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5
Apr

GDP Mapping Exercise – Illustrations from recent studies

By Surabhi Mall, IFMR Rural Finance

In the previous blog post of the KGFS Model Incubation series, we drew out the implications of mapping the GDP of a branch’s service area on strategic decisions related to district selection, branch potential, product suitability and customer centricity. In this context, some of the pertinent follow-up questions that arise are – How much will this activity cost? Should this capability be harnessed indigenously or simply outsourced? What is a statistically approved research design to follow? This post essentially focusses on understanding the ABC of executing this study with relevant elaborations and learning from the past.

The GDP exercise can serve different purposes and based on the objective use and nature of requirement the scope of the study can be defined. The following two studies conducted in two different districts in Tamil Nadu attempt to illustrate this.

The objective of the first study was to compare the gross potential of a branch in the Krishnagiri district, Tamil Nadu, to that of a model KGFS branch[1] wherein ‘comparable’ was defined as a range with an acceptable degree of error from the estimates of the model branch. In this case, focus of the study was to get the aggregate number and not necessarily its constituents. Simply put, this means that while sector-wise distribution of the branch economy could be insightful, it was not the focus of the study per se. Resources used as part of this included a full-time staff that spent one day in assimilating secondary data, three working days on the field to collect primary data as well as to validate the secondary data. The staff was actively supported by a Wealth Manager at the branch location.

The second study at Ellakuruchi village in Ariyalur district, Tamil Nadu, was done with an objective of profiling the district as well as the branch service area. District profiling required a thorough review of the district’s demography, geography, economic status, main crops, enterprises and occupations. Profiling the branch service area required field insights on aspects such as different occupations that thrive in the area so as to map each economic activity with its volume; cash-flows associated with the occupation so as to map business potential; formal and informal financial providers so as to understand current and potential gaps in the financial landscape among others. This objective required one data analyst responsible for secondary data collection, methodology design, primary data collection, data collation and presentation of the findings. Primary data collection was actively supported by 1-2 Wealth Managers in the field for 6-8 working days. The entire exercise was executed in one month. In order to add greater rigour and sanctity to the estimates, a similar study was executed in another branch in the district.

Below are pie-charts depicting the findings from both GDP studies in Krishnagiri and Ariyalur districts respectively.

One of the big challenges in initiating such a study is that the data records of this kind are not methodical, very contextual and mainly absent from conventional databases for any triangulation. The other concern may be related to the design of the research methodology for the intended purpose of the study. Often, simple doubts such as the size of a representative sample directly impact the resource requirements and rigour of the study. To address these issues simple project management tools such as defining the objective, scope and research design a priori through a clear project plan will be quintessential. In the first study, since the objective was clearly identified as “compare the gross potential of the branch to that of a model KGFS branch,” an exhaustive sector-distribution map of the branch’s economy was not required. Conversely, in the second study, to “profile the district and branch service area”, there was need for profound understanding of the demographic and economic constitution of the area. This in turn required information about the share of each activity in the total economic pie.

In cases wherein the objective of the exercise is to design a customer engagement strategy or an optimal capacity plan for the branch by projecting lean and peak cash requirement periods, activity mapping will need to be extremely exhaustive. This would imply that aspects such as the ‘crop net income’ component of the GDP pie be further broken down to list the main crops, their seasonality, and cash flow projections related to each crop.

It is also important to acknowledge the homogeneity in variables during the study. For example, if a service area constitutes of four villages of 300 households each[2], the economic map of one village can be multiplied to give the macroeconomic map of the branch. This adds to operational efficiency in the execution whilst minimizing scope for error.

Since the KGFS model is designed to entrench itself in the community it serves by developing a deep understanding of the geography, the local culture, the economic activities and dominant customer segments, the GDP exercise is perfectly tailored for the KGFS model. Understanding its benefits as a principal and inaugural step in model incubation and thereby budgeting for the costs involved will lead towards deepening the very foundation of the model.

The next blog post in the series will discuss the heuristics of site selection in a rural village context. By illustrating the KGFS’ experience across diverse geographies, it will attempt to showcase the various components that may play a foundational role in the science of site selection.

[1] The model KGFS branch is conceived and defined based on the KGFS mission, the business requirements and past learning experience.
[2] Unit of aggregation may vary from a household, a village, a panchayat or a block.

24
Mar

Designing a Framework for Event Risk & Loss Estimation: Understanding Natural Disasters

By Vaibhav Anand, IFMR Capital

In the previous post of this series on Event Risk & Loss Estimation, we discussed briefly the motivation and key modules of a framework for estimating capital against event risk. Before we discuss the approach and components of such a framework in detail, it is important to understand the nature of the extreme events and how such events impact borrowers, lenders and credit portfolios. In this post, we will discuss the former in the context of some of the extreme and not-so-extreme events. Not-so-extreme because either there may be some predictability in the timing of their occurrence or their impact may appear less extreme and more diffused- spatially as well as temporally. We will cover four natural disasters- Flood, Drought, Cyclone and Earthquake – in this post. We will discuss briefly the socio-political events in the next post along with a common investigation framework to evaluate the impact of events on borrowers and lenders.

Floods

India receives more than 70% of its rainfall over a period of four months1. In fact, the rainfall may not be evenly distributed during the season. This results in long dry spells followed by heavy rainfall in several geographies. Throughout history floods are the most recurrent natural disasters that cause havoc. A flood is defined as a phenomena in which water over flows its natural or artificial banks onto normally dry land, such as a river inundating its floodplain2. There are mainly four types of flood:

• Riverine Flood: This type of flooding is caused by overflow of water along the path of a river and mainly affects land area along the river bank. The frequent flooding in the Kosi River, also known as the sorrow of Bihar, is a typical example of riverine flood. The August 2008 flood in Kosi River was caused by a breach of an embankment of the river in Nepal. After the breach, the river changed its course to the one followed by it in 1930s in a very short time causing severe flooding over a large extant of land in Bihar3. Four major riverine flood prone regions are:
• Ganga basin
• Brahmaputra basin
• Krishna, Godavari, Mahanadi and Cauvery basin

Figure 1: Flood Hazard Map of India4

• Coastal flood: Flooding caused by the ocean water driven inland by another natural cause such as storms, cyclones, tidal waves caused by earthquake (tsunamis), etc.
• Urban flood: There could be several reasons for urban flooding such as heavy rainfall, sudden release of water from a bund or dam, tidal waves, etc. However, the main underlying cause is usually the slow absorption of water by the land. The 2005 flood in Mumbai was an example of urban flooding.
• Flash flood: Such floods manifest in a very small time and usually without warning, hence the name. These are usually caused by heavy rainfall or release of water from a dam. The disastrous floods in the North Indian geography in the last few years were examples of flash floods.

Droughts

Drought, along with flood, is one of the most recurring natural disasters in India. The duality is not surprising since both are linked to rainfall cycles to a large extant. However, unlike floods and other disasters that we will discuss shortly, it is very difficult to identify the onset and end of a drought. In fact, there are multiple definitions of drought proposed from time to time based on the specific ‘water content’ requirement (of soil, for example) for various human activities5.

There are mainly three types of droughts6 and the definitions vary based on the type:

• Meteorological drought: Situation when the deficiency of rainfall at a meteorological sub-division level is 25% or more of the long-term average (LTA) of that sub-division for a given period. If the rainfall deficiency is less than 50%, it is classified as “moderate” drought; else, it is termed as “severe” drought7.
• Hydrological drought: It is a prolonged meteorological drought resulting in depletion of surface water from various reservoirs causing severe shortage of water for human and livestock needs8.
• Agricultural drought: It is a situation when rainfall and soil moisture are inadequate to support healthy agricultural crop growth9. This may be caused by a meteorological drought followed by a hydrological one.

(It is very tempting to suggest a fourth type of drought here- Political Drought- but we will leave it out of this post!)

It can be concluded from the above definitions that a meteorological drought instance may not be disastrous in isolation. However, a series of meteorological droughts or mismanagement of water resources may result in a hydrological drought over time10. This coupled with lack of irrigation resources may result in agricultural drought. Further, it should be noted that agricultural drought is also a relative term- it would depend on the moisture requirement of the crop grown in the affected area. These factors make the impact evaluation exercise for a drought on the economic activities a very difficult task. We will discuss in detail this issue in the next post of the series.

National Climate Centre (NCC) provides a good literature on popular methods for measuring the drought severity over a spatial unit in a report published in 201011. The report also provides the drought indices for 458 districts using the southwest monsoon season rainfall time series over the period 1901-2003.

Cyclones

Cyclones, one of the most recurring extreme weather events across the globe, are weather systems with wind speed exceeding 62 km per hour. Though cyclones are known by different names in different regions (hurricane in North Atlantic and East Pacific, Typhoon in West Pacific, and cyclone in Indian Ocean), the classification is mainly based on the wind speeds. For a detailed classification, one may refer to the Wikipedia page on tropical storms which provides a great deal of information as well as useful references on cyclones12.

India with a coastline of more than 7500 kilometres is one of the worst affected regions in the world with on an average nearly 370 million people exposed to cyclone disasters annually13. Cyclones are multi-hazard systems, i.e. multiple hazards are associated with a cyclone– high speed winds, torrential rains and inland flooding, and storm tide14. However the development, eventual landfall and potential impact of a tropical cyclone can be estimated more accurately relative to other natural disasters such as floods, drought and earthquakes. Though the cyclones can change their course or dissipate suddenly, the modern forecasting systems have enabled the governments and disaster management bodies around the world to take preventive actions to minimize the damage due to cyclones and associated hazards.

