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% per month (3.6% per year) 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. Important Question. 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) Geomteric Mean = 0.3%
Geometric Mean = 0.8%
0.1
3
Arithmetic Mean = 0.94%
2.5
Density
0.08
Density
Arithmetic Mean = 0.3%
0.06
2
1.5
0.04 1
0.02
0 −30
0.5
−20
−10
0
% Returns
10
20
0 −0.8
30
−0.6
−0.4
−0.2
0
0.2
0.4
% Return
0.6
0.8
1
1.2
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. 42
3.1
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.
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: u0 (C) > 0 and u00 (C) < 0.
u
Utility
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 it is defined. The second condition is a little more subtle but this 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 it is defined. These properties are represented by the general form of the individual’s utility function in Figure 2.
Utility
0
Consumption
C
Figure 2: General Utility Function.
(3.1)
Example 1: Examples of utility functions are given by u(C) = ln C,
and u(C) =
1 1 1 C 1−γ , 0 < γ < 1 and U (C) = , γ > 1. 1−γ 1−γ 1−γC
Exercise 1. Check condition (3.1) for the utility functions in Example 1. Also, justify that 1 u(C) = ln C corresponds to γ = 1. More precisely, if you subtract the constant from 1−γ 1 u(C) = C 1−γ and let γ → 1 then you get the utility function u(C) = ln C. 1−γ
3.2
Risk Premiums and Actuarially Fair Gambles
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 z1 with probability p and a negative payoff z2 with probability 1 − p such that the expected value is zero, that is E(z) = pz1 + (1 − p)z2 = 0. 43
The variance is Var(z) = p(z1 − E(z))2 + (1 − p)(z2 − E(z))2 and the standard deviation is σ=
p
V ar(z).
The variance measures how close the random variable is to its expected value. Example 2: Suppose you have to pay $5,000 for a bet in which you win $10,000 with p = 12 , and you get $0 with 1 − p = 12 . As a result, z1 = 10, 000 − 5, 000 = 5, 000 and z2 = 0 − 5, 000 = −5, 000. Note here that the payoffs have the initial payment for the bet deducted. In this case, the expected gain is 1 1 E(z) = (5, 000) + (−5, 000) = 0. 2 2 The variance is 1 1 V ar(z) = (5, 000 − 0)2 + (5, 000 − 0)2 = 25, 000, 000. 2 2 As a result, the standard deviation is p σ = 25, 000, 000 = 5, 000. Exercise 2. Suppose the probability of a positive payoff in Example 2 was p = 0.9 instead. If the positive payoff stays at z1 = 5, 000, what is the new value of the negative payoff z2 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 u(C+z1)
u(C − ρ) = pu(C + z1 ) + (1 − p)u(C + z2 ), (3.2) B
u(C- ρ)
where C is the consumption without the gamble. 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:
pu(C+z1) +(1-p)u(C+z2)
u(C+z2)
0
E[u(C + z)] = u(C − ρ)
A
C+z2
C- ρ
C
C+z1
Figure 3: Risk Premium The calculation of the risk premium is illustrated in Figure 3. For simple utility functions the risk premium ρ can be computed explicitly by solving equation (3.2), like in the following example.
44
C
Example 3: Suppose an investor has a logarithmic utility function u(C) = ln C. What is the risk premium under the actuarially fair gamble of Example 2? Solution: Substituting the data from Example 2 into the implicit definition of risk premium from equation (3.2), we now have ln(C − ρ) = p ln(C + z1 ) + (1 − p) ln(C + z2 ),
(3.3)
so that ρ = C − (C + z1 )p (C + z2 )1−p . Since for the actuarially fair gamble in Example 1 with we have 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 3. Given the actuarially fair gamble in Exercise 2, find the risk premium for the investor in Example 3. Why do you think the risk premium changed? For more complicated utility functions the risk premium ρ can be computed approximately by solving an approximate version of equation (3.2).
