Internal Rate of Return
Basic Terminology When a corporation such as IBM or General Motors wants to raise funds or capital it often will borrow the money by selling a large issue of its bonds to the public. Corporate bonds are essentially IOU's of the firm, and they can be purchased by institutional investors such as insurance companies, mutual funds or other corporations or by individuals such as yourself.
Most corporate bonds state that the issuer (also called the borrower or the seller of the bond) agrees to pay the investor (also called the lender or the buyer of the bond) a series of fixed interest payments every six months plus a large sum when the bond matures. The amount of interest that a bond pays is determined by multiplying its COUPON RATE times the bond's PAR VALUE. Most corporate bonds have a PAR VALUE (also called the FACE VALUE or PRINCIPAL) of $1,000. This $1,000 par value is printed on the bond. The coupon rate is also printed on the bond and never changes over the life of the bond. For example, if IBM issues a bond whose coupon rate is 12%, this means that the bond pays the holder .12 x 1,000 or $120 per year. But since the interest payment is made semi-annually, the holder actually receives $60 every six months. The coupon rate is always stated on an annual basis even though the actual payment is semi-annual. If another bond's coupon rate is 20%, then an investor in these bonds would receive $100 every six months.
The bond also has a stated MATURITY DATE at which time the firm retires the bond by making the last coupon payment and also paying the holder the par value or principal of $1,000. If the IBM bond matures in 20 years and has a 12% coupon rate, the firm agrees to pay the investor $60 every six months for 20 years plus $1,000 at the end of the 20th year (or 40th period).
In exchange for this stream of coupon payments and single principal payment, the investor must, of course, buy the bond for the prevailing MARKET PRICE. This market price that investors are willing to pay for the bond is a function of the investors' REQUIRED INTEREST RATE. Don't assume that because the bond has a par value of $1,000 that this will be the price paid by the investors. This required interest rate is dependent upon several factors including the risk of the investment as perceived by the investors. Risk is affected by several factors: Because of its long history of profitability IBM will enjoy a better credit rating than unknown upstart Al's Business Machines. Because the soft drink industry is currently more profitable and stable than the steel industry, Coca Cola will be seen as being a safer investment than Bethlehem Steel. Consequently, the investor will be willing to accept a lower interest rate or return on IBM's and Coke's bonds than on Al's or Beth Steel's bonds. More can go wrong on a 30 year bond than on a 3 year bond. Therefore, the investor will usually demand a higher return on the long-term bonds. Although the interest rate is normally stated on an annual basis such as 10% per year, because bonds pay their coupons semi-annually, when doing computations you need to divide the stated annual interest rate in half and consider the rate per period and the number of six month periods. Thus, an interest rate of 10% per year is treated as 5% per period.
Because of the time value of money, a $60 coupon payment in one period's time is not worth $60 today. A $1,000 return of principal in 20 years (40 periods) is worth a lot less than $1,000 today. If the annual rate of interest is 10%, then $60 paid at the end of one period is worth only 60/(1+.05)1 or $57.14. The $1,000 payment is worth 1000/(1+.05)40 or only $142.09. Theoretically, the price that an investor would be willing to pay for a bond would be equal to the sum of all of the coupon payments and the par value paid at maturity when each payment is discounted at the appropriate rate of interest.
For example, the IBM bond that has an annual coupon rate of 12% and a maturity of 20 years would pay $60 each period for 40 periods and $1,000 at the end of the 40th period. If investors required an interest rate of 10% per year (5% per period), the bond would sell at $1,172. The price is equal to the sum of the coupon payments and the lump-sum par value when each is converted to its present value. The following equation shows this calculation. 1172 = 60/(1+.05)1 + 60/(1+.05)2 + 60/(1+.05)3 +...+ 60/(1+.05)40 + 1000/(1+.05)40
Let's verify that the 1172 is price of the bond. Use your calculator or click the button below to use the Fin 125 PV/FV Calculator. Input the following: PMT = 60, n = 40, FV = 1000 and i = 5 and then click the PV button. Remember that this calculation assumes that the FV occurs at the time of the last coupon payment and that n = 40.
As an alternative, you can do the problem using the cash flow keys of your calculator or click the button below to use the Fin 125 Cash Flow Calculator. Be careful! Input the following: discount rate = 5, CF1 = 60, n1 = 39, CF2 = 1060, n2 = 1. Click the NPV button. Notice that n1 = 39 and not 40. The final 60 coupon has to be included with the 1000. If you enter n1 as 40, then this assumes the 1000 principal payment occurs at the end of the period 41.
If an investor paid $1,172 for this bond, held it to maturity and received the 40 coupon payments and the par value of $1,000, she would have earned a return of 5% per period or 10% per year.
On the other hand, if investors perceive the firm as being riskier or if interest rates rise for other reasons and the required return is actually 14% a year, the price of the bond would be only $867. (Merely substitute .07 into the above equation in place of the .05.)