National Disaster Management Authority (NDMA) of India published a study in 2010 which suggested a cyclone hazard mapping of coastal districts in India based on the historical (1981-2008) occurrences of cyclones and the multiple hazards associated with them.

Figure 2: Cyclone hazard map based on multi-hazard model12

Earthquakes

Earthquakes are arguably the most fatal of the natural disasters. There is high amount of uncertainty attached to the timing, location and severity of an earthquake which makes preventive measures very difficult if not near impossible. Unlike floods, droughts or cyclones, there are no seasonal patterns or clustering in earthquake occurrences15. Earthquakes are caused by the sudden release of built up pressure in the earth’s crust in the form of an energy explosion that fractures the earth’s surface and creates seismic waves. The resulting ground acceleration is the main cause of damage caused by the earthquakes which impacts buildings, roads and other ground infrastructure. Like cyclones, earthquakes too have associated hazards, most common being the tidal waves, also known as tsunamis. The 2004 Tsunami was caused by one of the strongest earthquakes with magnitude of nearly 9.1 on Moment Magnitude Scale (MMS). The other two of the most disastrous earthquakes that India faced had magnitudes of 7.7 (Bhuj, 2001) and 6.2 (Latur, 1993).

Earthquake magnitude can be measured using seismograph (seismogram is the output graph of a seismograph!) which records the ground vibration. One of the most popular scales of measurement is the Richter scale, named after the seismologist Charles Richter16. The relation between the amount of energy and the scale reading is nonlinear. An increase of one magnitude signifies a ten times higher ground motion and nearly thirty times the energy17. Earthquakes with magnitude lower than 5.5 are usually not dangerous and may not cause any damage. An interesting fact is the frequency of earthquakes is far higher than intuition would allow us to guess; however, fortunately, most of these earthquakes have very low magnitude. A table based on the estimates of United States Geological Survey (USGS) is shown below.

Figure 3: Estimated Earthquake frequency

The earthquake hazard map of India divides the region in five seismic zones.

Figure 4: Seismic Zones of India18

Evaluating the impact on portfolio

Can we actually use a common investigation framework to evaluate the impact of such events? Though the extreme events listed and discussed above, and others not mentioned here, may differ significantly in nature, a common framework for investigation can be used to understand how such events impact portfolio performance of a credit institution. In the next post of the series we will discuss such an investigation framework.

We would like to thank Divyasree PK of IIT Madras who worked on the topic during her internship at IFMR Capital.

References:

1. http://agricoop.nic.in/DroughtMgmt/DroughtManual.pdf
2. http://saarc-sdmc.nic.in/pdf/flood.pdf
3. http://www.indiaenvironmentportal.org.in/files/The%2018%20August%202008%20Kosi%20river%20breach.pdf
4. http://www.cddrm-ncdc.org/e39621/e39678/
5. http://ijset.com/ijset/publication/v1s4/p%20149-157%20surendra%20published%20paper.pdf
6. National Commission of Agriculture in India
7. National Commission of Agriculture in India
8. http://www.nrsc.gov.in/pdf/Chap_13_Droght.pdf
9. http://www.nrsc.gov.in/pdf/Chap_13_Droght.pdf
10. http://sandrp.in/otherissues/Maharashtra_Drought_2012_13_worse_than_1972_March2013.pdf
11. http://www.imdpune.gov.in/ncc_rept/RESEARCH%20REPORT%2013.pdf
12. http://en.wikipedia.org/wiki/Tropical_cyclone#Hurricane_or_typhoon
13. http://ncrmp.gov.in/ncrmp/Cyclone_Impact.html
14. http://ndma.gov.in/images/cyclones/cyclonepronedistrict.pdf
15. However, there are some studies which suggest that weather changes or human activities may seasonally impact the seismic activity, e.g. https://www.sci.hokudai.ac.jp/grp/geodesy/top/research/files/heki/year03/Heki_EPSL2003.pdf
16. The Moment Magnitude Scale (MMS), developed in 1972, is commonly used now. However, both the scales are logarithmic and have similar characteristics for medium magnitude earthquakes. For a good account please refer to the Wikipedia page: http://en.wikipedia.org/wiki/Moment_magnitude_scale#Comparison_with_Richter_scale
17. http://saarc-sdmc.nic.in/pdf/earthquake.pdf
18. http://www.hpsdma.nic.in/ResourceList/Maps/EqIndia.pdf

9
Mar

An Initial Analysis of the Atal Pension Yojana

By Vishnu Prasad & Anand Sahasranaman, IFMR Finance Foundation

The Finance Minister, in his budget speech for 2015-16, has announced a new defined benefit pension scheme – Atal Pension Yojanai (APY, henceforth) – for unorganised sector workers who are not covered under any statutory social security scheme. It has been proposed that the existing subscribers of National Pension System-Swavalamban (NPS-S), the extant pension scheme for the unorganised sector, be automatically migrated to the APY unless they voluntarily opt out. Under the NPS-S, launched in September 2010, the Government of India currently contributes Rs. 1000 per year to every subscriber who makes a minimum contribution of Rs. 1000 per year towards building a post-retirement corpus.

The proposed APY differs from the NPS-S in two significant ways. First, the NPS-S is a defined contribution scheme where the subscriber’s contribution is invested in government securities, corporate bonds, and equity instruments-the scheme does not guarantee fixed returns to subscribers. In contrast, the APY is a defined benefit scheme that will provide the subscriber with fixed monthly incomes between Rs.1000 and Rs.5000 based on the respective monthly contribution amounts (which varies by age and saving potential of the subscriber; refer Table 2 below). Second, under the APY, the government will match either 50 per cent of the subscriber’s contribution or Rs. 1000, whichever is lower (initially for a five year period till 2019-20). This paves the way for the creation of a graded matching scheme where subscribers who contribute below Rs. 1000 also receive a (less than equal) matching contribution. For availing the same amount of co-contribution as under the NPS-S, a subscriber will need to contribute Rs. 2000 into APY (Rs. 1000 under NPS-S).

In order to judge the adequacy of the defined benefit proposed under APY, we compare the proposed benefit to the average monthly expenditure incurred by individuals in the lowest income quintile. Table 1 below presents the average monthly expenditure of an individual in each income quintileii. As the Table shows, the average monthly expenditure of an individual in the first income quintile is Rs. 628. As the APY is designed as a pension scheme for a household, we assume the average monthly expenditure for two people in our analysis.

Table 2 and Table 3 (below) present the defined pension amount (nominal and real) that will be paid to the subscriber and his spouse based on two sets of indicative monthly contributions so as to get nominal defined benefit pay-outs of Rs. 1,000 per month and Rs. 5,000 per month respectively. For example, in order to receive a nominal defined benefit of Rs. 1000 per month at the time of retirement at 60 years (Table 2, Column 4), an 18-year old needs to contribute Rs. 42 per month while a 40-year old needs to contribute Rs. 291 per month. Similarly, in order to receive a nominal defined benefit of Rs. 5000 per month (Table 3, Column 4), an 18-year old and a 40-year old need to contribute Rs. 210 per month and Rs. 902 per month respectively.

Source: http://pib.nic.in/newsite/PrintRelease.aspx?relid=116208 and author’s own calculations.

Table 2 and Table 3 also provide, in Columns 5 and 6, the real monthly pensions that households would receive and the shortfall in pension compared to their expenditure needs, which is a more realistic representation of the actual value of money households will actually be able to get. In order to calculate the real value of the pension we present value the cash flows at a discount rate of 5%, which is one per cent higher than the long-term inflation target, set by the Reserve Bank of India. According to the indicative tables, an 18-year old who contributes Rs. 42 per month will result in a real monthly pension of of Rs. 129 for the household. Assuming that the 18-year old’s income falls within the first income quintile, the defined benefit will cover 10.3% of her monthly expenditure. However, even if the 18-year is able to contribute Rs. 210 per month towards pensions, the defined benefit at retirement will yield a real monthly amount of Rs. 644 for the household (equivalent to a nominal monthly pension of Rs. 5,000) and will be sufficient to cover only 51.2% of her monthly expenditure.

Thus, an 18-year old in the first income quintile can hope to cover between 10% and 51% of her monthly expenditure upon retirement by contributing to the scheme. Tables 2 and 3 also make clear that there is a real risk of significant shortfalls for all age-groups, whether they contribute into the nominal defined benefit of Rs. 1,000 or Rs. 5,000.

It should also be noted that if inflation continues to hover around historical inflation of 8%, the defined benefit would cover a much lesser percentage of an individual’s expenditure per annum. For instance, for an 18-year old who contributes Rs. 42 a month, the monthly defined benefit would be sufficient to cover a mere 3.1% of her monthly expenditure upon retirement.