3.3
Computing the Risk Premium Approximately
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 3.1 A risk averse investor with utility function u(C) and consumption C ∗ has a risk premium ρ which is approximated by u00 (C ∗ ) 1 ρ ≈ − Var(z) 0 ∗ > 0, 2 u (C )
(3.4)
for a given actuarially fair gamble z. Proof. The idea of the proof is to replace the left-hand side of equation (3.2) by its linear approximation and the right-hand side of equation (3.2) by its quadratic approximation and then solve the resulting (linear in ρ) approximate equation for the risk free premium. To implement this idea, for a given actuarially fair gamble z we read equation (3.2) at C ∗ and we get u(C ∗ − ρ) = pu(C ∗ + z1 ) + (1 − p)u(C ∗ + z2 ). Using the linear approximation formula f (x) ≈ f (a) + f 0 (a)(x − a) 45
(3.5)
with f = u, a = C ∗ and x = C ∗ − ρ we replace the left-hand side of equation (3.5) by: u(C ∗ − ρ) ≈ u(C ∗ ) + u0 (C ∗ )(C ∗ − ρ − C ∗ ) ≈ u(C ∗ ) − ρu0 (C ∗ ).
(3.6)
Next, using the quadratic approximation formula 1 f (x) ≈ f (a) + f 0 (a)(x − a) + f 00 (a)(x − a)2 2 with f = u, a = C ∗ and x = C ∗ + z1 we replace the term u(C ∗ + z1 ) in the right-hand side of equation (3.5) by: 1 u(C ∗ + z1 ) ≈ u(C ∗ ) + u0 (C ∗ )(C ∗ + z1 − C ∗ ) + u00 (C ∗ )(C ∗ + z1 − C ∗ )2 2 1 ≈ u(C ∗ ) + z1 u0 (C ∗ ) + z12 u00 (C ∗ ). 2
(3.7)
Then, using the quadratic approximation formula with f = u, a = C ∗ and x = C ∗ + z2 we replace the term u(C ∗ + z2 ) by: 1 u(C ∗ + z2 ) ≈ u(C ∗ ) + u0 (C ∗ )(C ∗ + z2 − C ∗ ) + u00 (C ∗ )(C ∗ + z2 − C ∗ )2 2 1 ≈ u(C ∗ ) + z2 u0 (C ∗ ) + z22 u00 (C ∗ ). 2
(3.8)
Combining relations (3.7) and (3.8) gives 1 00 ∗ 2 ∗ ∗ ∗ 0 ∗ pu(C + z1 ) + (1 − p)u(C + z2 ) ≈ p u(C ) + u (C )z1 + u (C )z1 2 1 00 ∗ 2 ∗ 0 ∗ + (1 − p) u(C ) + u (C )z2 + u (C )z2 2 h i 1 h i ∗ 0 ∗ 00 ∗ 2 2 = u(C ) + u (C ) pz1 + (1 − p)z2 + u (C ) pz1 + (1 − p)z2 . 2 Recognizing that pz12 + (1 − p)z22 = V ar(z) and using the fact that for an actuarially fair gamble we have E(z) = pz1 + (1 − p)z2 = 0 we see that the last relation takes the simplified form: 1 pu(C ∗ + z1 ) + (1 − p)u(C ∗ + z2 ) ≈ u(C ∗ ) + u00 (C ∗ )V ar(z). 2
(3.9)
Thus, replacing the left-hand side of equation (3.5) by its linear approximation (3.9) and the righthand side of equation (3.5) by its quadratic approximation (3.9) we obtain the following approximate equation 1 u(C ∗ ) − ρu0 (C ∗ ) ≈ u(C ∗ ) + u00 (C ∗ )V ar(z). (3.10) 2 Solving equation (3.10) for ρ gives the following approximate value 1 u00 (C ∗ ) ρ ≈ − Var(z) 0 ∗ 2 u (C ) 46
of the risk premium. Since, by the definition of the utility function, we have u00 (C ∗ ) < 0 and u0 (C ∗ ) > 0, it follows that 1 u00 (C ∗ ) ρ ≈ − Var(z) 0 ∗ > 0, 2 u (C ) which completes the proof of the theorem. Note: 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 3 using the Arrow measure of the risk premium (3.4). Solution: From Example 2 the variance is 25, 000, 000. The derivatives of the logarithmic utility function are d ln(C) 1 d2 ln(C) 1 = and = − 2. 2 dC C dC C As a result, the risk premium is given by 1 u00 (C ∗ ) 1 1 1 1 ρ ≈ − Var(z) 0 ∗ = 25, 000, 000 = 25, 000, 000 = 250. 2 u (C ) 2 C 2 50, 000 Exercise 4. Suppose that S1 (H) = 100, S1 (T ) = 25, p = 0.5, and r = 0.05. Let the investor’s preferences be given by u(C) = ln(C). How much does the stock price have to fall from S0 = 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. For those interested in probability. Suppose we have a general probability density function for the fair gamble with a known variance. Prove Arrow’s theorem for approximating the risk premium. Exercise 6. (a.) Suppose the utility function is linear in Figure 5. What happens to the risk premium in Theorem 3.1? 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 3.1? What would you call an individual with this type of preferences? Given Theorem 3.1, Arrow introduced the following definitions of risk aversion. Definition 5: The measure of an investor’s absolute risk aversion is . u00 (C) A(C) = − 0 > 0, u (C)
(3.11)
where C is the investor’s consumption without undertaking the actuarially fair gamble. As a result, according to Theorem 3.1, the risk premium is given by 1 ρ ≈ Var(z)A(C ∗ ). 2 Definition 6: The measure of an investor’s relative risk aversion is . u00 (C) R(C)) = − 0 C, u (C) 47
