3 Introduction to Assessing Risk

3 Introduction to Assessing Risk Important Question. How do we assess the risk an investor faces when choosing among assets? In this discussion we examine how an investor would assess the risk associated with investments. This assessment is carried out in terms of how an individual is affected by taking on additional risk. In Figure 1 we plot the monthly returns on stocks (left graph) and short term government bonds (right graph) from 1926 2008. The smooth line is a standard normal distribution with a standard deviation which matches the data for stock and bond returns, respectively. The stock return is the value of the weighted average of all stocks traded in the United States. The arithmetic (geometric) average return on stocks is 0.94% (0.80%) or 11.28% (9.6%) per year. The inflation rate over this same period was 0.25% per month or 3.0% per year. As a result, the return on stocks adjusted for inflation was 8.28% per year. The short-term government rate is the 3-month Treasury bill rate. The average short term rate is 0.3% (3.6% per year) per month or a rate of return after adjusting for inflation of 0.25% per month, the real return on short-term government bonds over this period was 0.6% per year. As a result, the real return on stocks was greater than the return on short term bonds by 7.68% per year. Given such an advantage to holding stocks why do people hold short term bonds? Monthly Stock Returns (1926 2008) Monthly Risk Free Returns (1926 2008) 0.1 Geometric Mean = 0.8% Arithmetic Mean = 0.94% 3 Geomteric Mean = 0.3% Arithmetic Mean = 0.3% 0.08 2.5 Density 0.06 Density 2 1.5 0.04 1 0.02 0.5 0 30 20 10 0 10 20 30 % Returns 0 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 1.2 % Return Figure 1: Monthly Returns on Stocks and Bonds. In this lecture we will show that the excess return on stocks stems from the fact that stocks are riskier assets than bonds, and so investors must be paid an additional amount known as a risk premium to compensate for the added uncertainty and volatility. One way to see the risk is to look at the distribution of monthly returns on stocks and the short term interest rate. In particular, the range of the short term rate was [ 0.1, 1.2]% per month, while the fluctuation in stock returns was [ 25, 25]% per month. In fact there are several months in which the investor would lose over 10% of stock value such as in October and November 2008. As a result, we refer to the rate on short term bonds as the risk-free rate. We will show in the rest of the lecture that investors demand a higher return on stocks to compensate for the risk of losing part of their wealth when they invest in stocks. 41

4 Utility Functions To measure how an individual feels we introduce a utility function which has two properties that are common to all individuals: 1. An individual feels better if she has more goods. 2. An individual s utility increases at a decreasing rate as she obtains more goods. The first condition means that consumers gain utility as they acquire more things, so that the utility function has a positive first derivative at every point for which the utility function is defined. The second condition is a little more subtle, but is analogous to the principle of diminishing returns in economics. If you are a poor person, then giving you more goods increases your utility a lot. However, a rich person, like Bill Gates, would have a smaller increase in utility when given a similar increase in goods, since his consumption is already quite high. This second condition means that the utility function has a negative second derivative at every point for which the utility function is defined. These properties are represented by the general form of the individual s utility function in Figure 2. Definition 1: A utility function u = u(c) is a correspondence which assigns to each consumption choice C a real number u(c) (indicating the satisfaction from the consumption C) such that: u (C) > 0 and u (C) < 0. (4.1) 5 Risk Premiums and Actuarially Fair Gambles Utility 0 u Utility Consumption Figure 2: General Utility Function. We now examine how an individual responds to the uncertainty associated with investment in the stock market. First we consider the concept of a fair gamble to represent uncertainty. Definition 2: An actuarially fair gamble is a random variable z with a positive payoff z 1 with probability p and a negative payoff z 2 with probability 1 p. In addition, the expected value is zero, E(z) = pz 1 + (1 p)z 2 = 0, while the variance is Var(z) = pz 2 1 + (1 p)z 2 2. The standard deviation is σ = V ar(z). Example 1: Suppose you have to pay $5,000 for a bet in which you win $10,000 with p = 1 2, and you get $0 with 1 p = 1 2. As a result, z 1 = 10, 000 5, 000 = 5, 000 and z 2 = 0 5, 000 = 5, 000. Note here that payoffs have the initial payment for the bet deducted. In this case, the expected gain is E(z) = 1 2 (5, 000) + 1 ( 5, 000) = 0. 2 C The variance is V ar(z) = 1 2 5, 0002 + 1 2 5, 0002 = 25, 000, 000. 42