The need for inflation indexation and a more optimal investment mix

If we assume that the government contribution will be discontinued at the end of five years, the annualised rate of return required to guarantee Rs.1000 per month (or Rs.1.7 lakh corpus) for an 18-year old contributing Rs. 42 a month is 7.58%. If we relax the assumption that government contribution will be discontinued and that the matching contribution from the government will continue in perpetuity, the annualised rate of return required to guarantee the defined benefit drops to 6.68%. Depending on the evolution of inflation rates over time, these nominal rates of return could end up resulting in very low real return or even the risk of erosion in capital due to inflation. The APY could benefit from addressing two key design limitations of the NPS-S:

1. Lack of inflation indexation: In order to ensure that the pension corpus of low-income customers is not eroded over time, the government needs to index both the subscriber contribution and the matching contribution to inflation. Assuming that the matching contribution continues in perpetuity and that the government invests the corpus in the current NPS-S investment mix (upto 85% in government bonds and upto 15% in equity), indexing both the subscriber and government contributions to inflation on an annual basis could provide the subscriber with a substantially higher corpus (as much as 4 times higher than the guaranteed amount). Indexing social security benefits to inflation would also be in line with international best practices. For instance, the US Social Security Administration makes an annual cost of living adjustment by linking social security benefits to the Consumer Price Index.
2. Conservative investment mix: The current NPS-S investment mix invests upto 85% of the corpus in bonds and upto 15% in equity, the remaining comprising of corporate bonds. In contrast, the NPS-Main follows a life-cycle investment mix which invests 50% of a 20-year old subscriber’s corpus in equity, 30% in corporate bonds, and 20% in government bonds. As the subscriber ages, the share of equity and corporate bonds is reduced and transferred to the less-volatile government bonds. According to our calculations, investing in the life cycle fund mix could provide returns that are 49% higher than the returns on the current NPS-S mix. While guaranteeing a minimum amount through investments in approved fixed income instruments, the government could ensure that subscribers can accumulate corpuses that vastly exceed the guaranteed benefit by shifting to a more equity-heavy investment mix depending on age of the subscriber.

It is indeed heartening that there is a lot of policy attention on these very important questions of old age income security, and the emergence of the NPS and now the APY are testament to this. If some of the design limitations of the NPS-S were to be addressed in the APY, it would mean that the most vulnerable households would be able to build pension corpuses that could meaningfully provide them with old age income security.

—-

i – http://pib.nic.in/newsite/PrintRelease.aspx?relid=116208
ii – This analysis has been performed on data from a financial services firm that is operational across rural districts in three states of India for a sample of over 200,000 households.

2
Mar

District selection and estimating the potential of the KGFS entity through GDP Mapping

By Surabhi Mall, IFMR Rural Finance

This is the first blog post in the KGFS (Kshetriya Gramin Financial Services) Model Incubation series. The objective of the series is to methodically conceptualize an approach to build the branch network while incubating a new KGFS entity or expanding to contiguous districts. The posts focus on themes that range from district selection to identification of branch locations and optimization of the distribution network.

In this post we start with a brief discussion on the choice of a district and later discuss geography-specific questions that influence the economics of the model. Finally, the post stresses on GDP mapping as a heuristic solution to those questions.

Among the early steps of setting up a KGFS model is identifying a district. District selection is mostly a matter of organisational strategy and choice. The choice may be based on parameters such as district’s rural population, its population density, credit to GDP ratio, forest cover, road density, among others. Reviewing these attributes prior to finalising the district provides an elementary sense of the KGFS’s business potential, composition of the product basket in the branches as well as the degree of customisation that may be required to set up the model. Following this, reconnaissance (Recce) of the district provides insights to questions such as – Does the model need any fundamental changes or customisation in the given geography? What is the degree of competition it is likely to face? What is the current status of customers – i.e. their access to credit, income earning potential, and savings or repayment behaviour. This step serves as a final validation to the district selection strategy.

Recce gives field-level insights on the geography and competition in the district while secondary data helps understand the demography and infrastructure availability. Since geography, demography and the availability of credit in a place have significant influence on how the local economy develops, this should provide reasonable understanding of village-level economics and perhaps its influence on the branch’s business. It may even be leveraged to formulate the entity’s business plan, its competitive strategy and the network of branches. However, is this sufficient to understand the requirements and provide suitable comprehensive financial services? Is this information adequate for each branch to realise and achieve its potential volume of business at every life-stage? In fact, how does one quantify potential of a typical service branch in the chosen district? Finally, can the impact that the branch or the entity would have in the economic well-being of people in the long-run be measured? Estimating the branch service area’s Gross Domestic Product (GDP) provides powerful insights in unpacking such questions.

Figure 1: Three steps prior to setting up a new KGFS branch

Why the GDP?

To start with, what do we mean by estimating the GDP of a branch service area? GDP of an area is essentially the summation of different economic activities that thrive and contribute to the economy of that area.

Now, a KGFS stands to ensure the financial well-being of ‘every’ customer and enterprise in the area. This dictates that each customer’s needs be identified, acknowledged and serviced. In this context, the GDP exercise provides valuable insights on the composition of different sectors that thrive in the area. This can then be used to segment customers, identify their respective needs, customise the product basket at the branch and design an apt pricing as well as marketing strategy for them. A deep dive into the village economics through this exercise enables the branch to identify and prioritize customers from occupations with high degree of cash-flow mismatches, i.e. customers who are most likely to benefit from financial services. In effect, all this reinforces the KGFS’ geographic commitment by accounting for all possible households and economic drivers in the area.

The exercise gives insights on the share of existing financial players (from the interest income generated in the area), the median profile of a customer’s household and her debt servicing capacity. At the very outset, these can serve as a filter (post the Recce) to re-assert the choice of the district and the working estimates of the business plan. These estimates can then be built into the annual and monthly business targets for the branch. The activity can also be designed to give a rich sense of actual business that the branch is expected to do, i.e. its market share1. By capturing the share of other formal and informal financial institutions in the area, one can assess the volume of competition. This can then be netted off from the estimated demand for credit, thereby giving the potential market share of the branch.

From an operational point of view, data from GDP can be used to add greater granularities to the branch’s customer management database. It also enables scope for new business development2. For the product development team, it helps estimate occupation-based credit requirement and decisions regarding risk-exposure limits.

More generally, the GDP map helps visualise what kind of financial services are required to increase the size of the pie in the first place. It is a quantifiable measure that may be used as baseline for a ‘village profile’ in order to assess the financial viability and/or impact of a branch.

When – Ex-ante or ex-post?

While the above arguments advocate executing this prior to branch opening, this may not be a binding proposition. One can customize the scope of the exercise based on the objective sought at different life stages of a branch. For example, for an existing, low-performing branch where the enrolled database isn’t representative of the area’s population, the study’s objective can be to capture the economic drivers, re-estimate the branch’s potential and identify untapped business avenues. It may also aid to gauge the share of competition of other players’ vis-à-vis that of the KGFS branch.

However, the maximum utility of the exercise would lie in leveraging it as a diagnostic tool to acclimatize with the new geography of business. Such information, when captured at the very inception stage of branch set-up can greatly aid in understanding the branch’s gross potential and scale of operations, relevant needs of potential customer and perhaps even insights into strategies to thwart current or budding competition. If the GDP study in a branch is conducted at the proposed stage of branch, re-estimation and scoping for new business development through primary studies may then become redundant. Another added advantage of doing this prior is that this can be a part of the branch staff’s training track aimed at familiarizing them to the landscape of the area.

In the long run, a time-series GDP exercise process– prior to branch opening and ‘x years’ post branch opening is perhaps going to be the only real indicator of branch performance. Branch performance in this case will not mean the business at the branch but perhaps the change in (composition of) GDP of the service area and the impact of the KGFS branch on the lives of the people3.

KGFS Impact Long Run = f (actual business and type of customers served by the branch, increase in GDP share contributed by those served in the area that can be exclusively attributed to KGFS’ operations between KGFS0 & KGFS1).
[KGFS0 – is the GDP estimate from the study in time period 0 (prior to branch opening) & KGFS1 – Is the GDP estimate from the study in time period 1 (post x years of operation)]

The next blog in the series is on the Activity-based Costing of the GDP exercise. Through the lens of GDP studies done in the past, it will attempt to provide indicative answers to questions such as time and resource costs of the exercise.

Notes:

1. Market share of a branch estimates the actual served market/sales of a branch. This addresses concerns such as market cannibalization.
2. Based on forward and backward linkage of value chains in the area, loan purpose and occupation type, unexplored segments, etc,.
3. This estimation needs to account for fixed effects.

We recently hosted a series of knowledge management sessions (Spark Spring Edition 2015), as part of which Surabhi presented on this topic. In her session titled “Gross Domestic Product Mapping”, she spoke on how GDP Mapping can provide an intensive diagnosis of a defined location and how it can be used in a research process aimed at providing richer, robust and relevant information about markets, variables and potential.

View the presentation from her session below:

4
Feb

Estimating the Diversity Score of a Portfolio across Multiple Correlated Sectors: Generalized Herfindahl-Hirschman Index

By Vaibhav Anand and Ramasubramanian S V, IFMR Capital

Diversification is an effective risk mitigation strategy for portfolio risk management. It helps to mitigate risk arising from various factors, including extreme events, except factors which are systemic in nature. Often diversification across counterparties, sectors or geographies is the only risk mitigation tool available to a credit portfolio manager. It is important to measure and monitor the degree of diversification periodically to ensure that concentration risk remains low. However, quantifying the diversification may not be straightforward. Herfindahl-Hirschman Index (HHI) is one such measure but has limitations as it does not take into account correlation among underlying assets. In this post, we discuss a more general and effective measure of diversity score, Generalized-HHI (GHHI), to quantify diversification of a portfolio across multiple correlated sectors and sub-sectors.