(3.12)
(3.13)
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 ez C¯ rather than an additive random variable C¯ + z leads to the risk premium formula 1 ρ ≈ Var(z)R(C ∗ ). 2 Exercise 7. The example in Figure 5 is a multiplicative random shock where ez1 = u and ez2 = d. As a result consumption is given by Cu = ez1 C ∗ in the good state and Cd = ez2 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) = C2¯ . 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 1 1 2 ρ ≈ Var(z)A(C ∗ ) = 25, 000, 000 = $500 2 2 50, 000 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 over the next year you would have an accident in which your car will be destroyed, since you are a bad driver. If your absolute risk aversion is as in 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, the insurance company makes money by reducing risk for individuals in exchange for an insurance premium.
3.4
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 A0 (C) < 0.
(3.14)
Taking the derivative of the formula for absolute risk aversion (using the quotient rule) we get: A0 (C) =
u00 (C)2 − u(3) (C)u0 (C) . u0 (C)2
Therefore A0 (C) < 0 is equivalent to 2
u00 (C) − u(3) (C)u0 (C) < 0.
(3.15)
Individuals with a higher standard of living are more willing to take on gambles, and so their utility function should satisfy condition (3.15). 48
Example 6: If A0 (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 prove that A0 (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) =
−b 1 , so that A0 (C) = , a + bC (a + bC)2
(3.16)
where a and b are constants. Note that b > 0 would correspond to decreasing absolute risk aversion. Theorem 3.2 An investor with absolute risk aversion given by (3.16) has a utility function given by one of the following three formulae, where K and L are any constants. Case 1: If b 6= 0 and b 6= 1 and γ = 1b , then u(C) = K
1 [a + bC]1−γ + L; 1−γ
Case 2: If b = 1, then u(C) = K ln(a + C) + L; Case 3: If b = 0, then C
u(C) = −Kae− 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/a). Also, the case a = 0 is called 1 and so constant relative risk aversion since R(C) is constant. In this case A(C) = bC R(C) = A(C)C =
1 1 C= . bC 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) = −
u00 (C) 1 = , where a and b are fixed parameters. 0 u (C) a + bC
With this in mind, let us prove each of the cases of the theorem individually. Case 1: In this case, b 6= 0, b 6= 1, and γ = 1b . Also, we have the following two equations: Z Z u00 (C) 1 ln[a + bC] 0 − 0 dC = − ln[u (C)] + K1 and dC = + K2 , u (C) a + bC b 49
where K1 and K2 are constants of integration. However, both integrals are representations of R A(C)dC, so ln[a + bC] − ln[u0 (C)] = + K3 , b where we have altered the constant of integration K3 without losing equality. After some manipulation of this equation, and once again simplifying the constant of integration, we arrive at 1
u0 (C) = K[a + bC]− b . Integrating both sides of this equation with respect to C, we get u(C) = K
1 1− 1b + L, 1 [a + bC] 1− b
where L is another constant of integration. Then, substituting in γ, we arrive at the correct formula for Case 1: 1 u(C) = K [a + bC]1−γ + L. 1−γ 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 1 have A(C) = a+C , giving us the following two representative integral equations: Z
u00 (C) − 0 dC = − ln[u0 (C)] + K1 u (C)
Z and
1 dC = ln[a + C] + K2 . a+C
These equations can be combined to give ln[u0 (C)] = − ln[a + C] + K, which simplifies by formal manipulation to u0 (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) = a1 . Therefore, by taking a step similar to the first step in the proofs of Cases 1 and 2, we get two integral equations of forms of A(C): Z Z u00 (C) 1 C 0 − 0 dC = − ln[u (C)] + K and dC = + K. u (C) a a Accepting a change in the constant of integration K, these equations can be combined as ln[u0 (C)] = −
C + K, a
which by manipulation simplifies to C
u0 (C) = Ke− a . 50
Then, through a simple integration of this equation, we arrive at the following formula for the third case: C u(C) = −Kae− a . This completes the proof of Theorem 3.2. Remark: Note that it is traditional to set L = 0 in Cases 1 and 2 above, since this additive constant of integration does not affect 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.
