Investments, tenth edition

  Optimal Portfolios and Nonnormal Returns 

 The portfolio optimization techniques we have used so far assume normal distributions 

of returns in that standard deviation is taken to be a fully adequate measure of risk. 

However, potential nonnormality of returns requires us to pay attention as well to risk 

measures that focus on worst-case losses such as value at risk (VaR) or expected short-

fall (ES). 

 In Chapter 6 we suggested that capital allocation to the risky portfolio should be recon-

sidered in the face of fat-tailed distributions that can result in extreme values of VaR and 

ES. Specifically, forecasts of greater than normal VaR and ES should encourage more 

moderate capital allocations to the risky portfolio. Accounting for the effect of diversifica-

tion on VaR and ES would be useful as well. Unfortunately, the impact of diversification 

on tail risk cannot be easily anticipated. 

 A practical way to estimate values of VaR and ES in the presence of fat tails is boot-

strapping (described in Section 5.9). We start with a historical sample of returns of each 

asset in our prospective portfolio. We compute the portfolio return corresponding to a draw 

of one return from each asset’s history. We thus calculate as many of these random port-

folio returns as we wish. Fifty thousand returns produced in this way can provide a good 

estimate of VaR and ES values. The forecasted values for VaR and ES of the mean-variance 

optimal portfolio can then be compared to other candidate portfolios. If these other portfo-

lios yield sufficiently better VaR and ES values, we may prefer one of those to the mean-

variance  efficient  portfolio.    

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P A R T   I I

  Portfolio Theory and Practice

    7.5 

Risk Pooling, Risk Sharing, and the Risk 

of  Long-Term  Investments *   

  Diversification means that we spread our investment budget across a variety of assets and 

thus limit overall risk. Sometimes it is argued that spreading investments across time, so 

that average performance reflects returns in several investment periods, offers an analo-

gous benefit dubbed “time diversification.” A common belief is that time diversification 

can make long-term investing safer. 

 Is this extension of diversification to investments over time valid? The question of how 

risk increases when the horizon of a risky investment lengthens is analogous to risk pool-

ing, the process by which an insurance company aggregates a large portfolio (or pool) of 

uncorrelated risks. However, the application of risk pooling to investment risk is widely 

misunderstood, as is the application of “the insurance principle” to long-term investments. 

In this section, we try to clarify these issues and explore the appropriate extension of the 

insurance principle to investment risk.