Investments, tenth edition

o

i

l

o

f

t

r

o

P

:

g

n

i

l

o

o

P

k

s

i

R

Z

o

i

l

o

f

t

r

o

P

:

g

n

i

r

a

h

S

k

s

i

R

V

R

Z

 2 R

R

V

 2(y/2)R

yR

Z

2

5

5

2y

2 2

V

2

5

5

2 y /2

2 2

5

5

y

2 2

/2

Z

5

Z

2

5 y

2

V

5

V

2

5 y / 2

S

(

)

Z

5 R

Z

/

Z

5 2yR/ y

2 5 2R/

S

V

5 R

V

/

V

5

2R/

y

 We observe that portfolio  V  matches the attractive Sharpe ratio of portfolio  Z,  but with 

lower volatility. Thus risk sharing  combined  with risk pooling is the key to the insurance 

industry. True diversification means spreading a portfolio of  fixed size  across many assets, 

not merely adding more risky bets to an ever-growing risky portfolio. 

 To control his total risk, Warren had to sell off a fraction of the pool of assets. This 

implies that a portion of those assets must now be held by someone else. For example, 

if the assets are insurance policies, other investors must be sharing the risk, perhaps by 

buying shares in the insurance company. Alternatively, insurance companies commonly 

“reinsure” their risk by selling off portions of the policies to other investors or insurance 

companies, thus explicitly sharing the risk. 

 

We can easily generalize Warren’s example to the case of more than two assets. 

Suppose the risky pool has  n  assets. Then the volatility of the risk-sharing portfolio will 

be     s

V

5 ys/"n,  and its Sharpe ratio will be    "nR/s.  Clearly, both of these improve as  n  

increases. Think back to our gambler at the roulette wheel one last time. He was wrong to 

argue that diversification means that 100 bets are less risky than 1 bet. His intuition would 

be correct, however, if he shared those 100 bets with 100 of his friends. A 1/100 share of 

100 bets is in fact less risky than one bet. Fixing the amount of his total money at risk as 

that money is spread across more independent bets is the way for him to reduce risk.  

14

    

  

14

 For the Las Vegas gambler, risk sharing makes the gambles ever more certain to produce a  negative  rate of 

return, highlighting the illness that characterizes compulsive gambling. 

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  C H A P T E R  

7

  Optimal Risky Portfolios 

233

 With risk sharing, one can set up an insurance company of any size, amassing a large 

portfolio of policies and limiting total risk by selling shares among many investors. As the 

Sharpe ratio steadily increases with the number of policies written, while the risk to each 

diversified shareholder falls, the size of ever-more-profitable insurance companies appears 

unlimited. In reality, however, two problems put a damper on this process. First, burdens 

related to problems of managing very large firms will sooner or later eat into the increased 

gross margins. More important, the issue of “too big to fail” may emerge. The possibility of 

error in assessing the risk of each policy or misestimating the correlations across losses on 

the pooled policies (or worse yet, intentional underestimation of risk) can cause an insur-

ance company to fail. As we saw in Chapter 1, too big to fail means that such failure can 

lead to related failures among the firm’s trading partners. This is similar to what happened 

in the financial crisis of 2008. The jury is still out on the role of lack of scruples in this affair. 

It is hoped that future regulation will put real limits on exaggerated optimism concerning 

the power of diversification to limit risk, despite the appealing mitigation of risk sharing.