3.5
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. Assume that the payoff from investing in either stocks or bonds are given by the decision trees in Figure 4. p
S1(H)
p
S0
1+r
B0= $1 1-p
1-p
S1(T)
1+r
Figure 4: Payoffs from Stocks and Bonds. For stocks, there is a probability of a heads (good state) p in which the stock price increases to S1 (H) > S0 . The chance of a tail (poor state) is 1−p where the stock price decreases to S1 (H) < S0 . The bond is safe in that the payoff is always 1 + r whether the flip of the coin results in a head or tail. Example 7. Suppose S1 (H) = 100, S1 (T ) = 25, p = 0.5, and r = 0.05, the expected future payment on the stock is E(F V ) = (0.5)100 + (0.5)25 = 62.5 The expected present value of the stock’s payoffs is E(P V ) =
62.5 = 59.52, 1.05
or
E(P V ) = 0.5
100 25 + 0.5 = 59.52. 1.05 1.05
Suppose the bond also had an expected (certain!) future value of 62.5. Then we know the expected present value is also 59.52. Question. What should the stock price be so that a risk averse individual to be willing to purchase a stock? Obviously, if an investor cares about risk then this person would not buy the stock at a price of 59.52 since she can get the same 5% rate of return from the bond. Suppose the individual has logarithmic utility function u(c) = ln C. Then, we can use the concept of a risk premium to determine the price of a stock. We can look at these calculations from the point of view of present value or future value. The comparison can be made relative to the 51
payment from the bond so that z1 = 100 − 62.50 = 37.50 and z2 = −37.50. This comparison gives a fair gamble since it has an expected value of zero. If C = 62.50, then (3.3) becomes ln(62.50 − ρ) = 0.5 ln(62.50 + 37.50) + 0.5 ln(62.50 − 37.50). Solving this equation for the risk premium gives 1
1
ρ = 62.50 − (100) 2 (25) 2 = 12.50. This risk premium is a payment in one year so that it is a future value. To determine the stock price we have to calculate the present value of the risk premium, which is 12.50/1.05 = 11.90. Thus, the stock price at the present time t = 0 should be reduced from the present value of the bond B0 = 62.50/1.05 = 59.52 by this risk premium so that the new stock price that rewards risk is S0 = 59.52 − 11.905 = 47.615. This gives a very simple and intuitive explanation of the reason for which expected stock returns are higher than bond returns. Exercise 9. Calculate the stock price in Example 7 using present values rather than future values. Exercise 10. Show that for a logarithmic utility function and general stock prices in Figure 4 that the risk premium on stocks is the same whether you use present or future values. Given the payoffs from stocks and bonds in Figure 4 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 uB = u[(1 + r)X0 ] where X0 is the amount invested. If she uses all her wealth to buy stocks, such that S0 = X0 , then her utility is uncertain. Before the random event occurs, her expected utility from buying stocks is Es [u(X0 )] = pu(uX0 ) + (1 − p)u(dX0 ). (Warning: The symbol u inside the utility function u stands 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, Es [u(X0 )] = u[(1 + r)X0 ], (3.17) 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 = 21 so the expected utility is given by the point S. 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.
52
u uu uB
B onds
uS
S Stocks
ud
0
B
dX0
(1+r)X 0
uX 0
C
Figure 5: All stocks versus all bonds.
Exercise 11. As in Example 7, assume that the price of the stock at the end of the first year is a random variable with S1 (H) = 100 and S1 (T ) = 25. However the probability of the good state is 0.9. Also, let the risk free interest rate be r = 0.05. 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? References Arrow, Kenneth J. (1971). Essays in the Theory of Risk-Bearing. North-Holland, Amsterdam. Cvitani´ c, J. and F. Zapetatero, Introduction to the Economics and Mathematics of Financial Markets, MIT Press, Boston (2004) pp.104-110.