In this post we first give a brief overview of how diversification helps to mitigate risk. We also discuss briefly the effect of correlation on portfolio risk. Next, we give a quick introduction to the classical HHI measure. In the last sections we present the GHHI formulation and illustrate its usage to identify a best diversified portfolio. In this blog post, we do not discuss in detail the derivation of GHHI and request interested readers to refer to the Working Paper for a detailed discussion.

How Diversification Helps

We discuss the benefit of diversification using a hypothetical credit portfolio of INR 1000 which can be lent to a single or multiple borrowers. Let us assume that all borrowers have an annual default probability (PD) of 5%. The portfolio manager has three options to lend the entire amount to:

(a) Scenario 1- A single borrower – No diversification
(b) Scenario 2- Equally to ten borrowers with no default correlation among them
(c) Scenario 3- Equally to ten borrowers with a pairwise default correlation of 0.33 among them

Assume no recovery.

Assuming 5% PD for each borrower, the expected loss of the portfolio in all scenarios is same and is equal to INR 50. However, the unexpected risk, measured as the standard deviation of loss here, for the three portfolios will be different.

The loss distributions of Scenario 1 and Scenario 2 are shown in the Figure 1. Scenario 1 has higher standard deviation because it has probabilities bunched only towards the ‘INR 0 loss’ and ‘INR 1000 Loss’ events which is intuitive for a one borrower portfolio- either all good or all bad, whereas a diversified portfolio, under scenario 2, has a better distribution with a very thin tail. For example, under Scenario 2 the probability of INR 1000 loss is nearly 1 in 10,000,000,000,000 as opposed to 1 in 20 in Scenario 1. Remember that Scenario 2 assumes no correlation among borrowers.

Figure 1: Loss Distribution: Scenario 1 and Scenario 2

Let’s see how correlation impacts the loss distribution. The loss distribution of Scenario 2 and Scenario 3 are compared in Figure 2. It can be seen that the latter’s loss distribution has fatter tails. The probability under Scenario 3 of INR 1000 loss is nearly 1 in 240, much higher than that under Scenario 2 but lower than that under no diversification Scenario 1. This tells us that greater than zero default correlation among borrowers increase risk in the portfolio and should be taken into account while quantifying the degree of diversification.

Figure 2: Loss Distribution: Scenario 2 and Scenario 3

Herfindahl-Hirschman Index (HHI)

One commonly used method of measuring the degree of diversification is HHI. HHI is defined as sum of the squares of the portfolio proportions. Consider a loan portfolio P with exposure across 3 counterparties, $C_i$, with corresponding proportions, $c_i$, where $i$ = 1 to 3. Then the degree of diversification for P across counterparties can be measured using HHI, where HHI is:

$HHI =\sum\limits_{i=1}^3 c_i^2$

However, the HHI assumes the sectors are independent and does not take into account the correlation among them. We propose a more general metric, the Generalized Herfindahl-Hirschman Index (GHHI), which incorporates correlation among underlying assets.

Generalized Herfindahl-Hirschman Index (GHHI)

The GHHI formulation for a portfolio P with across n assets having pair wise correlation of $\rho_{ij}$ can be written as:

$GHHI= {\sum\limits_{i=1}^n} c_i^2 + \sum\limits_{i}^n \sum\limits_{j\neq i}^n 2c_ic_j\rho_{ij}$

The derivation of the above formulation is not discussed here for the sake of brevity. We request the reader to refer to the Working Paper for a detailed account.

GHHI Vs HHI – Illustration

As an illustration the HHI and GHHI are estimated for four hypothetical credit portfolios, A, B, C, and D, with exposure in 12 counterparties across three sectors S1, S2 and S3. The counterparties are pairwise correlated within a sector with correlations of 0.05, 0.25 and 0.5 respectively. It is assumed that sectors are pairwise uncorrelated. Column 4 to 6 of Table 1 shows the exposure of the four portfolios in different counterparties.

Portfolio A is a seemingly perfectly diversified portfolio with equal exposure to all the available counterparties. In fact, a comparison based on the classic HHI yields a similar conclusion. However, it is shown using GHHI that a more diversified portfolio can be created taking into account the correlation among the counterparties in a sector. The last two rows of Table 1 show the calculated value of diversity scores of each portfolio as well as the effective number of counterparties in each portfolio. Based on HHI, the degree of diversification of the portfolios follows the order: A > B = C = D. Whereas GHHI takes into account the correlation and provides a more accurate order: D > A > C > B, i.e. a portfolio with exposures skewed towards a highly correlated sector, such as S3, will have lower risk mitigation.

Finally, we minimize the GHHI to identify the optimum proportions across assets for the best diversification. The optimal proportions across different sector are indicated in the last column of Table 1.

For a more detailed perspective on this subject please do refer our working paper which you can access here.


text:
Terminate one or more cluster nodes using process signaling

killNode(x, signal = tools::SIGTERM, ...)

## Arguments

x

cluster or cluster node to terminate.

signal

An integer that specifies the signal level to be sent to the parallel R process. It's only tools::SIGINT (2) and tools::SIGTERM (15) that are supported on all operating systems (i.e. Unix, macOS, and MS Windows). All other signals are platform specific, cf. tools::pskill().

...

Not used.

## Value

TRUE if the signal was successfully applied, FALSE if not, and NA if signaling is not supported on the specific cluster or node. Warning: With R (< 3.5.0), NA is always returned. This is due to a bug in R (< 3.5.0), where the signaling result cannot be trusted.

## Details

Note that the preferred way to terminate a cluster is via parallel::stopCluster(), because it terminates the cluster nodes by kindly asking each of them to nicely shut themselves down. Using killNode() is a much more sever approach. It abruptly terminates the underlying R process, possibly without giving the parallel worker a chance to terminate gracefully. For example, it might get terminated in the middle of writing to file.

tools::pskill() is used to send the signal to the R process hosting the parallel worker.

## Known limitations

This function works only with cluster nodes of class RichSOCKnode, which were created by makeClusterPSOCK(). It does not work when using parallel::makeCluster() and friends.

Currently, it's only possible to send signals to parallel workers, that is, cluster nodes, that run on the local machine. If attempted to use killNode() on a remote parallel workers, NA is returned and an informative warning is produced.

Use isNodeAlive() to check whether one or mode cluster nodes are alive.

## Examples

if (.Platform\$OS.type != "windows" || interactive()) {
cl <- makeClusterPSOCK(2)
print(isNodeAlive(cl))  ## [1] TRUE TRUE

res <- killNode(cl)
print(res)

## It might take a moment before the background
## workers are shutdown after having been signaled
Sys.sleep(1.0)

print(isNodeAlive(cl))  ## [1] FALSE FALSE
}
#> [1] TRUE TRUE
#> [1] TRUE TRUE
#> [1] FALSE FALSE

format: plain


text:
# Would going underground into a planet offer protection from a supernova? [duplicate]

I am considering a scenario where people are anticipating the star of their system going supernova, which would be somehow triggered by an advanced, hostile alien race. However, the people cannot predict when it will happen. Their best option is to prepare some temporary shelter they can stay in until people from other systems can come extract them after the brunt of the explosion has passed.

I considered them building bunkers deep underground that could shield them from radiation for a few days or weeks until help arrives.

Is this a realistic way at all to temporarily protect yourself from your own star’s explosion? Or, would the planet heat up all the way through, making the underground unsafe? Or, would the planet just be completely destroyed? Taking into account planet size, distance from the star, size of the star, and radiation shielding technologies.

In addition, as the supernova would be triggered by a hostile alien force, there is the possibility that its power may not be equal to that of a natural supernova.

If not, any ideas for more realistic options of escape/defense in this situation? (Or some other stellar phenomenon that an underground bunker might be reasonable for?)

• Comments are not for extended discussion; this conversation has been moved to chat. – L.Dutch - Reinstate Monica Jul 8 '20 at 7:35
• It might be possible to survive by going underground, if the star is not the sun of their own system, but a star somewhere far enough that adding some 1km of solid earth between you and incoming radiation helps. that would also make it possible that the hostile aliens are far away, too and can't come destroy planet directly. how far should supernova be that a 1km underground distance makes a difference, is another question though. – Gnudiff Jul 8 '20 at 20:47

# No.

A typical supernova will perhaps not destroy the planet, and "just" leave it as a burnt-out cinder. If that were all, then a refrigerated, shielded, very deep underground hideout could survive the onslaught. Indeed, such a hideout would survive the onslaught.

But the supernova explosion will also lash out with a shock wave of neutrinos. While mostly harmless, neutrinos cannot be shielded by any ordinary matter, because their interaction probability is just too low. By the same token, a huge enough number of them will interact with matter; the whole planet will be flashed inside and out, and a significant portion of its constituent elements will undergo inverse beta decay.

The necessary calculations are found here and yield a fluence of 8.4E+22 events per square meter at a distance of one parsec. Assuming a neutrino energy range of 5 to 15 MeV, the average dose at that distance would be between 1.4E-9 Sv and 1.6E-10 Sv. Randall Munroe derives an estimate of 0.5E-9 Sv, giving a LD at 2.3 AU; from the original figures, we can re-derive an absorbed LD (for a human-mass organism) between 1.3 and 4 AU depending on neutrino energy spectrum.

