1 ASSET ALLOCATION

2 With Riskless Asset

3 Mean Variance Relative to a Benchmark

4 Hedging First Asset If lambda is large or mu is small, then this is least squares

5 With No Riskless Asset

6 Allocation subject to VaR

7 But if constraint is binding: The constraint reduces expected utility Higher generalized Sharpe ratios still lead to improved utility

8 STATIC ALLOCATIONS

9 Static Allocations

10 Derivatives of Optimal Allocation

11 Derivatives of Portfolio Risk

12 Dynamic Asset Allocation Maximize the Expected Utility of Terminal Wealth Ignore hedging demands and simply chose portfolios to be myoptically optimal For all t choose portfolio weights so that

13 Measure of Success If one dollar is invested, what should expected utility be? A good model for mean and variance should achieve a higher level of realized utility

14 Active Portfolio Strategies Typically involve efforts to estimate expected returns Covariance matrix is taken as constant Constraints imposed on short positions or ranges of asset shares Tilts toward some factor may be done as an overlay Rebalancing is periodic

15 Volatility Based Asset Allocation Keep expected returns constant to isolate the effects of volatility Volatilities can be estimated at high frequencies so this gives potential for fast acting asset allocation Risk management can be implemented in this way so that the trade-off between risk and return is explicitly considered

16 Results for Stock Bond Problem Use Multivariate GARCH model of component form Rebalance every day in response to new volatility and correlation information Keep expected returns equal to their realized values to isolate volatility effects

17

18

19 Strategic Rebalancing To reduce transaction costs and trading, only rebalance when the expected gain exceeds a threshold Rebalance on day t if Expected Utility at t+1 with optimal weights exceeds Expected Utility with existing weights by more than a fixed number

20