53
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% per month (3.6% per year) 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. Important Question. 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) Geomteric Mean = 0.3%
Geometric Mean = 0.8%
0.1
3
Arithmetic Mean = 0.94%
2.5
Density
0.08
Density
Arithmetic Mean = 0.3%
0.06
2
1.5
0.04 1
0.02
0 −30
0.5
−20
−10
0
% Returns
10
20
0 −0.8
30
−0.6
−0.4
−0.2
0
0.2
0.4
% Return
0.6
0.8
1
1.2
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. 42
3.1
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.
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: u0 (C) > 0 and u00 (C) < 0.
u
Utility
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 it is defined. The second condition is a little more subtle but this 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 it is defined. These properties are represented by the general form of the individual’s utility function in Figure 2.
Utility
0
Consumption
C
Figure 2: General Utility Function.
(3.1)
Example 1: Examples of utility functions are given by u(C) = ln C,
and u(C) =
1 1 1 C 1−γ , 0 < γ < 1 and U (C) = , γ > 1. 1−γ 1−γ 1−γC
Exercise 1. Check condition (3.1) for the utility functions in Example 1. Also, justify that 1 u(C) = ln C corresponds to γ = 1. More precisely, if you subtract the constant from 1−γ 1 u(C) = C 1−γ and let γ → 1 then you get the utility function u(C) = ln C. 1−γ
3.2
Risk Premiums and Actuarially Fair Gambles
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 z1 with probability p and a negative payoff z2 with probability 1 − p such that the expected value is zero, that is E(z) = pz1 + (1 − p)z2 = 0. 43
The variance is Var(z) = p(z1 − E(z))2 + (1 − p)(z2 − E(z))2 and the standard deviation is σ=
p
V ar(z).
The variance measures how close the random variable is to its expected value. Example 2: Suppose you have to pay $5,000 for a bet in which you win $10,000 with p = 12 , and you get $0 with 1 − p = 12 . As a result, z1 = 10, 000 − 5, 000 = 5, 000 and z2 = 0 − 5, 000 = −5, 000. Note here that the payoffs have the initial payment for the bet deducted. In this case, the expected gain is 1 1 E(z) = (5, 000) + (−5, 000) = 0. 2 2 The variance is 1 1 V ar(z) = (5, 000 − 0)2 + (5, 000 − 0)2 = 25, 000, 000. 2 2 As a result, the standard deviation is p σ = 25, 000, 000 = 5, 000. Exercise 2. Suppose the probability of a positive payoff in Example 2 was p = 0.9 instead. If the positive payoff stays at z1 = 5, 000, what is the new value of the negative payoff z2 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 u(C+z1)
u(C − ρ) = pu(C + z1 ) + (1 − p)u(C + z2 ), (3.2) B
u(C- ρ)
where C is the consumption without the gamble. 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:
pu(C+z1) +(1-p)u(C+z2)
u(C+z2)
0
E[u(C + z)] = u(C − ρ)
A
C+z2
C- ρ
C
C+z1
Figure 3: Risk Premium The calculation of the risk premium is illustrated in Figure 3. For simple utility functions the risk premium ρ can be computed explicitly by solving equation (3.2), like in the following example.
44
C
Example 3: Suppose an investor has a logarithmic utility function u(C) = ln C. What is the risk premium under the actuarially fair gamble of Example 2? Solution: Substituting the data from Example 2 into the implicit definition of risk premium from equation (3.2), we now have ln(C − ρ) = p ln(C + z1 ) + (1 − p) ln(C + z2 ),
(3.3)
so that ρ = C − (C + z1 )p (C + z2 )1−p . Since for the actuarially fair gamble in Example 1 with we have 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 3. Given the actuarially fair gamble in Exercise 2, find the risk premium for the investor in Example 3. Why do you think the risk premium changed? For more complicated utility functions the risk premium ρ can be computed approximately by solving an approximate version of equation (3.2).