While not too worrisome for mechanical systems and most electronics, that level of inverse beta decay is potentially lethal to anything organic.

The only chance to avoid destruction is to escape - which will be a considerable endeavour, because you'll need to be at least 300 million kilometer distant to be barely out of the lethality envelope. In all likelihood you will want to achieve a distance at least two orders of magnitude greater, which means completely escaping the doomed solar system.

(This heavily relies on human absorbed doses. Even for humans, 5Sv is only a ballpark estimate of lethality; it is the single-exposure dose that kills 50% of exposed humans within one month. Even a 1Sv dose, which would happen at a distance of approximately 5 AU, has a death rate of around 5% for humans. On the other hand, other organisms would probably tolerate it nicely).

• from the XKCD mentioned in L.Dutch's answer: "The idea of neutrino radiation damage reinforces just how big supernovae are. If you observed a supernova from 1 AU away—and you somehow avoided being being incinerated, vaporized, and converted to some type of exotic plasma—even the flood of ghostly neutrinos would be dense enough to kill you." The problem - also mentioned by that XKCD - is that you'd be inside the exploding star to be within 1 AU from its center. – John Dvorak Jul 5 '20 at 14:04
• “If it’s going fast enough you can absolutely knock someone down with a feather” - Randall Munroe, What If. – Joe Bloggs Jul 5 '20 at 15:26
• @skyprezzel I'm afraid so. The problem being that if the supernova was caused by an alien race (see e.g. The Dark Forest by Liu Cixin, or Calculating God by Robert J. Sawyer), then the same race could make short work of an escape fleet... – LSerni Jul 5 '20 at 21:14
• Sedna has an aphelion of 140 Tm or 140 billion km. Deep underground in Sedna during its aphelion may be far enough? OP didn't specify how far away the planet was orbiting. – gerrit Jul 6 '20 at 7:09
• @gerrit absolutely, yes. Sedna would likely survive the supernova. It is basically the plot of Arthur C. Clarke's The Star, ( en.wikipedia.org/wiki/The_Star_(Clarke_short_story) ), with a redoubt instead of a time vault. The problem remains that I referred to in my previous comment - anyone wielding supernovas could probably make short work of Sedna. – LSerni Jul 6 '20 at 11:33

A supernova is an hazardous object to be within in a radius of several light years because it's very, very, very energetic.

To quote Randall Munroe

his rule of thumb for estimating supernova-related numbers: However big you think supernovae are, they're bigger than that.

Here's a question to give you a sense of scale: Which of the following would be brighter, in terms of the amount of energy delivered to your retina:

• A supernova, seen from as far away as the Sun is from the Earth, or

• The detonation of a hydrogen bomb pressed against your eyeball?

Applying the physicist rule of thumb suggests that the supernova is brighter. And indeed, it is ... by nine orders of magnitude.

The radiation shower following a supernova explosion is dangerous not only because it will exceed the tanning dose by several orders of magnitude, but also because it will strip the planet from its atmosphere and maybe even from a considerable amount of whatever it's below that atmosphere.

Hiding underground to get shelter from a supernova looks like when Wile E Coyote opens an umbrella to get repair from the rocks about to drop on him.

• Might it partially do a nuke-in-space? Spreading the force to the empty space (getting filled by the explosion but still) and get deflected enough not to kill people underground? I'll not contest the atmosphere being unliveable afterwards (if it's even there) but anyone in closed-system bunkers deep underground could be safe? – Demigan Jul 5 '20 at 17:23
• @Demigan - nope. The amount of energy involved here is ridiculously staggering. – jdunlop Jul 5 '20 at 22:32
• that comparison makes me think that supernovas are universe's way of saying "you have made things explode? that's cute" – conquistador Jul 6 '20 at 13:58
• @Demigan: The problems with bunkers deep underground is that they still need to vent the heat upwards. And when waht remains of the surface of the earth is now also boiling lava, you're sandwiched with no place left for the heat to go. – MSalters Jul 6 '20 at 14:59
• @MSalters That depends on how deep you are. If your shelter isn't also swimming in boiling lava, then there's a buffer of relatively cooler rock between you and the surface. – chepner Jul 6 '20 at 21:09

This is an attempt to elaborate on L. Dutch’s answer with a simplified mathematical model and some calculations.

To take a simplified view of a supernova, let’s assume that all of the gravitational binding energy of the star is converted into kinetic energy, radiation, etc during the blast. Assuming that the star has radius $$R$$, mass $$M$$, and uniform density $$\delta=3M/4\pi R^3$$, its gravitational binding energy is equal to $$\frac{3GM^2}{5R}$$ For our sun, this is about $$2.3\cdot 10^{41}$$ joules. Wow!

Not all of this energy would be directed straight towards Earth. Let’s assume the supernova explodes symmetrically in all directions. Then, by the time this energy reaches Earth, it will be spread out over the surface of a sphere with radius equal to the distance between the Earth and the sun. The fraction of this energy that hits the Earth is approximately equal to $$\frac{\pi R_E^2}{4\pi \text{AU}^2}\approx 4.5\cdot 10^{-10}$$ where $$R_E$$ is the radius of the Earth and $$\text{AU}$$ is the Astronomical Unit (distance between the Earth and sun). Thus, the amount of energy received by the Earth is about $$(2.3\cdot 10^{41}\space\text{J})(4.5\cdot 10^{-10})\approx 1.04\cdot 10^{32}\space\text{J}$$ To estimate how much damage that would inflict, let’s calculate the binding energy of the Earth: $$\frac{3GM_E^2}{5R_E}\approx 2.24\cdot 10^{32}\space\text{J}$$ Surprisingly, that’s actually greater than the amount of supernova energy that would reach Earth! This means that the Earth wouldn’t be completely obliterated.

However, as mentioned in L. Dutch’s answer, it would be very nearly obliterated. Although there wouldn’t be enough energy to completely disintegrate Earth, it would probably rip off the atmosphere and the upper layers of the Earth’s crust. It’s doubtful that your people would be able to dig deep enough underground to protect themselves.

EDIT: According to @GaryWalker in the comments, my estimate for the energy output of a supernova is off by a factor of $$1000$$. So if there were any doubts before about whether a human bunker could survive, they can be safely dismissed.

• "Mr President, shall we get roasted from the supernova blast or from the mantle heat?" – L.Dutch - Reinstate Monica Jul 5 '20 at 15:13
• @L.Dutch-ReinstateMonica An interesting follow-up to the OP’s question might be “is it practically possible to create a strong-box capable of protecting people inside from a supernova (even if it is flung out into space by the blast)?” – Franklin Pezzuti Dyer Jul 5 '20 at 15:21
• You underestimate the power of a supernova, minimum size is about 1.5E44 Joules, so nearly another factor of 1000 more powerful than your guess. – Gary Walker Jul 5 '20 at 17:06
• @skyprezzel My simplified model would suggest that energy drops off as the inverse square of the distance from the star $\sim 1/D^2$. (For example, if you double the distance, the amount of energy reaching the planet is cut by a factor of $4$.) Unfortunately, the farther the planet is away from the star, the less solar energy it enjoys before the supernova blast, and the less likely it is to host life. – Franklin Pezzuti Dyer Jul 5 '20 at 21:29
• @SteveJessop fair point. Star explosion != supernova – Gensys LTD Jul 7 '20 at 6:58

Everybody isnthinking only about radiation. You probably did too, since you are considering whether people can shelter from it.

When a star explodes as supernova it sheds mass. From the Wikipedia article about supernova remnants:

A supernova remnant (SNR) is the structure resulting from the explosion of a star in a supernova. The supernova remnant is bounded by an expanding shock wave, and consists of ejected material expanding from the explosion, and the interstellar material it sweeps up and shocks along the way.

(...) the resulting supernova explosion expels much or all of the stellar material with velocities as much as 10% the speed of light (or approximately 30,000 km/s). These speeds are highly supersonic, so a strong shock wave forms ahead of the ejecta. That heats the upstream plasma up to temperatures well above millions of K. The shock continuously slows down over time as it sweeps up the ambient medium, but it can expand over hundreds or thousands of years and over tens of parsecs before its speed falls below the local sound speed.

Even if you could shield yourself from the radiation in a planet (which you can't, as everybody else is showing you), the planet will be vaporized. The oncoming plasma that hits it will be millions of degrees hot in any scale, with each particle packing a hulk punch due to the immoral oncoming speed. And remember that supernova come from stars way more massive than the sun. The oncoming stream of stellar material will be more massive than any planet, so even if the planet could stay whole and solid after the initial blast it would still be surrounded by plasma hotter than a star; Hot enough to melt the planet.

Just in case I haven't made myself clear enough, let's put this all in scale:

• The dino killing asteroid had a mass estimated around at most 4.6 × 1017 kg, with impact speed estimated at around 20 km/s.

• For all we know, for the Moon to form the Earth had to be hit by Theia, with a mass of approximately 6.4 × 1023 kg, and an impact speed of around 4 km/s. This impact re-melted the crust of the Earth. By the way, the kinect energy input from this is five orders of magnitude greater than the dino killing one (i.e.: like 100,000 dino asteroids hitting at once).