As a result, the standard deviation is σ = 25, 000, 000 = 5, 000. Exercise 1. Suppose the probability of a positive payoff is p = 0.9 in Example 1. If the positive payoff stays at z 1 = 5, 000, what is the new value of the negative payoff z 2 so that the random variable is an actuarially fair gamble. Definition 3: An investor is strictly risk averse if she will not accept an actuarially fair gamble. Definition 4: The risk premium ρ is implicitly defined as the amount an investor is willing to pay to avoid an actuarially fair gamble. The risk premium ρ is implicitly defined such that u(c ρ) = pu(c + z 1 ) + (1 p)u(c + z 2 ), (5.2) u(c+z 1 ) u where C is the consumption without the gamble. u(c- ρ) B A pu(c+z 1 ) +(1-p)u(C+z 2 ) The right hand side is the expected utility of the investor from the actuarially fair gamble, that is, the expected value of the utility function. On the other hand, the left-hand side is the utility from the sure payment of C ρ. We can rewrite this equation in probabilistic terms, recalling Definition 2, we can write: E[u(C + z)] = u(c ρ) The calculation of the risk premium is illustrated in Figure 3. u(c+z 2 ) Example 2: Suppose an investor has a logarithmic utility function u(c) = ln(c). 0 C+z 2 What is the risk premium under the actuarially fair gamble of Example 1? C- ρ Figure 3: Risk Premium C C+z 1 C Solution: Substituting the data from Example 1 into the implicit definition of risk premium from equation(5.2), we now have so that ln(c ρ) = p ln(c + z 1 ) + (1 p) ln(c + z 2 ), ρ = C (C + z 1 ) p (C + z 2 ) 1 p Suppose we use the actuarially fair gamble in example 1 with initial consumption C = 50, 000. The risk premium becomes ρ = 50000 (55000) 0.5 (45000) 0.5 = 250.63. This means that this investor would pay $250.63 to avoid the actuarially fair gamble with standard deviation of $5, 000. Exercise 2. Given the actuarially fair gamble in exercise 1, find the risk premium for the investor in example 2. Why do you think the risk premium changed? 43

6 Relating Stocks and Bonds to an Actuarially Fair Gamble We can relate an actuarially fair gamble to the uncertainty in stocks relative to investing in risk free bonds. Remember that the payoff from investing in either stocks or bonds are given by the decision trees in Figure 4. p S 1 (H)= us 0 p 1+r S 0 B 0 = $1 1-p S 1 (T)= ds 0 Figure 4: Payoffs from Stocks and Bonds. 1-p 1+r Given these payoffs from stocks and bonds we can compare the utility of the investor when she buys bonds and stocks. If she purchases bonds, then her utility is certain, and is measured at the point B in Figure 5. Her utility at this point is u B = u[(1 + r)x 0 ] where X 0 is the amount invested. If she uses all her wealth to buy stocks, such that S 0 = X 0, then her utility is uncertain. Before the random event occurs, her expected utility from buying stocks is E s [u(x 0 )] = pu(ux 0 ) + (1 p)u(dx 0 ). (Warning: The symbol u inside the utility function u stand for up value of the stock.) In order for the trade-off between stocks and bonds to be an actuarially fair gamble, we require that, under the conditions described above, E s [u(x 0 )] = u[(1 + r)x 0 ], (6.3) so that the expected utilities of the value of the investor s wealth when investing in the stock and the value of the wealth after accruing interest for one period are the same. This is demonstrated in Example 3 below. In Figure 5, we take p = 1 so the expected utility is given by the point S. 2 Thus, the investor is expected to lose utility when the stock is an actuarially fair gamble. We show in subsequent lectures that this leads to a higher expected return on stocks to convince risk averse investors to hold the stock. u u u u B Bonds B u S u d S Stocks 0 C d = dx 0 C= (1+r)X 0 C u = ux 0 C 44

