1. L5: Dynamic Portfolio Management1 Lecture 5: Dynamic Portfolio Management The following topics will be covered: Will risks be washed out over time? Solve the Dynamic Investment Problem –Invest in period 0, consume in period 1 Time diversification –Simply the case of consumption in multiple periods Dynamic investment with predictable returns –Investment in multiple periods Uncertain Distribution of excess returns (Is there a general form in solving dynamic investment problems?)

2. L5: Dynamic Portfolio Management2 Will Time Wash out Risk? Investing in a long run is equivalent to engaging in a series of risks x i (i=1, …, n) Would the sum of n risks have a lower risk?

3. L5: Dynamic Portfolio Management3 Dynamic Investment An investor endowed with wealth w 0 lives for two periods. He will observe his loss or gain on the risk he took in the first period before deciding how much risk to take in the second period How would the opportunity to take risk in the second period (Period 1) affect the investor’s decision in the first period (period 0)? –In other words, would dynamic investment attract more risk taking? To solve this problem, we apply backward induction. That is, to solve the second period maximization first taking the first period investment decision as given. To be specific x α0 Period 0α1 Period 1 Note: this is not the general form

4. L5: Dynamic Portfolio Management4 Backward Induction Assuming the first period payoff is z(α 0, x) The second objective function is Then solve for the first period Ev(z(α 0, x)) Good examples of the backward induction application: –Froot, K. A., David S. Scharfstein, and J. Stein. "Risk Management: Coordinating Corporate Investment and Financing Policies." Journal of Finance 48, no. 5 (December 1993): 1629-1658.Journal of Finance –Froot, K. A., and J. Stein. "Risk Management, Capital Budgeting and Capital Structure Policy for Financial Institutions: An Integrated Approach." Journal of Financial Economics 47, no. 1 (January 1998): 55-82.

5. L5: Dynamic Portfolio Management5 Two-Period Investment Decision Assume the investor has a DARA utility function. –The investor would take less risk in t+1 if he suffered heavy losses in date t The investor makes two decisions In period 1, the investor invests is an AD portfolio decision, In period 0, the investor invests in risky portfolio (selecting α 0 ), which decides z. He attempts to optimize his expected utility which contingent on period 1 allocation.

6. L5: Dynamic Portfolio Management6 Implicit Assumptions Investment decision is made only in period 0 Only two periods No return in risk-free assets The key is to compare the investment in risky asset, α 0, for this long term investors with that of a short-lived investor This is to compare the concavity of these two utility functions

7. L5: Dynamic Portfolio Management7 Solution

8. L5: Dynamic Portfolio Management8 So, It states that the absolute risk tolerance of the value function is a weighted average of the degree of risk tolerance of final consumption. If u exhibits hyperbolic absolute risk aversion (HARA), that is T is linear in c (see HL chapter 1 for discussions on HARA), then v has the same degree of concavity as u – the option to take risk in the future has no effect on the optimal exposure to risk today If u exhibits a convex absolute risk tolerance, i.e., T is a convex function of z, or say T’’>0, then investors invest more in risky assets in period 0. Opposite result holds for T’’<0 Proposition 7.2: Suppose that the risk-free rate is zero. In the dynamic Arrow-Debreu portfolio problem with serially independent returns, a longer time horizon raises the optimal exposure to risk in the short term if the absolute risk tolerance T is convex. In the case of HARA, the time horizon has no effect on the optimal portfolio. If investors can take risks at any time, investors risk taking would not change if HARA holds.

9. L5: Dynamic Portfolio Management9 Time Diversification What would there are multiple consumption dates? This is completely different setting from the previous one The setup the problem is as following:

10. L5: Dynamic Portfolio Management10 Solution

11. L5: Dynamic Portfolio Management11 Liquidity Constraint Time diversification relies on the condition that consumers smooth their consumption over their life time The incentive to smooth consumption would be weakened if consumers are faced with liquidity constraints Conservative How about other considerations regarding saving and consumption decisions listed in Chapter 6?

12. L5: Dynamic Portfolio Management12 Dynamic Investment with Predictable Returns What if the investment opportunity is stochastic with some predictability Two period (0, 1); two risk (x 0, x 1 ), where x 1 is correlated with x 0 Investors invest only for the wealth at the end of period 1. i.e., there is no intermediate consumption E(x 0 )>0

13. L5: Dynamic Portfolio Management13 More on this Case

14. L5: Dynamic Portfolio Management14 More … The result depends on the effect of x 0 on x 1 –An in crease in x 0 may cause either a deterioration or an improvement in the distribution of x 1 in the sense of FSD. –Take the case deterioration of x1 in FSD (a specific case is mean reversion x 1 |x 0 =kx 0 +ε) –We have –h’ 0 when γ>1 –We can show

15. L5: Dynamic Portfolio Management15 Result Proposition 7.3: Suppose that an increase in the first-period return deteriorates the distribution of the second-period return in the sense of FSD. Then, the hedging demand for the risky asset is positive (resp. negative) if constant risk aversion is large (resp. smaller) than unity. In other words, longer time horizon should induce investors to take more risk when risk aversion is high, but the opposite when risk aversion is low For the case of mean reversion –Increase in x0, leads to lower x1 and wealth, thus higher U’(α). Investors would take more risk –Precautionary effect

16. L5: Dynamic Portfolio Management16 Uncertain Distribution of Excess Returns What if investors do not have perfect information on the distribution of risky portfolio returns? Standard intuition suggests investors would be more prudent. Proposition 7.4: Suppose that, as in the Bayesian learning process, an increase in the first-period return improves the distribution of the second- period return in the sense of first-order stochastic dominance. Then, the hedging demand for the risky asset is negative (resp. positive) if CRRA is larger (resp. smaller) than unity.

17. L5: Dynamic Portfolio Management17 Example There are two equally probable distributions of excess returns: (2, 0.9; -1, 0.1) and (2, 0.1; -1, 0.9); (2) Investors are uncertain about future distributions of excess returns. (3) Their prior is (2, 0.5; -1, 0.5) Observing x 0 =2, then the posterior distribution is (2, ; -1, ) We can apply backward induction to solve this problem ( γ=2) Solving the portfolio choice problem in the second period based on the posterior distribution  v(z) Solving the portfolio choice problem in the first period

18. L5: Dynamic Portfolio Management18 Exercises EGS, 7.1; 7.3