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Fin305f: LeBaron, 2019 Campbell notes. Chapter 2: Static Portfolio Choice. Small risks: Linear approximation. Section 2.1.2. Simple 2 period investment problem. Maximum problem. Simple portfolio decision (part 1). Add controlled mean to excess return:. Simple portfolio decision (part 2).
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Fin305f: LeBaron, 2019Campbell notes Chapter 2: Static Portfolio Choice
Small risks: Linear approximation Section 2.1.2
Simple portfolio decision (part 1) • Add controlled mean to excess return:
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
CARA utility and normal returns • Big assumptions • Big payoff in structure • Analytic results
Back to portfolio problem • Using feature that log(x) is monotonic, min (which was max) problem becomes: • This is a mean/variance optimization
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
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
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
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)
Maximization (wealth) • This again looks like a mean/variance tradeoff is coming
Maximization (wealth) • This again looks like a mean/variance tradeoff is coming
Maximization (log normal wealth) • Second case looks a little weird. • What’s going on?
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
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
Portfolio returns in log return space • Excess returns in logs (exact), but messy
Taylor series approximation • This useful Campbell equation takes a portfolio into components in log space. • This turns out to be useful
Solving portfolio problem • Approximate optimal portfolio for two assets, log normal, CRRA(power) utility, two period problem.
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)
Two risky assets Section 2.2.1
N risky assets Section 2.2.3
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
Costs and relationships • Constraint is cost of keeping expected return on target • Relaxing this a little would allow going to smaller variance
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
Global minimum variance portfolio • This is also an important data point • Nondiversifiable + Diversifiable risk • N to infinity gives equal weighted index
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
Excess standard deviation versus number of stocks in portfolio
Mutual fund theorem Section 2.2.5
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
One riskless and N risky assets Section 2.2.6
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
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
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