Investments, tenth edition

Risk Pooling and the Insurance Principle 

    Risk  pooling   

 means merging uncorrelated, risky projects as a means to reduce risk. 

Applied to the insurance business, risk pooling entails selling many uncorrelated insurance 

policies. This application of risk pooling has come to be known as the insurance principle. 

Conventional wisdom holds that risk pooling reduces risk, and that such pooling is the 

driving force behind risk management for the insurance industry. 

 But even brief reflection should convince you that risk pooling cannot be the entire 

story. How can  adding  bets that are independent of your other bets reduce your total expo-

sure to risk? This would be little different from a gambler in Las Vegas arguing that a few 

more trips to the roulette table will reduce his total risk by diversifying his overall “port-

folio” of wagers. You would immediately realize that the gambler now has more money at 

stake, and his overall potential swing in wealth is clearly wider: While his average gain or 

loss  per bet  may become more predictable as he repeatedly returns to the table, his total 

proceeds become less so. As we will see, the insurance principle is sometimes similarly 

misapplied to long-term investments by incorrectly extending what it implies about  aver-

age  returns to predictions about  total   returns. 

 Imagine a rich investor, Warren, who holds a $1 billion portfolio,  P.  The fraction of the 

portfolio invested in a risky asset,  A,  is  y,  leaving the fraction 1  2   y  invested in the risk-

free rate. Asset  A ’s risk premium is  R,  and its standard deviation is  s . From Equations 6.3 

and 6.4, the risk premium of the complete portfolio  P  is  R  

 P 

   5   yR,  its standard deviation is 

 s  

 P 

   5   y  s , and the Sharpe ratio is  S  

 P 

   5   R / s . Now Warren identifies another risky asset,  B,  

with the same risk premium and standard deviation as  A.  Warren estimates that the correla-

tion (and therefore covariance) between the two investments is zero, and he is intrigued at 

the potential this offers for risk reduction through diversification. 

 Given the benefits that Warren anticipates from diversification, he decides to take a 

position in asset  B  equal in size to his existing position in asset  A.  He therefore transfers 

another fraction,  y,  of wealth from the risk-free asset to asset  B.  This leaves his total port-

folio allocated as follows: The fraction  y  is still invested in asset  A,  an additional invest-

ment of  y  is invested in  B,  and 1  2  2 y  is in the risk-free asset. Notice that this strategy is 

 *The material in this section is more challenging. It may be skipped without impairing the ability to understand 

later chapters. 

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  C H A P T E R  

7

  Optimal Risky Portfolios 

231

analogous to pure risk pooling; Warren has taken on additional risky (albeit uncorrelated) 

bets, and his risky portfolio is larger than it was previously. We will denote Warren’s new 

portfolio as  Z.  

We can compute the risk premium of portfolio  Z  from Equation 7.2, its variance from 

Equation 7.3, and thus its Sharpe ratio. Remember that capital  R  denotes the risk premium 

of each asset and the risk premium of the risk-free asset is zero. When calculating portfolio 

variance, we use the fact that covariance is zero. Thus, for Portfolio  Z:   

R

Z

5 yR 1 yR 1 (1 2 2y)0 5 2yR

(double R

P

)

s

Z

2

5 y

2

s

2

1 y

2

s

2

1 0 5 2y

2

s

2

(double the variance of P)

s

Z

5 "s

Z

2

5 ys"2

(

"2 5 1.41 times the standard deviation of P)

S

Z

5 R

Z

/s

Z

5 2yR/(ys"2) 5 "2R/s

(

"2 5 1.41 times Sharpe ratio of P)

 The good news from these results is that the Sharpe ratio of  Z  is higher than that of  P   by 

the  factor     

"2.  Its excess rate of return is double that of  P,  yet its standard deviation is only 

   

"2  times larger. The bad news is that by increasing the scale of the risky investment, the 

standard deviation of the portfolio also increases by    

"2.  

 We might now imagine that instead of two uncorrelated assets, Warren has access to 

many. Repeating our analysis, we would find that with  n  assets the Sharpe ratio under 

strategy  Z  increases (relative to its original value) by a factor of    

"n   to     "n 3 R/s.  But the 

total risk of the pooling strategy  Z  will increase by the same multiple, to    s

"n.  

 This analysis illustrates both the opportunities and limitations of pure risk pooling: 

Pooling increases the scale of the risky investment (from  y  to 2 y ) by adding an additional 

position in another, uncorrelated asset. This addition of another risky bet also increases the 

size of the risky budget. So risk pooling  by itself  does not reduce risk, despite the fact that 

it benefits from the lack of correlation across policies. 

 The insurance principle tells us only that risk increases less than proportionally to the 

number of policies insured when the policies are uncorrelated; hence profitability—in this 

application, the Sharpe ratio—increases. But this effect does not actually reduce risk. 

 This might limit the potential economies of scale of an ever-growing portfolio such 

as that of a large insurer. You can interpret each “asset” in our analysis as one insurance 

policy. Each policy written requires the insurance company to set aside additional capital 

to cover potential losses. The insurance company invests its capital until it needs to pay out 

on claims. Selling more policies entails increasing the total position in risky investments 

and therefore the capital that must be allocated to those policies. As the company invests 

in more uncorrelated assets (insurance policies), the Sharpe ratio continuously increases 

(which is good), but since more funds are invested in risky policies, the overall risk of the 

portfolio rises (which is bad). As the number of policies grows, the risk of the pool will 

certainly grow—despite “diversification” across policies. Eventually, that growing risk 

will overwhelm the company’s available capital. 

 Insurance analysts often think in terms of probability of loss. Their mathematically cor-

rect interpretation of the insurance principle is that the probability of loss declines with risk 

pooling. This interpretation relates to the fact that the Sharpe ratio (profitability) increases 

with risk pooling. But to equate the declining probability of loss to reduction in total risk 

is erroneous; the latter is measured by overall standard deviation, which increases with 

risk pooling. (Again, think about the gambler in Las Vegas. As he returns over and over 

again to the roulette table, the probability that he will lose becomes ever more certain, but 

the magnitude of potential dollar gains or losses becomes ever greater.) Thus risk pooling 

allows neither investors nor insurance companies to shed risk. However, the increase in 

risk can be overcome when risk pooling is augmented by  risk sharing,  as discussed in the 

next subsection.  

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P A R T   I I

  Portfolio Theory and Practice