Measures of risk in the life of an investor. Taught in masters for IBEF course

Talks about the measures of risk

1. Measures of Risk

2. One of the most commonly used absolute risk   metrics   is  standard deviation , a statistical measure of  dispersion  around a central tendency. The field of  behavioral finance  has contributed an important element to the risk equation, demonstrating asymmetry between how people view gains and losses. In the language of  prospect theory , an area of behavioral finance introduced by Amos Tversky and  Daniel Kahneman  in 1979, investors put more weight on the pain associated with a loss than the good feeling associated with a gain

3. Thus, what investors really want to know is not just how much an asset deviates from its expected outcome, but how bad things look way down on the left-hand tail of the  distribution curve .  One reason might be the aggravation that one experiences in losing a sum of money appears to be greater than the pleasure associated with gaining the same amount.

4. A hypothetical value function

5. Measures that help inventors in assessing the “riskiness” of their portfolios. If an investor has the choice between two mutually exclusive portfolios, A and B, which of these two portfolios does he prefer? Can we specify certain general patterns of investment choice, whereby all “sensible” investors can be expected to prefer A over B if A has certain well-defined properties relative to B.

6. Risk is defined in terms of loss. Domar and Musgrave formulated a quantitative index of risk that takes into account all possible negative income. RI= As x i ≤ 0, RI is a positive number. Domar Musgrave Risk Index

7. Many investors feel that they have failed in their investment if they earn less than the riskless interest rate. So modified version , RI= Modified Domar Musgrave Index

8. Suppose the following investment is there: Riskless interest rate=6 % Example x -50% -10% 5% 50% 100% P(x) 1/5 1/5 1/5 1/5 1/5

9. The index will be = –[1/5(-50%)+ 1/5(-10%)]=12 % Modified RI = –[1/5(-50%-6%)+ 1/5(-10%-6%)+ 1/5(5%-6%)]=14.6%

10. Using either equation 1 or 2 it is possible to calculate a risk index for all alternative investments. Higher the riskless interest rates higher the index given by equation 2.

11. All investors will not agree with resultant risk ranking that Domar and Musgrave measure produce. Using equation 1both RIs will be 5%. But for many investors 50% loss is more disaster . Drawbacks of Domar Musgrave Index Rates of Return Probability Investment A -50% 0.1 Investment B -10% 0.5

12. References: E Domar and R.A Musgrave,”Proportional Income Taxation and Risk Bearing”, Quarterly Journal of Economics, May 1944

13. CHEBYSHEV’S INEQUALITY, Let X be a random variable with finite expected value μ and finite non –zero variance Nearly all values are close to mean. For k<1 or k=1 , it is very obvious as R.H.S becomes >1 CHEBYSHEVE’S INEQUALITY

14. Using k=√2 As an example using k=√2 shows that at least half of values lie in the interval

15. Roy’s Safety First Rule: According to A.D. Roy investors are mainly concerned with avoiding the probability of “disaster” Risk is measured in terms of probability that future income will be lower than “d”. “d” is the disaster level as perceived by investor. Roy’s Safety Index

16. Roy’s Index (RI) : ELABORATION:- Let x denotes future income (random variable) with mean μ and variance

17. Using CHEBYSHEVE’S INEQUALITY,

18. Thus, is the upper bound of probability of disaster . If the whole distribution is known Can be calculated. Risk index can be assigned to each investment. If only mean and variance are known the measure suggests selecting the investment which minimizes

19. MV:- Variance which takes into account both positive and negative deviations from mean may be misleading. Mean – Variance Index of Risk

20. SV:- A is some constant , earning less than which is considered as failure. Generally A is selected to be=E(x) Semi- Variance Index of Risk

21. W. Baumol claims that the risk is due to earning less than some critical value or “floor”. According to him, “an investment with relatively high s.d will be relatively safe , if its expected value is sufficiently high”. Baumol’s Risk Measure

22. Consider the following investment, According to variance risk index Investment B would be ranked as riskier. Investment A Investment B mean 2 20 Variance 1 2

23. However, Therefore he proposes the risk Index, Where k is some constant selected by the investor such that return is unlikely to fall below it.

24. Higher is i,e lower bound safer is the investment. A return below is also possible but probability is less than 1/k 2

25. VaR ( α ) represents the maximum possible loss , when α percent of left tail distribution is ignored.(As they are considered very very unlikely). VaR on a portfolio is the maximum loss we might expect over a given period at a given level of confidence. It is therefore defined over two parameters :-the period concerned , usually known as holding period and the confidence level, whose values are arbitrarily chosen. Value at Risk :- VaR ( α )

26. Example: 5% VaR of Rs 1 Lakh .05 probablility that portfolio will fall in value by more than 1 lac . 1% VaR of Rs 2 Lakh .01 probability that portfolio will fall in value by more than Rs 2 lakh .

27. If a portfolio of stocks has a one-day 1% VaR of $1 million, that means that there is a 0.01 probability that the portfolio will fall in value by more than $1 million over a one-day period if there is no trading. Alternatively loss of $1 million or more on this portfolio is expected on 1 day out of 100 days (because of 1% probability). A loss which exceeds the VaR threshold is termed a " VaR break."