Figure 5: All stocks versus all bonds. Example 3: Suppose that the the price of the stock at the end of the first period (year) is: S 1 (H) = 100 and S 1 (T ) = 25. Also let the risk free interest rate be r = 0.05 and the probability of the good state p = 0.5. Suppose the investor s wealth is X 0 = 59.52. After one year the investor s wealth is 62.50. If S 0 = 59.52, then the stock price is a fair gamble. To see this calculate the expected stock price The proceeds from the bond is ps 1 (H) + (1 p)s 1 (T ) = 0.5 100 + 0.5 25 = 62.5. (1 + r)x 0 = (1.05) 59.52 = 62.5. As a result, the investor ends up with $62.50 on average if she invests in either the stock or the bond. However, the expected utility from the stock as illustrated in Figure 5 is less than the expected utility from investing in the risk free bond. Thus, for any investor to buy stock at time t = 0 its price must be less than 59.52 giving the investor a reward for taking risk. Exercise 3. Suppose the probability of the good state is 0.9, what should the stock price be for the stock to be an actuarially fair gamble relative to the risk-free bond? Let the individual investor have logarithmic preferences. What does the stock price have to be so that the investor receives the same utility from the stock and bond? 7 Measuring the Risk Premium We now show how to relate the risk premium to the shape of the utility function following the argument by Kenneth Arrow, the 1972 winner of the Nobel Prize in Economics. Theorem 7.1 A risk averse investor with utility function u(c) and consumption C has a risk premium ρ which is approximated by for an actuarially fair gamble. ρ 1 2 Var(z)u (C ) u (C ) > 0 Proof: To start, we can define the risk premium at the specific consumption C according to Definition 4, i.e., u(c ρ) = pu(c + z 1 ) + (1 p)u(c + z 2 ). (7.4) The proof applies a first order Taylor approximation to the left hand side of this definition near the point C. Second, we take a second order Taylor approximation of the right hand side near the point C. We then equate these two results and solve for the unknown risk premium ρ. Recall that the notation T (n) (x 0, x) refers to the n-th Taylor polynomial of x estimated at the point x 0, which has known values. Take a first order Taylor s polynomial expansion of the left hand side of (7.4) near the point C to find u(c ρ) T (1) (C, C ρ) = u(c ) + u (C )((C ρ) C ), 45

so that u(c ρ) u(c ) ρu (C ). (7.5) Next, find the second order Taylor s polynomial expansion of the right hand side of (7.4) near the point C to obtain pu(c + z 1 ) + (1 p)u(c + z 2 ) T (2) (C, C + z 1 ) + T (2) (C, C + z 2 ) [ = p u(c ) + u (C )z 1 + 1 ] 2 u (C )z1 2 + [ (1 p) u(c ) + u (C )z 2 + 1 ] 2 u (C )z2 2 = u(c ) + u (C ) [pz 1 + (1 p)z 2 ] + 1 2 u (C ) [ pz 2 1 + (1 p)z 2 2]. Note that in the first step above we took the Taylor expansion of only the parts of the formula containing z 1 and z 2 resepectively. Now recognize that the random variable is an actuarially fair gamble so that E(z) = pz 1 + (1 p)z 2 = 0 and V ar(z) = pz 2 1 + (1 p)z 2 2. As a result the approximation of the right hand side of (7.4) is pu(c + z 1 ) + (1 p)u(c + z 2 ) u(c ) + 1 2 u (C )V ar(z). (7.6) Thus, the approximation of (7.4) is given by comparing (7.5) with (7.6) The final step is to solve for the risk premium u(c ) ρu (C ) u(c ) + 1 2 u (C )V ar(z). ρ 1 2 Var(z)u (C ) u (C ). Lastly, note that by the definition of the utility function, we have u (C ) < 0 and u (C ) > 0, so the formula implies that ρ 1 2 Var(z)u (C ) u (C ) > 0. This completes the proof. Exercise 5 asks you to employ an argument similar to the one used in this proof to find more precise approximations of the risk premium using higher order Taylor polynomials. Example 4: Redo Example 2 using the Arrow measure of the risk premium. Solution: From Example 1 the variance is 25, 000, 000. The derivatives of the logarithmic utility function are d ln(c) dc = 1 C and d2 ln(c) = 1 dc 2 C. 2 As a result, the risk premium is given by ρ 1 2 Var(z)u (C ) u (C ) = 1 2 2, 500, 000 1 C = 1 2 2, 500, 000 1 50, 000 = 250. 46