3.3
Computing the Risk Premium Approximately
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 3.1 A risk averse investor with utility function u(C) and consumption C ∗ has a risk premium ρ which is approximated by u00 (C ∗ ) 1 ρ ≈ − Var(z) 0 ∗ > 0, 2 u (C )
(3.4)
for a given actuarially fair gamble z. Proof. The idea of the proof is to replace the left-hand side of equation (3.2) by its linear approximation and the right-hand side of equation (3.2) by its quadratic approximation and then solve the resulting (linear in ρ) approximate equation for the risk free premium. To implement this idea, for a given actuarially fair gamble z we read equation (3.2) at C ∗ and we get u(C ∗ − ρ) = pu(C ∗ + z1 ) + (1 − p)u(C ∗ + z2 ). Using the linear approximation formula f (x) ≈ f (a) + f 0 (a)(x − a) 45
(3.5)
with f = u, a = C ∗ and x = C ∗ − ρ we replace the left-hand side of equation (3.5) by: u(C ∗ − ρ) ≈ u(C ∗ ) + u0 (C ∗ )(C ∗ − ρ − C ∗ ) ≈ u(C ∗ ) − ρu0 (C ∗ ).
(3.6)
Next, using the quadratic approximation formula 1 f (x) ≈ f (a) + f 0 (a)(x − a) + f 00 (a)(x − a)2 2 with f = u, a = C ∗ and x = C ∗ + z1 we replace the term u(C ∗ + z1 ) in the right-hand side of equation (3.5) by: 1 u(C ∗ + z1 ) ≈ u(C ∗ ) + u0 (C ∗ )(C ∗ + z1 − C ∗ ) + u00 (C ∗ )(C ∗ + z1 − C ∗ )2 2 1 ≈ u(C ∗ ) + z1 u0 (C ∗ ) + z12 u00 (C ∗ ). 2
(3.7)
Then, using the quadratic approximation formula with f = u, a = C ∗ and x = C ∗ + z2 we replace the term u(C ∗ + z2 ) by: 1 u(C ∗ + z2 ) ≈ u(C ∗ ) + u0 (C ∗ )(C ∗ + z2 − C ∗ ) + u00 (C ∗ )(C ∗ + z2 − C ∗ )2 2 1 ≈ u(C ∗ ) + z2 u0 (C ∗ ) + z22 u00 (C ∗ ). 2
(3.8)
Combining relations (3.7) and (3.8) gives 1 00 ∗ 2 ∗ ∗ ∗ 0 ∗ pu(C + z1 ) + (1 − p)u(C + z2 ) ≈ p u(C ) + u (C )z1 + u (C )z1 2 1 00 ∗ 2 ∗ 0 ∗ + (1 − p) u(C ) + u (C )z2 + u (C )z2 2 h i 1 h i ∗ 0 ∗ 00 ∗ 2 2 = u(C ) + u (C ) pz1 + (1 − p)z2 + u (C ) pz1 + (1 − p)z2 . 2 Recognizing that pz12 + (1 − p)z22 = V ar(z) and using the fact that for an actuarially fair gamble we have E(z) = pz1 + (1 − p)z2 = 0 we see that the last relation takes the simplified form: 1 pu(C ∗ + z1 ) + (1 − p)u(C ∗ + z2 ) ≈ u(C ∗ ) + u00 (C ∗ )V ar(z). 2
(3.9)
Thus, replacing the left-hand side of equation (3.5) by its linear approximation (3.9) and the righthand side of equation (3.5) by its quadratic approximation (3.9) we obtain the following approximate equation 1 u(C ∗ ) − ρu0 (C ∗ ) ≈ u(C ∗ ) + u00 (C ∗ )V ar(z). (3.10) 2 Solving equation (3.10) for ρ gives the following approximate value 1 u00 (C ∗ ) ρ ≈ − Var(z) 0 ∗ 2 u (C ) 46
of the risk premium. Since, by the definition of the utility function, we have u00 (C ∗ ) < 0 and u0 (C ∗ ) > 0, it follows that 1 u00 (C ∗ ) ρ ≈ − Var(z) 0 ∗ > 0, 2 u (C ) which completes the proof of the theorem. Note: 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 3 using the Arrow measure of the risk premium (3.4). Solution: From Example 2 the variance is 25, 000, 000. The derivatives of the logarithmic utility function are d ln(C) 1 d2 ln(C) 1 = and = − 2. 2 dC C dC C As a result, the risk premium is given by 1 u00 (C ∗ ) 1 1 1 1 ρ ≈ − Var(z) 0 ∗ = 25, 000, 000 = 25, 000, 000 = 250. 2 u (C ) 2 C 2 50, 000 Exercise 4. Suppose that S1 (H) = 100, S1 (T ) = 25, p = 0.5, and r = 0.05. Let the investor’s preferences be given by u(C) = ln(C). How much does the stock price have to fall from S0 = 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. For those interested in probability. Suppose we have a general probability density function for the fair gamble with a known variance. Prove Arrow’s theorem for approximating the risk premium. Exercise 6. (a.) Suppose the utility function is linear in Figure 5. What happens to the risk premium in Theorem 3.1? 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 3.1? What would you call an individual with this type of preferences? Given Theorem 3.1, Arrow introduced the following definitions of risk aversion. Definition 5: The measure of an investor’s absolute risk aversion is . u00 (C) A(C) = − 0 > 0, u (C)
(3.11)
where C is the investor’s consumption without undertaking the actuarially fair gamble. As a result, according to Theorem 3.1, the risk premium is given by 1 ρ ≈ Var(z)A(C ∗ ). 2 Definition 6: The measure of an investor’s relative risk aversion is . u00 (C) R(C)) = − 0 C, u (C) 47