• To blow up as supernova, a star needs to have at least eight solar masses (so the exploding star will have at least ~1.6 × 1031 kg). And while the explosion is omnidirectional, potentially most of the mass will be blown across the star's plane of rotation (I believe clouds such as the Veil Nebula provide evidence for this), which is where planet orbits will tend to be. We are therefore talking about multiple planetary masses coming onto your planet at 30,000 km/s.

How many planetary masses will depend on your planet's size and distance from the star, with such data we might be able to calculate. But the mass and speeds involved don't bode well for anyone trying to escape the apocalypse in a vault. For a single Earth mass (already factoring in relativistic mass) in plasma hitting your planet at 10% the speed of light, the impact is nine orders of magnitude greater than the impact that created the Moon - or, in approximate terms, the same kinetic energy of a billion planets the size of Mars falling onto Earth.

And that's still less energy than what you'll get in the form of gamma radiation from the supernova.

• Back of envelope: the radius of the earth is 6300 km, so the area of its disk is 1.25E8 km². Distance to Sun is 150 million km, so the area of the shell at Earth's orbit is 2.8E17 km². Therefore even in the optimistic assumption that it's ejected spherically we'd get 4.4E-10 of whatever mass is coming. If only 50% of a Solar mass is ejected (1E30 kg), that means we take 4.4E20kg to the face. 1/1000 of the mass of Theia, but going 7000 times as fast, so 49000 times the kinetic energy using the non-relativistic equation. – Steve Jessop Jul 7 '20 at 2:48

If the planet is Earth-like, e.g. it has a hot molten core, then you can't burrow very deeply without it getting too hot. That's why human's haven't managed to dig through the Earth's crust. The Kola Superdeep Borehole is 12,226 m deep and it's 180 C down there.

• Good call-out. At the point where you're talking a strong enough bunker to (potentially) withstand a supernova, a little thing like a planet's molten core probably won't be a problem. But if it's just a misguided attempt that can't work, this would be a valid problem during construction. – Bobson Jul 7 '20 at 20:58

An issue as well is your "until help arrives" comment. It suggest you think someone could shield themselves from a supernova, but there exists a spaceship which could whitstand said supernova for long enough to evacuate a colony during the worst part of it. according to this time line you have hours, if that, until your solar system is screwed, and it will get worse over the next few weeks. If somehow you could withstand said supernova through all that by bunkering, you're not looking at a few weeks until help can arrive, but more likely a good few years. During which the surface of your planet will instantly go ozonless, then boil and liquify, then slowly freeze solid. Survival during that time with absolutely no access to natural biological resources will be enough trouble, but the inability to replenish your air makes this even worse.

Practically, you'd need to build the equivalent of a space station underground close to the core of your planet. If you're going through all that effort, it would be much better to build an actual space ship and leave. And I mean leave now, because even as soon as you notice the start of a supernova, you're already too late to launch a space ship and escape.

Your basic problem from step one is 'anticipating' that a hostile Alien race 'cause' a Supernova to occur in the first place. That and the fact not every star is capable of becoming a supernova (aliens igniting it or not). Only the most massive stars can do it this. And there aren't many within reasonable range.

It also implies contact with the race in question, so either we have traveled out into interstellar space and met them or they have traveled here. If its the latter case they literally don't need a supernova to kill us. By definition if they are that advanced (compared to us) and hostile we are DOA Supernova or not. A race with that kind of power would simply not have to bother with Supernova to deal with a species that wasn't at least near their (God Like) tech level.

That leaves us meeting them out there. And if that the case we have not only have advance warning we have also probably spread out in the opposite stellar direction anyway. So taking out Earth, while a tremendous blow doesn't kill off all the annoying apes anyway - we're still out there in the stars somewhere.

In terms of the effect on Earth. Bunkers would protect humanity and any species we selected from the worst effects of a Supernova which primarily involve the destruction of the ozone layer. Humanity could remain underground for a few decades/generations until the worst effects have past. So if the aliens are intending to kill us all off they have gone to an awful lot of effort for very little effect.

A better scenario would be a simper one. Human astronomers a few years from now (pick your date) start to detect preliminary changes in local large star (there is a list of candidates) that indicates it has or will go Supernova shortly. Of course Einstein gets in the way (AGAIN) as it could have gone BOOM already and we wouldn't know sans early signs until the first signs/wave front arrived. Giving humanity a precise period of time to prepare based on how many light years the chosen star is away and what advance warning (changes in the star) there are.

• While this is a hard science question the means by which the aliens trigger is can't be as we have no hard science answers to it. Thus we can't rule out the possibility that the aliens can trigger one in a too-small star. How about introducing a core collapse by means of artificial gravity? Squeeze hard enough, it's going to blow even if it's an itsy-bitsy red dwarf. – Loren Pechtel Jul 5 '20 at 18:43
• Good points, but about 'only most massive stars': according to wikipedia, also binary stars may be capable of a supernova. So, for ignition, the enemy aliens might throw another star on ours :-) – Roland Jul 5 '20 at 23:46
• An alternate might simply be to drop a 'very' tiny black hole (BH) into a local star. I think (need to check) that BH will then do what all BHs do - pull in all the matter available in its immediate locality growing as it does so. Once it has consumed the star it does not however have the mass available to sustain itself. The smaller the BH the shorter its life time. Having exhausted all available local matter it degrades shedding radiation/mass as it does. And the end is highly explosive - involving a series of intense gamma ray bursts. – Mon Jul 6 '20 at 3:48
• An alternate might simply be to drop a 'very' tiny black BH into a tight orbit around the outer edge of a local star. Dropping it 'in' won't help. That BH will then do what all BHs do - pull in all the matter available in its immediate locality growing as it does so. In this case starting with the outer layers of the stars structure and working inwards. Once the it has absorbed all available mass it will start to degrade. The smaller a BH is the shorter its life time. And the end is highly explosive - involving a series of intense gamma ray bursts. – Mon Jul 6 '20 at 3:59
• The problem is matching exactly the right sized BH with exactly the right sized star and exactly the right distance from Earth. All very complex. Plus you then have to wait for the whole system to reach its end point - which could be thousands of years. If the enemy civilization is really that high up the Kardashev Scale all they really need to do is map Earths Orbit around the Sun then send one or more small probes our way at a high % of C all timed to arrive when Earth is in the correct position. Even a at a low % of C we would only have at best a few days/hours warning. At highers speeds? – Mon Jul 6 '20 at 4:14

Of course it's possible. With the right assumptions just about anything is possible.

Assume: the planet is massive, but has a cold core and is made of something rigid, so it doesn't flow like a liquid under its own weight. And it's far from the exploding sun. And the race living on it has advanced technology with access to vast amount of unobtanium, which reflects neutrinos, doesn't conduct heat well, and is entirely shatterproof.

Then they tunnel to the core, build their shelter from unobtanium, install their hyperfusion reactors, load up on food sources, pile in and close the hatch. Dimly, they hear a boom, and then they just wait, because the environment outside their shelter isn't going to be healthy for quite some time. Occasionally they burp out any waste heat so they don't cook themselves. Time passes.

Eventually they open the hatch. What do they find? You're the author, so you get to decide, but it's reasonable to assume there's no planet left there anymore, or at least nothing that can support what we usually think of as life. So they'd do best to rig their reactor to generate propulsion and move their habitat to some other star. Volia, they (well, probably their descendants if they aren't immortal) survived.

If they have all that tech, though, they could drive off the attacking aliens with unobtanium bullets and green death-star rays and keep their planet safe that way.

Realistic? Of course not. But you posited aliens that can make a sun go supernova so you're already off the map, in terms of realism. There are a lot of simpler, entirely feasible ways of taking out life on a planet, if that's the goal. A large number of messy nuclear bombs will likely do it. Why get all high tech?


text:
To submit this assignment, upload the full document on blackboard, including the original questions, your code, and the output. Submit both your .Rmd source file and a knitted .html version of the same file.

For questions that ask you to plot something by hand, indicate which assignment and question the plot is for and either include a photo or scan of it in the markdown document or hand in the hard copy.

1. Quantitatively analyzing a one-dimensional model (2 marks).

Consider the differential equation $$\frac{dx}{dt} = x(1-x)$$ from last week’s assignment.

1. Find the fixed points (equilibrium values) of the system (analytically, on paper) (0.5 marks).

2. Use R to simulate this equation and plot $$x$$ vs. $$t$$ for $$0 \le t \le 10$$. Plot trajectories for several different starting values of $$x$$ (initial conditions) using ggplot and the do function we used in class. Based on the long-time behaviour of the system, which fixed point(s) are stable and which are unstable? (1 mark)

3. Analyze the stability of the fixed points on paper. Since this system is one-dimensional, we only need a single derivative: $$\frac{\partial f(x)}{\partial x}$$, where $$f(x) = x(1-x)$$. Calculate this derivative, evaluate it at each of the fixed points, and use the result to classify each point as stable or unstable (0.5 marks).

1. The Lotka-Volterra competition model (6 marks).

Consider the Lotka-Volterra competition model. Unlike the predator-prey model considered last week, in this model the growth rate of each species ($$x$$ and $$y$$) depends on the per-capita effect the competing species have on each other, which can be thought of as the amount of overlap in resource use. The amount of overlap for each species is given by $$\alpha_1$$ and $$\alpha_2$$, respectively. $$\alpha_1$$ and $$\alpha_2$$ don’t need to be equal.

