Fin305f: LeBaron, 2019 Campbell notes

1. Fin305f: LeBaron, 2019Campbell notes Chapter 2: Static Portfolio Choice

2. Small risks: Linear approximation Section 2.1.2

3. Simple 2 period investment problem

4. Maximum problem

5. Simple portfolio decision (part 1) • Add controlled mean to excess return:

6. Simple portfolio decision (part 2)

7. Wealth share

8. Two very important utility/cases • Constant Absolute Risk Aversion(CARA) (exponential utility), and normal returns: • Exact solutions • Many useful properties • Constant Relative Risk Aversion (CRRA) (power utility) • Taylor series approximation • Very common tool in asset pricing • CRRA always makes more sense

9. CARA utility and normal returns • Big assumptions • Big payoff in structure • Analytic results

10. Big equation in Finance (Campbell is right! Memorize this!)

11. Back to portfolio problem • Using feature that log(x) is monotonic, min (which was max) problem becomes: • This is a mean/variance optimization

12. Easy solution for optimal portfolio (on your own) • You will see many equations which look like this • Very simple, three inputs (mean, variance, risk aversion) • Dollar demand is linear in expected return • This is often important

13. Good features of the CARA format • Multiple asset extension easy (can be used to derive CAPM) • Heterogeneous agents • Linear demands allow for adding up diverse preferences • Information/Risk aversion • Demands don’t require knowledge of wealth distribution

14. Bad features of the CARA format • Wealth irrelevant to risky investments • Bounded utility • CARA bounded above • Large gains may not offset large losses • No lower bounds • Wealth and consumption can go negative • Trending risk premia • Assume economy grows with multiplicative risks • Risk premia grows as absolute risk grows • Very counter factual to financial data • Does not generalize to multiple periods • (1+R(1))*(1+R(2))-1 • Not normal even when R(1) and R(2) are normal • This is particularly troubling

15. CRRA-Log normal • Generalize to CRRA (power) utility • Assume wealth is log normal • Remember, all of these are pretty simple problems • One period • Two assets (we’ll relax this soon)

16. Maximization (wealth) • This again looks like a mean/variance tradeoff is coming

17. Maximization (wealth) • This again looks like a mean/variance tradeoff is coming

18. Maximization (log normal wealth) • Second case looks a little weird. • What’s going on?

19. Mapping to expected arithmetic returns • Use the magic formula again to map to arithmetic • Then sub into max problem • This shows that in arithmetic returns we have a standard MV trade off • This shows a little bit about how log returns and preferences can be tricky

20. Arithmetic versus log returns (the big dilemma) • Log returns easier in many situations (multiplication -> addition) • Most involve time • This may look trivial, but it is a deeply important issue in finance • Arithmetic returns are better for portfolios • Portfolios are linear weighted sums of individual arithmetic returns • This is messy in logs • Kind of a quandary • Finally, investors eventually care about arithmetic (not log) objects

21. Portfolio returns in log return space • Excess returns in logs (exact), but messy

22. Taylor series approximation • This useful Campbell equation takes a portfolio into components in log space. • This turns out to be useful

23. Solving portfolio problem • Approximate optimal portfolio for two assets, log normal, CRRA(power) utility, two period problem.

24. Approximate arithmetic solution

25. Growth-optimal portfolio • What about g=1? • Log preferences, “growth optimal” • Very special case • Evolutionarily important • Strategies behaving as if log utility will acquire more wealth than any other strategy in the long run • Big debate in the 1960’s • Samuelson shuts it down • Good gambling strategy • See popular book “Fortune’s Formula” • Also, related to information theory (Kelly/Shannon/Thorpe at the casino) • Mathematically very interesting • In macro it can be a point where income/substitution effects cancel • Not a very typical situation • We will see this again (Problem 2.1)

26. Two risky assets Section 2.2.1

27. Combining risky assets (means and variances)

28. Geometry of portfolio risk

29. Minimum variance portfolios

30. Sharpe Ratios: Very important ratio

31. N risky assets Section 2.2.3

32. Portfolio optimization • Find “mean/variance” efficient portfolios • Minimize portfolio variance for a given expected return target • Fits general preference framework • Useful tool in portfolio problems (nice graphs) • Common tool for investment managers • Somewhat complicated (not elegant) • Remember that all inputs are measured with error • In real world examples many more constraints are added

33. Minimum variance portfolios (this is messy)

34. Costs and relationships • Constraint is cost of keeping expected return on target • Relaxing this a little would allow going to smaller variance

35. Global minimum variance portfolio • Forget expected return target • Just find portfolio that minimizes variance • This has been useful in real finance problems • One reason is that expected returns are hard to measure • Also, another useful point for understanding portfolio problems

36. Global minimum variance portfolio • This is also an important data point • Nondiversifiable + Diversifiable risk • N to infinity gives equal weighted index

37. Mean/variance efficient set

38. Changing return correlations

39. Experiments in correlations and portfolios • Std(portfolio)-std(ew index, N big) • Diversifications gains • For N (a classic plot about portfolios) • But changing over time • Gains getting generally smaller • A little different from last figure

40. Excess standard deviation versus number of stocks in portfolio

41. Mutual fund theorem Section 2.2.5

42. Mutual Fund Theorem (Tobin, 1958) • All mean/variance efficient portfolios can be built as a portfolio from two different portfolios • One is the minimum variance portfolio • Other is a little strange

43. One riskless and N risky assets Section 2.2.6

44. Risk free portfolio/setup • w’s don’t need to sum to 1 • w’s can all be positive or negative • Short sales allowed • Borrowing/lending at Rf allowed

45. Maximization problem

46. More maximization

47. Investment Opportunity Set

48. Classic picture • With risk free asset one asset is risk free • Other is “tangency portfolio” • Every holds risky assets in same proportions (think market portfolio) • Then combines with risk free in some combination • Mathematically, the lower line is relevant too, but no one would ever buy it • Without risk free, the frontier is in the darker area

49. Mutual funds • This figure, and Tobin’s two fund theorem have big empirical issues • Basically, there are many, many mutual funds (about 10,000) • Number of stocks = 3,500-4,000 • The market is not simplifying this down • Some movement toward this, but not much

50. Asset allocations: Bond/stock ratios