(3.12)
(3.13)
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 ez C¯ rather than an additive random variable C¯ + z leads to the risk premium formula 1 ρ ≈ Var(z)R(C ∗ ). 2 Exercise 7. The example in Figure 5 is a multiplicative random shock where ez1 = u and ez2 = d. As a result consumption is given by Cu = ez1 C ∗ in the good state and Cd = ez2 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) = C2¯ . 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 1 1 2 ρ ≈ Var(z)A(C ∗ ) = 25, 000, 000 = $500 2 2 50, 000 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 over the next year you would have an accident in which your car will be destroyed, since you are a bad driver. If your absolute risk aversion is as in 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, the insurance company makes money by reducing risk for individuals in exchange for an insurance premium.
3.4
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 A0 (C) < 0.
(3.14)
Taking the derivative of the formula for absolute risk aversion (using the quotient rule) we get: A0 (C) =
u00 (C)2 − u(3) (C)u0 (C) . u0 (C)2
Therefore A0 (C) < 0 is equivalent to 2
u00 (C) − u(3) (C)u0 (C) < 0.
(3.15)
Individuals with a higher standard of living are more willing to take on gambles, and so their utility function should satisfy condition (3.15). 48
Example 6: If A0 (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 prove that A0 (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) =
−b 1 , so that A0 (C) = , a + bC (a + bC)2
(3.16)
where a and b are constants. Note that b > 0 would correspond to decreasing absolute risk aversion. Theorem 3.2 An investor with absolute risk aversion given by (3.16) has a utility function given by one of the following three formulae, where K and L are any constants. Case 1: If b 6= 0 and b 6= 1 and γ = 1b , then u(C) = K
1 [a + bC]1−γ + L; 1−γ
Case 2: If b = 1, then u(C) = K ln(a + C) + L; Case 3: If b = 0, then C
u(C) = −Kae− 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/a). Also, the case a = 0 is called 1 and so constant relative risk aversion since R(C) is constant. In this case A(C) = bC R(C) = A(C)C =
1 1 C= . bC 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) = −
u00 (C) 1 = , where a and b are fixed parameters. 0 u (C) a + bC
With this in mind, let us prove each of the cases of the theorem individually. Case 1: In this case, b 6= 0, b 6= 1, and γ = 1b . Also, we have the following two equations: Z Z u00 (C) 1 ln[a + bC] 0 − 0 dC = − ln[u (C)] + K1 and dC = + K2 , u (C) a + bC b 49
where K1 and K2 are constants of integration. However, both integrals are representations of R A(C)dC, so ln[a + bC] − ln[u0 (C)] = + K3 , b where we have altered the constant of integration K3 without losing equality. After some manipulation of this equation, and once again simplifying the constant of integration, we arrive at 1
u0 (C) = K[a + bC]− b . Integrating both sides of this equation with respect to C, we get u(C) = K
1 1− 1b + L, 1 [a + bC] 1− b
where L is another constant of integration. Then, substituting in γ, we arrive at the correct formula for Case 1: 1 u(C) = K [a + bC]1−γ + L. 1−γ 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 1 have A(C) = a+C , giving us the following two representative integral equations: Z
u00 (C) − 0 dC = − ln[u0 (C)] + K1 u (C)
Z and
1 dC = ln[a + C] + K2 . a+C
These equations can be combined to give ln[u0 (C)] = − ln[a + C] + K, which simplifies by formal manipulation to u0 (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) = a1 . Therefore, by taking a step similar to the first step in the proofs of Cases 1 and 2, we get two integral equations of forms of A(C): Z Z u00 (C) 1 C 0 − 0 dC = − ln[u (C)] + K and dC = + K. u (C) a a Accepting a change in the constant of integration K, these equations can be combined as ln[u0 (C)] = −
C + K, a
which by manipulation simplifies to C
u0 (C) = Ke− a . 50
Then, through a simple integration of this equation, we arrive at the following formula for the third case: C u(C) = −Kae− a . This completes the proof of Theorem 3.2. Remark: Note that it is traditional to set L = 0 in Cases 1 and 2 above, since this additive constant of integration does not affect 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.