Exercise 4. Suppose the payoffs from the stock and the risk free bond are as in Example 3. Let the investor s preferences be given by u(c) = ln(c). How much does the stock price have to fall from S 0 = 59.52, so that this investor purchases the stock? Exercise 5. Suppose we have a utility function with derivatives of order n+1. Find a more accurate formula for the risk premium in terms of the higher order derivatives of the utility function. Exercise 6. (a.) Suppose the utility function is linear in Figure 5. What happens to the risk premium in Theorem 1.1? Why do we call this person risk neutral? (b.) Now suppose the utility function in Figure 5 is curved toward the origin rather than away from the origin. What happens to the risk premium in Theorem 1.1? What would you call an individual with this type of preferences? Given Theorem 1.1, Arrow introduced the following definitions of risk aversion. Definition 5: The measure of an investor s absolute risk aversion is A(C) = u (C) u (C), where C is the investor s consumption without undertaking the actuarially fair gamble. As a result, according to Theorem 1.1, the risk premium is given by ρ 1 2 Var(z)A(C ). Definition 6: The measure of an investor s relative risk aversion is R(C) = u (C) u (C) C, where C is the investor s consumption without undertaking the actuarially fair gamble. Remark: This measure of risk is called relative risk aversion since a multiplicative random variable e z C rather than a additive random variable C + z leads to the risk premium formula ρ 1 2 Var(z)R(C ). Exercise 7. The example in Figure 5 is a multiplicative random shock where e z 1 = u and e z 2 = d. As a result consumption is given by C u = e z 1 C in the good state and C d = e z 2 C in the bad state. Prove Theorem 1.1 for the case of these multiplicative shocks. Example 5: Suppose the investor s absolute risk aversion is a decreasing function of consumption. In particular, let A(C) = 2 C. Suppose an average investor has $50,000 to consume per year. For the actuarially fair gamble in Example 1, the investor is willing to pay ρ 1 2 Var(z)A(C ) = 1 2 25, 000, 000 = $500. 2 50, 000 47

to avoid this gamble. We can use this result to explain why a risk averse individual purchases various types of insurance such as health and life insurance. Suppose you have a car worth $5,000 and you have a 50% chance that you would have an accident in which your car will be destroyed over the next year, since you are a bad driver. If your absolute risk aversion is like example 5, then you are willing to pay an insurance company $500 per year for an auto insurance policy, which pays for the automobile if it is destroyed. Thus, insurance company makes money by reducing risk for individuals in exchange for an insurance premium. 8 Risk Aversion and the Investor s Utility Function We now want to investigate the relation between risk aversion and the functional form of the utility function of an investor. We want to know how risk aversion changes with the standard of living of a person in order to gain some insight into investor behavior. Since an investor has less absolute risk aversion as his consumption increases, A(C) must be a decreasing function, that is A (C) < 0. (8.7) Taking the derivative of the formula for absolute risk aversion (using of the quotient rule) we get: Therefore A (C) < 0 is equivalent to A (C) = u (C) 2 u (3) (C)u (C) u (C) 2. u (C) 2 u (3) (C)u (C) < 0. (8.8) Individuals with a higher standard of living are more willing to take on gambles, and so their utility function should satisfy condition (8.8). Example 6: If A (C) < 0 for people, then investors like Warren Buffett or Bill Gates would be willing to take on more risk than the average person. This helps to explain why Berkshire Hathaway is willing to insure other insurance companies against catastrophes such as the fall of the World Trade Center. It also helps to explain why Hedge funds, who invest for wealthy investors, would take on more risk. Exercise 8. For the logarithmic utility function find A (C) < 0. Does this utility function lead to increasing or decreasing absolute risk aversion. We now want to consider a class of utility functions which would allow for decreasing absolute risk aversion. Below, we use the simplest functional form for absolute risk aversion such that the first derivative is negative. We consider the following general form: A(C) = 1 a + bc, so that A (C) = b (a + bc) 2, (8.9) where a and b are constants. aversion. As a result, b > 0 would correspond to decreasing absolute risk 48

