Investments, tenth edition

  Investment for the Long Run 

 Now we turn to the implications of risk pooling and risk sharing for long-term investing. 

Think of extending an investment horizon for another period (which adds the uncertainty 

of that period’s risky return) as analogous to adding another risky asset or insurance policy 

to a pool of assets. 

 Examining the impact of an extension of the investment horizon requires us to clarify 

what the alternative is. Suppose you consider an investment in a risky portfolio over the 

next 2 years, which we’ll call the “long-term investment.” How should you compare this 

decision to a “short-run investment”? We must compare these two strategies over the same 

period, that is, 2 years. The short-term investment therefore must be interpreted as invest-

ing in the risky portfolio over 1 year and in the risk-free asset over the other. 

 Once we agree on this comparison, and assuming the risky return on the first year is 

uncorrelated with that of the second, it becomes clear that the “long-term” strategy is 

analogous to portfolio  Z.  This is because holding on to the risky investment in the sec-

ond year (rather than withdrawing to the risk-free rate) piles up more risk, just as selling 

another insurance policy does. Put differently, the long-term investment may be considered 

analogous to risk pooling. While extending a risky investment to the long run improves the 

Sharpe ratio (as does risk pooling), it also increases risk. Thus “time diversification” is not 

really diversification. 

 The more accurate analogy to risk sharing for a long-term horizon is to spread the risky 

investment budget across each of the investment periods. Compare the following three 

strategies applied to the whole investment budget over a 2-year horizon:

    1.  Invest the whole budget at risk for one period, and then withdraw the entire 

proceeds, placing them in a risk-free asset in the other period. Because you 

are invested in the risky asset for only 1 year, the risk premium over the whole 

investment period is  R,  the 2-year SD is  s , and the 2-year Sharpe ratio is 

 S   5   R / s .  

   2.  Invest the whole budget in the risky asset for both periods. The 2-year risk premium 

is 2 R  (assuming continuously compounded rates), the 2-year variance is 2 s  

2

 ,  the 

2-year SD is    s

"2,  and the 2-year Sharpe ratio is    S 5 R"2/s.  This is analogous to 

risk pooling, taking two “bets” on the risky portfolio instead of one (as in Strategy 1).  

   3.  Invest half the investment budget in the risky position in each of two periods, plac-

ing the remainder of funds in the risk-free asset. The 2-year risk premium is  R,  

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P A R T   I I

  Portfolio Theory and Practice

the 2-year variance is 2  3  (½   s ) 

2

   5   s  

2

 /2, the SD is    s/

"2,  and the Sharpe ratio is 

   S

5 R"2/s.  This is analogous to risk sharing, taking a fractional position in each 

year’s investment return.   

Strategy 3 is less risky than either alternative. Its expected total return equals Strategy 1’s, 

yet its risk is lower and therefore its Sharpe ratio is higher. It achieves the same Sharpe 

ratio as Strategy 2 but with standard deviation reduced by a factor of 2. In summary, its 

Sharpe ratio is at least as good as either alternative and, more to the point, its total risk is 

less than either. 

 We conclude that risk does not fade in the long run. An investor who can invest in an 

attractive portfolio for only one period, and chooses to invest a given budget in that period, 

would find it preferable to put money at risk in that portfolio in as many periods as allowed 

 but will decrease the risky budget in each period.  Simple risk pooling, or in this case, time 

diversification, does not reduce risk.