$\frac{dx}{dt} = r_1 x \left(1-\frac{x+\alpha_1 y}{K_1}\right)$

$\frac{dy}{dt} = r_2 y \left(1-\frac{y+\alpha_2 x}{K_2}\right)$

$$r$$ and $$K$$ are growth rates and carrying capacities, respectively.

1. Find the fixed points and nullclines of the system (analytically, on paper). The fixed points can be found by setting $$\frac{dx}{dt} = 0$$ and $$\frac{dy}{dt} = 0$$ simultaneously, and the nullclines can be found by setting each derivative to zero individually and solving for $$y$$ as a function of $$x$$ in each case so that they can be plotted on a $$y$$ vs. $$x$$ phase portrait (0.75 marks).

2. Use R to simulate these equations and plot $$x$$ and $$y$$ vs. $$t$$. Use the symmetric case: $$\alpha_1 = \alpha_2 = \alpha$$, $$K_1 = K_2 = K$$, and $$r_1 = r_2 = r$$. Choose $$\alpha = 0.2, K = 50$$, and $$r = 1$$. Choose a starting value where $$x$$ is different from $$y$$ (0.75 marks).

3. Create a phase portrait for this system in R using phaseR. Plot the following:
• all the fixed points.
• the nullclines found in (a). Plot all the nullclines yourself instead of using nullcline from phaseRnullcline doesn’t always plot all the nullclines correctly.
• three to four trajectories (1 marks).
4. Analyze the stability of each fixed point using the same assumptions and parameter values as in part (b). Calculate the four partial derivatives needed for the Jacobian: $$\frac{\partial f(x,y)}{\partial x}$$, $$\frac{\partial f(x,y)}{\partial y}$$, $$\frac{\partial g(x,y)}{\partial x}$$, $$\frac{\partial g(x,y)}{\partial y}$$, where $$f(x) = r_1 x \left(1-\frac{x+\alpha_1 y}{K_1}\right)$$ and $$f(y) = r_2 y \left(1-\frac{y+\alpha_2 x}{K_2}\right)$$.

For example, for the fixed point $$x^* = 0$$, $$y^* = K$$ ($$x^*$$ notation indicates fixed point):

# Fixed point
x_star <- 0;
y_star <- K;

And we can use our solutions for the partial derivatives above, plugging in the equilibrium values. For example, the partial derivative of $$f(x)$$ with respect to x evaluated at the fixed point above is:

# Partial derivatives / elements of the Jacobian
df_dx <- r * (1 - (x_star + alpha * y_star) / K) - (1 / K) * r * x_star

Now create the Jacobian matrix and then calculate the eigenvalues:

J <- matrix(c(df_dx, df_dy,
dg_dx, dg_dy),
ncol = 2, nrow = 2,
byrow = TRUE)

e_vals <- eigen(J)$values For each point, give the eigenvalues and explain whether the fixed point is stable or unstable. Will it oscillate? Explain why or why not (1 marks). 5. Simplify the fixed points of the system on paper in the case when $$\alpha_1 = \alpha_2 = \alpha$$, $$r_1 = r_2 = r$$, and $$K_1 \neq K_2$$. Create a plot in R showing $$x^*$$ vs. $$\alpha$$, $$y^*$$ vs $$\alpha$$, and $$x ^ * + y ^ *$$ vs. $$\alpha$$ for$ -1 < < 1.5\$ for the non-trivial fixed point where both $$x^*$$ and $$y^*$$ are not zero. Note that $$\alpha = -1$$ and $$\alpha = 1$$ will give infinity, so skip those two points when plotting. Label $$K_1$$, $$K_2$$, and $$K_1 + K_2$$ on the y-axis. (2.5 marks total)

• Thinking of $$\alpha$$ as the parameter that controls “overlap” between the two competing species, what is the biological meaning of $$\alpha = 0$$? (Hint: compare this fixed point to the non-trivial fixed point from the logistic model: $$N^* = K$$.)

• What is the biological meaning of $$0 < \alpha < 1$$? What is the biological meaning of $$\alpha = 1$$, and of $$\alpha > 1$$? (Hint: what is the sum of the non-trivial fixed points $$x^*$$ and $$y^*$$ in each of these cases in terms of $$K_1$$ and $$K_2$$?)

• What would happen to the system if $$\alpha$$ was negative? (Hint: is “competition” still an appropriate name for the model if $$\alpha < 0$$?)


text:
# What did you get for Christmas?

me neither, I remember my dad having one. And I played around with one a couple of months ago. It was a round one though with three disks and a clear marker. All I could do was multiply and divide though and still had to write down the numbers so I could remember the decimal places.


text:
# Separating Variables in Differential Equations

By Kathleen Knowles, 07 Oct 2020

Separation of Variables, widely known as the Fourier Method, refers to any method used to solve ordinary and partial differential equations. To apply the separation of variables in solving differential equations, you must move each variable to the equation's other side.

## Get to Understand How to Separate Variables in Differential Equations

As indicated in the introduction, Separation of Variables in Differential Equations can only be applicable when all the y terms, including dy, can be moved to one side of the equation. All the other x terms, including dx, are taken to the other side of the equation. Kindly follow these steps in Separating Variables involving Differential Equations.

Step One: Move all the y terms, including dy, to one side of the equation

Step Two: Move all the x terms, including dx, to the other side of the equation

Step Three: Integrate the y side.

Step Four: Integrate the x side. Kindly note that "+C" is the constant of the integration.

Step Four: Simply the equation.

### Illustration

Task: Assuming that k is constant, solve the following differential equation.

dy/dx = ky

Solution

Move all the y terms to one side of the equation and the x terms to the other side of the equation.

Thus

dy/dx = ky

dy = ky × dx ( You are simply multiplying both sides by dx)

You should then divide both sides of the equation by y.

y:dy/y = k dx

Now integrate both sides of the equation. Kindly note that the integration should be done separately.

Thus;

∫ dy/y = ∫ k dx ( note the placement of the integration sign)

Now integrate the left side of the equation:

(y) + C = ∫ k dx

Then integrate the right side of the equation:

(y) + C = kx + D

Remember that "C" is the constant of integration and that D is used for the other. Therefore it is a different constant.

You can now simplify the equation. To simplify, roll the two constants into one.

Thus;

(a=D−C): ln(y) = kx + a

e(ln(y)) = y

It would help if you now took the exponents on both sides

Thus;

y = ekx + a

And ekx + a = ekx ea

Therefore:

y = ekx ea

Assuming that you have keenly followed these steps and the example given, you can now use the formula and the procedure to solve real-life examples. It is prudent to state here that though we used y and x in our example, you are at liberty to use other variable names.

### Other illustrations

Take rhinos, for example. The more rhinos you have, the more baby rhinos you will have. When these baby rhinos grow, they automatically sire babies. Therefore there is an increase in population.

This information can translate into a differential equation.

(N) population any time be t;

the growth rate be r and;

the rate of change in population is dNdt.

Please note that the rate of change at any given time is equal to the growth rate times the population. Simply put:dN dt = rN.

If you note keenly, the resulting equation is the same as the previous one we solved, in example (1) above. The only difference is that this time, different letters have been used. That is:

Therefore N = cert

Further Illustration

Solve this differential equation by separating the variables:

dy/dx = 1y

### Solution

First, separate the y and x variables. Then you will have to move all the y variables to one side of the equation. Then move all the x terms to the other side of the equation.

Thus;

Multiply both sides by dx:

dy = (1/y) dx

Multiply both sides by y:

y dy = dx

It would be best if you then integrated both sides of the equation separately. Please be keen to put the integral sign (∫ ) in front of the equation.

Thus;

∫ y dy = ∫ dx

Now integrate each side:

Thus;

(y2)/2 = x + C

You have now integrated both sides of the equation in one line.

Next, simplify the equation:

Multiply both sides by 2

y2 = 2(x + C)

Find the square root of both sides:

y = ±√(2(x + C))

Note that y = ±√(2(x + C)) is not the same as y = √(2x) + C. The difference is as a result of the addition of C before finding the square root. You realize that this is common in many differential equations. C is not just added at the end of the process. You should add the C only when integrating.

Thus;

y = ±√{2(x + C)}

### Solution

First, learn how to separate the Variables.

Multiply both sides by dx, then divide both sides by y:

Thus:

1y dy = 2x1+x2dx

Next, integrate the two sides of the equation separately.

Thus:

∫1y dy = ∫2x1+x2dx

Note that while the left side is a simple logarithm, the right side can be integrated by the substitution method.

Let u = 1 + x2, so du = 2x dx:∫1y dy = ∫1udu

You can then integrate(y) = ln(u) + C

Then make C = ln(k):ln(y) = ln(u) + ln(k)

Therefore you get:

y = uk

Lastly please put u = 1 + x2 back again.

Hence;

y = k(1 + x2)

Lastly, simplify the equation.

Therefore y = k(1 + x2)

### Verhulst Logistics Equation

The previous example on growth Differential Equation (dNdt = rN) is closely related overhaul Logistics Equation. The equation also referred to as the Verhult's model was introduced in 1838 and works with the assumption that growth may not last forever because the population will soon run of food.