3.5
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. Assume that the payoff from investing in either stocks or bonds are given by the decision trees in Figure 4. p
S1(H)
p
S0
1+r
B0= $1 1-p
1-p
S1(T)
1+r
Figure 4: Payoffs from Stocks and Bonds. For stocks, there is a probability of a heads (good state) p in which the stock price increases to S1 (H) > S0 . The chance of a tail (poor state) is 1−p where the stock price decreases to S1 (H) < S0 . The bond is safe in that the payoff is always 1 + r whether the flip of the coin results in a head or tail. Example 7. Suppose S1 (H) = 100, S1 (T ) = 25, p = 0.5, and r = 0.05, the expected future payment on the stock is E(F V ) = (0.5)100 + (0.5)25 = 62.5 The expected present value of the stock’s payoffs is E(P V ) =
62.5 = 59.52, 1.05
or
E(P V ) = 0.5
100 25 + 0.5 = 59.52. 1.05 1.05
Suppose the bond also had an expected (certain!) future value of 62.5. Then we know the expected present value is also 59.52. Question. What should the stock price be so that a risk averse individual to be willing to purchase a stock? Obviously, if an investor cares about risk then this person would not buy the stock at a price of 59.52 since she can get the same 5% rate of return from the bond. Suppose the individual has logarithmic utility function u(c) = ln C. Then, we can use the concept of a risk premium to determine the price of a stock. We can look at these calculations from the point of view of present value or future value. The comparison can be made relative to the 51
payment from the bond so that z1 = 100 − 62.50 = 37.50 and z2 = −37.50. This comparison gives a fair gamble since it has an expected value of zero. If C = 62.50, then (3.3) becomes ln(62.50 − ρ) = 0.5 ln(62.50 + 37.50) + 0.5 ln(62.50 − 37.50). Solving this equation for the risk premium gives 1
1
ρ = 62.50 − (100) 2 (25) 2 = 12.50. This risk premium is a payment in one year so that it is a future value. To determine the stock price we have to calculate the present value of the risk premium, which is 12.50/1.05 = 11.90. Thus, the stock price at the present time t = 0 should be reduced from the present value of the bond B0 = 62.50/1.05 = 59.52 by this risk premium so that the new stock price that rewards risk is S0 = 59.52 − 11.905 = 47.615. This gives a very simple and intuitive explanation of the reason for which expected stock returns are higher than bond returns. Exercise 9. Calculate the stock price in Example 7 using present values rather than future values. Exercise 10. Show that for a logarithmic utility function and general stock prices in Figure 4 that the risk premium on stocks is the same whether you use present or future values. Given the payoffs from stocks and bonds in Figure 4 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 uB = u[(1 + r)X0 ] where X0 is the amount invested. If she uses all her wealth to buy stocks, such that S0 = X0 , then her utility is uncertain. Before the random event occurs, her expected utility from buying stocks is Es [u(X0 )] = pu(uX0 ) + (1 − p)u(dX0 ). (Warning: The symbol u inside the utility function u stands 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, Es [u(X0 )] = u[(1 + r)X0 ], (3.17) 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 = 21 so the expected utility is given by the point S. 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.
52
u uu uB
B onds
uS
S Stocks
ud
0
B
dX0
(1+r)X 0
uX 0
C
Figure 5: All stocks versus all bonds.
Exercise 11. As in Example 7, assume that the price of the stock at the end of the first year is a random variable with S1 (H) = 100 and S1 (T ) = 25. However the probability of the good state is 0.9. Also, let the risk free interest rate be r = 0.05. 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? References Arrow, Kenneth J. (1971). Essays in the Theory of Risk-Bearing. North-Holland, Amsterdam. Cvitani´ c, J. and F. Zapetatero, Introduction to the Economics and Mathematics of Financial Markets, MIT Press, Boston (2004) pp.104-110.
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