time zero and one. Answers are given in millions of dollars ($m), with the first cash flow at time zero, and the second at time one.

A company has:

• 50 million shares outstanding.
• The market price of one share is currently $6. • The risk-free rate is 5% and the market return is 10%. • Market analysts believe that the company's ordinary shares have a beta of 2. • The company has 1 million preferred stock which have a face (or par) value of$100 and pay a constant dividend of 10% of par. They currently trade for $80 each. • The company's debentures are publicly traded and their market price is equal to 90% of their face value. • The debentures have a total face value of$60,000,000 and the current yield to maturity of corporate debentures is 10% per annum. The corporate tax rate is 30%.

What is the company's after-tax weighted average cost of capital (WACC)? Assume a classical tax system.