Now solve the Verhaust's Equation using Differential Equation by Separating the Variables

dNdt = rN(1−N/k)

First, separate the two variables by multiplying both sides of the equation by dt, then divide again by N.

Thus;

dN = rN(1−N/k) dt

N(1-N/k):1N(1−N/k)dN = r dt

Next, please integrate the resulting expression.

Thus;

∫1N(1−N/k)dN = ∫ r dt

You realize that the left side of the equation seems difficult to integrate. You don't have to worry, though. With a little knowledge of Partial Fractions, you don't have to spend a whole day looking for a solution. Kindly rearrange the equation like this:

It would help if you began with:1N(1−N/k) then multiply both top and bottom by k:kN(k−N). You should then add N and -N to the top.

Thus;

N+k−NN(k−N)

You should then split the expression into two fractions.

Thus:

NN(k−N) + k−NN(k−N)

You can then simplify each fraction

1k−N + 1N.

You can now quickly solve the problem. Go ahead and separate each term separately.

Thus;

∫1k−NdN + ∫1NdN = ∫ r dt

Then, Integrate:−ln(k−N) + ln(N) = rt + C

Simplify the equation:

In(k−N) − ln(N) = −rt − C

Combine ln():ln((k−N)/N) = −rt − C

Then, take exponents on both sides:(k−N)/N = e−rt−C

Now, separate the powers of e:(k−N)/N = e−rt e−C

Remember that e−C is constant. You should therefore replace it with A:(k−N)/N = Ae−rt.

You are now close to the answer. With a little more algebra, you are assured of getting N on its own. To do this:

Separate the fraction terms:(k/N)−1 = Ae−rt

Then add 1 to both sides:k/N = 1 + Ae−rt

Next,divide both by k:1/N = (1 + Ae−rt)/k

Find the reciprocal of both sides: N = k/(1 + Ae−rt)

Therefore the solution is N = k1 + Ae−rt

### Final Thoughts

Separating Variables in Differential Equations, though often understood by a few people and widely feared by many, can be a straightforward concept to grasp if you adhere to the procedures discussed in this article.

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# Does reputation correlate with the question-to-answer ratio?

Reputation is a rough measure of a user's experience with the site. It can also be used as a (not very accurate) proxy for their overall expertise in physics.

Our user population consists of people who ask lots of questions and answer few of them, experts who answer lots of questions and typically ask few of them, and people in between. A good measure of a given user's position in this spectrum is the number of questions they've asked over the number of answers they've posted.

Is there a correlation between these two metrics?

• I just got a silver badge for tenacious which is proof that it's not universally true. To have such a badge defined suggests that there is at least a bimodal distribution. – JDługosz Mar 17 '15 at 18:51
• – bwDraco Mar 21 '15 at 1:06

### Yes, it does.

I tried exploring this using this SEDE query, whose results go something like this:

This is a log-log plot and it shows two distinct regimes, with a definite linear correlation in the higher-rep side. This means that the site's user base really does consist of two different populations with very different behaviours: novice users with reps below ~200 for whom there's no correlation between reputation and question/answer ratio, and more experienced users who ask noticeably less questions at higher reputations.

The dependence in this regime is roughly Q/A = 1/rep, with r squared of about 0.25. I'll leave more detailed statistics to people who can do them right.

One important thing to note is that the query returns results for only 3579 users, which is about 5% of our total of ~72k users, because it only reports on users that have ≥1 answer and ≥1 question. This could be a bug in the SQL, and it's something to keep an eye on. Maybe someone can extend this to cover those cases? It's not clear to me how you'd include those users in the log-log plot below, where they'd form a 'ceiling' at Q/A=∞ of pure askers and a 'floor' at Q/A=0 of pure answerers.

• More interesting is why there are those horizontal lines. Also, I'd wager that a lot of people probably signed up to the site just to upvote HNQ here. – Kyle Kanos Mar 10 '15 at 12:59
• The horizontal lines correspond to ratios of small integers, so I don't really find it that surprising. They're a consequence of the discrete possibilities for question numbers and answers. They taper out after a certain rep as Q/A~2 becomes possible via ratios like 19/10 or 21/10, as opposed to 2/1. – Emilio Pisanty Mar 10 '15 at 13:18
• Ah, that makes sense now. – Kyle Kanos Mar 10 '15 at 13:23
• Maybe it's just me, but I see a hint of a third population -- in the mid-rep range, there is a branch of the distribution with a positive Q/A-rep correlation. Put another way, if you created some sort of contour on the right-hand side (e.g. the 90th percentile in rep at each Q/A bin), there appears to be a statistically significant minimum in rep at Q/A of ~2. – user10851 Mar 12 '15 at 2:49
• @ChrisWhite That definitively sounds plausible, but I don't really have the stats chops to disentangle that. Maybe some of our more statistically minded users could have a look at the data and see what they can find? – Emilio Pisanty Mar 12 '15 at 12:30
• I am one of the Q=0 people who don't get represented on your otherwise excellent graph. You could use a statistical measure of log(Q/A) to deal with the singularity. A simple approach might be (I did not fully think this through): if you can say "next post is either a question or an answer with probability p", then you find a limit on p given the "asked 0 answered N" data, where $(1-p)^N < 0.5$ (for example). Then you solve the equality for p and plot log(p/(1-p)) instead. – Floris Mar 12 '15 at 15:48
• What accounts for the rep disadvantage Questioners are at by only getting +5? – Mazura Mar 13 '15 at 7:26
• I just fiddled around a bit with the idea I floated yesterday, and if you use $ln(2)/N$ as your Q/A ratio for the case where $Q=N, A=0$ you will come out to about the right place. Obviously you would use the reciprocal when $Q=0, A=N$ – Floris Mar 13 '15 at 16:55
• How about just plotting the $(Q+1)/(A+1)$ ratio (or its logarithm)? – hmakholm left over Monica Mar 15 '15 at 16:19
• @ChrisWhite: If you ignore a single outlier (the highest-rep dot with Q/A>10), it begins to look more like the "third population" is just the faint outer end of a symmetric trumpet-shaped "novice" population that overlaps the "experienced" population between roughly 500 and 2000 rep. – hmakholm left over Monica Mar 15 '15 at 16:27
• For Reputation <= 200, there is a positive correlation of 0.087, which according to the standard parametric hypothesis test is highly significant, p = 5e-5. Spearman correlation 0.084 and nonparametric test p = 9e-5. – A. Donda Mar 19 '15 at 18:00

I think the story is both more complicated and more simple, and it helps to look at questions and answers separately:

The superimposed curves are very strongly smoothed versions of the data, using a cubic smoothing spline (Matlab's csaps with parameter p = 0.1).

According to this, with increasing reputation a user has more and more questions, but at some point (about 2000) this increase levels off towards about 10 questions. The smoothed curve is not a particularly good description of the data though. (I'm ignoring the little dip for reputation < 3, which is driven by users with 1 reputation and several questions. Similar for answers.)

For answers vs reputation, the initial increase is similar, but instead of leveling off it becomes a linear relationship in the log-log plot, which implies a power law behavior. The exponent is about 1.11, which means the increase is only slightly faster than linear.

Comparison of the two smooth curves suggests that increase in Reputation is more driven by questions for small reputation, but more driven by answers for high Reputation, with a crossing point at about 370 Reputation and 4.5 both Questions and Answers.

This explains the picture when looking at questions/answers vs reputation. Until a reputation of about 100, the ratio increases slowly up to a value of 1.4, and then decreases faster and faster.

The "third population" speculated about in the comments might appear due to the fact that both the number of Questions and the number of answers vary wildly in the "medium" range of Reputation around 1000. It would be interesting to see whether these relations observed across users also describe the evolution of users over time.

To me this suggests that the number of answers alone is a better match for reputation that the ratio questions/answers, since the smoothed relationship is monotonous starting from a reputation of about 8.5, i.e. across 94% of users.

• Somehow, "a fool can ask more questions than seven wise men can answer" sprang to mind when I saw your analysis. – Floris Mar 21 '15 at 21:42
• @Floris, I can't figure out whether that's intended as an insult or not. – A. Donda Mar 26 '15 at 16:32
• If you knew me, you would realize that it is never my intention to insult. The point of the quote (where "fool" is used in the sense of the "fool of the court", or court jester) is that there is a never ending source of questions, and relatively few people capable of answering. So that high reputations tend to be built by a handful of people with the answers. I suspect that if you plotted "number of people who have asked at least N questions" vs "people who answered N" as a parametric function of N you would also see something interesting. – Floris Mar 26 '15 at 16:38
• @Floris, thanks for clarifying, your comment was completely opaque to me. Particularly so since I don't really know you, though I believe we have crossed paths on SO one or two times. Thanks for the suggestion, I'll have a look. – A. Donda Mar 26 '15 at 17:55
• No problem - glad you asked so I could clear it up. And yes I believe we have on occasion visited the same post... although these days I spend more time on the physics site and less on SO. Incidentally - the Fool was an extremely important person at Court, since he was the only one that could tell the king the truth with impunity - wrapping his criticism of the king in a joke, getting a laugh from the courtiers, and keeping his head. And a smart king would actually pay attention... – Floris Mar 26 '15 at 18:27