Callable Bonds Many corporate bonds contain a CALL FEATURE that enables the issuing firm to buy back or redeem the bonds prior to maturity at a stated price called the CALL PRICE. Corporations like to include this feature with their bonds because it gives them flexibility should interest rates drop over the life of the bond. The call feature is an option and it does not have to be exercised by the firm.
A firm that issued 20 year bonds five years ago when the interest rate was 14% would like to be able to replace them with new bonds today when the interest rate is only 10%. Rather than have to go into the open market and pay high prices, by making the bonds callable the corporation retains the option of buying them back from investors at a stated call price. The investors have no choice but to comply. This is a risk the investors assumed when they originally purchased the bonds. When a bond is called, the firm ceases to make any additional coupon payments and sends the investor a single terminal amount, the call price.
Convertible Bonds Some corporate bonds also contain a convertible feature that gives the investor the option of exchanging the bond for a specified number of shares of the corporation's common stock. It is an option and it need not be exercised if the investor does not desire. By exercising the option, the investor returns the bond to the company, thereby canceling the firm's IOU, and receives shares of the company's common stock. The investor changes his legal position from lender or creditor of the company to one of owner. The terms of the conversion is given by the CONVERSION PRICE of the stock. For example, if the conversion price is $40 per share, then the bond can be swapped for 1000/40 or 25 shares. 25 shares to 1 bond is the CONVERSION RATIO. The conversion ratio is always equal to the face value of the bond divided by the conversion price. If the conversion price is $100 per share, the conversion ratio is 1000/100 or 10 shares of the common stock. Investors find the conversion feature attractive because if the firm shows promise and is expected to grow and become more profitable over time, the bond holder can start out as a creditor with a very strong legal claim on the firm should anything go wrong and then switch at the best time and become an owner. Conversion is not reversible. If the firm's profits grow and its stock price also keeps pace, the bond holder will likely want to convert and become a stock holder. If a year later profits fall off and the stock price plummets, the investor cannot get his bond back.
Now consider the possibility of combining the call and convertible features. If the conversion ratio is 10 shares per bond and the stock is selling for $115 per share, the bond would be worth 10 x 115 or $1,150 if it were converted to stock. If the firm notifies the bond holder that the bond is being called for a call price of $1,080, the investor will quickly convert it to $1,150 worth of stock rather than sell it back to the firm for the call price of $1,080.
Depending upon the features of the bond, there are four ways for an investor to get rid of a bond: turn it into the firm at the maturity date (would receive a terminal value of $1,000), sell it on the open market to another investor prior to maturity (the terminal value would be the current market price), have it called by the firm (terminal value would equal the call price) or convert it to common stock (terminal value of the conversion ratio times the stock's price).
Internal Rate of Return After following one of the above four courses of action, the investor would want to compute his actual realized rate of return over the time he held the bond. This actual rate of return is often called the INTERNAL RATE OF RETURN and is a very important concept in finance but is difficult to visualize. It is the value of "r" in equation  below: Price Paid = C/(1+r)1 + C/(1+r)2 + C/(1+r)3 +...+ C/(1+r)n + TV/(1+r)n where: C = the coupon in dollars per six-month period; n = the length of holding period in six-month periods; TV = the terminal value; r = the internal rate of return;
Thus, the actual or internal rate of return is the value of the discount rate that equates the price initially paid for the bond with all of the cash flows received by the investor over the course of his holding period. Without a calculator or a computer, this would have to be found by trial and error. Fortunately, Excel has a built-in IRR function to simplify the calculations. If the investor holds the bond to maturity and receives the par value of $1,000 from the firm, then the value of the internal rate of return would be equal to the required interest rate used in equation  to determine the market price. If, however, the terminal value is something other than the $1,000 par value, the actual or internal rate of return will be different from the required interest rate.
Here's a typical problem that you should expect: Six years ago Bunky's Burgers issued some 20 year convertible bonds that have a coupon rate of 12%, paid semi- annually. The annual interest rate was 12.41%. The conversion price is $50 a share and the common stock is currently trading at $87.90 a share. The bond is callable at $1,080. Use your model to show that an investor would have paid $970 six years ago and that he would have earned an internal rate of return of almost 20% if the bond were called today.
Use your calculator or click the button below to use the Fin 125 Cash Flow Calculator and first solve for the price paid by the investor. To find the initial price paid, enter the following: discount rate = 12.41/2 = 6.205; CF1 = 60; n1 = 39; CF2 = 1060; n2 = 1 and click on NPV. The price should be 969.94. Now find the IRR earned over the 6 years. Enter the following: CF1 = 60; n1 = 11; CF2 = 1818 (87.90 x 20 shares at conversion + 60 coupon); n2 = 1 and click on IRR. The return is about 9.99% a periods or 20% for the year, compounded semiannually.