Theorem 8.1 An investor with absolute risk aversion given by (8.9) has a utility function given by one of the following three formulae, where K and L are any constants. Case 1: If b 0 and b 1 and γ = 1 b, then u(c) = K 1 1 γ [a + bc]1 γ + L; Case 2: If b = 1, then Case 3: If b = 0, then u(c) = K ln(a + C) + L; u(c) = Kae C a. Remark: Note that we use the general constant k in each of the three formulae since multiplication by any constant will not change the optimal behavior of the investor. Case 3 is called constant absolute risk aversion, since A(C) is constant (A(C) = 1). Also, the case a = 0 is called a constant relative risk aversion since R(C) is constant. In this case A(C) = 1, and so bc R(C) = A(C)C = 1 bc C = 1 b. Note that we cannot have a = 0 = b.remember also that we are considering the case of decreasing absolute risk aversion, so b 0 in all the cases of Theorem 1.2 by assumption. Proof: Recall that A(C) = u (C) u (C) = 1, where a and b are fixed parameters. a + bc With this in mind, let us prove each of the cases of the theorem individually. Case 1: In this case, b 0, b 1, and γ = 1. Also, we have the following two equations: b u (C) u (C) dc = 1 ln[a + bc] ln[u (C)] + K and dc = + K, a + bc b where K is the constant of integration. However, both integrals are representations of A(C)dC, so ln[u ln[a + bc] (C)] = + K, b where we have altered the constant of integration K without losing equality. After some manipulation of this equation, and once again simplifying the constant of integration, we arrive at u (C) = K[a + bc] 1 b. Integrating both sides of this equation with respect to C, we get u(c) = K 1 1 1 [a + bc] 1 1 b + L, b where L is another constant of integration. Then, substituting in γ, we arrive at the correct formula for Case 1: u(c) = K 1 1 γ [a + bc]1 γ + L. 49

Case 2: In this case, we have b = 1, so clearly substituting this into the equation from the first case would yield an undefined solution. Therefore, we must start again with the first step. So, we have A(C) = 1, giving us the following two representative integral equations: a+c u (C) u (C) dc = 1 ln[u (C)] + K and dc = ln[a + C] + K. a + C These equations can be combined to give which simplifies by formal manipulation to ln[u (C)] = ln[a + C] + K, u (C) = K 1 a + C. Integrating both sides of this equation, we get the final formula for Case 2: u(c) = K ln(a + C) + L. Case 3: This case is the simplest. Since b = 0, we have A(C) = 1. Therefore, by taking a step a similar to the first step in the proofs of Cases 1 and 2, we get two integral equations of forms of A(C): u (C) u (C) dc = ln[u (C)] + K and 1 a dc = C a + K. Accepting a change in the constant of integration K, these equations can be combined as which by manipulation simplifies to ln[u (C)] = C a + K, u (C) = Ke C a. Then, through a simple integration of this equation, we arrive at the following formula for the third case: u(c) = Kae C a. This completes the proof of Theorem 1.2. Remark: Note that it is traditional to set L = 0 in Cases 1 and 2 above, since this additive constant of integration does not effect optimal investment decisions. Also, it is standard to set K = 1 in all cases for simplicity. However, we have chosen to be explicit in this lecture by including them. References Arrow, Kenneth J. (1971). Essays in the theory of risk-bearing. North-Holland, Amsterdam. Cvitanić, J. and F. Zapetatero, Introduction to the economics and mathematics of financial markets, MIT Press, Boston (2004) pp.104-110. 50