Download Risk-Spreading via Financial Intermediation: Life Insurance

XII. RISK-SPREADING VIA FINANCIAL INTERMEDIATION: LIFE INSURANCE
As discussed briefly at the end of Section V, financial assets can be traded directly in the
capital markets or indirectly through financial intermediaries.
In general, "standardized"
securities are traded in markets (e.g., government bonds, wheat futures, shares of IBM) while
"custom" contracts (e.g., individual mortgage, personal loan, or insurance) are handled through
financial intermediaries. In this section, the classical case of pure life insurance is examined to
show how efficient risksharing can be achieved using a combination of financial intermediation
and the capital market.
The Life Insurance Company
Suppose that there are N people in the economy each with wealth (per capita) W.
Hence, national wealth ≡ W = NW. Suppose further that each person purchases a one-year term
life insurance policy which pays $c in the event of death and we define q ≡ c/W
to be the
amount of insurance coverage purchased by each person as a fraction of his wealth. Let
yi
random variable describing the death of the i
th
be a
person where yi = 0 if person i survives the year
and yi = 1 if person i dies during the year. Assume that the mortality tables are such that
E( y i ) = ρ , the same for all people, i = 1,2,..., N and Var( y i ) = V 2
which is also the same for all
people. Hence, ρ is the expected number of deaths per person (0 < ρ < 1) . Let Y N =
N
∑y
i
be the
i=1
random variable for deaths of all people and it is equal to the number of deaths in the economy. If the
death of one person is independent of another (a crucial but reasonable assumption), then
E[ Y N ] = Nρ ; Var( Y N ) = NV 2
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Robert C. Merton
If a single competitive insurance company writes all the policies, then the analysis will determine
the:
Premium per policy charged, P N
The amount of equity capital required by the company to do business, K N .
The required (expected) return on the equity by investors in the insurance
company.
•
•
•
Premiums are received at the beginning of the year in the amount, NP N . Benefits are paid at the
end of the year in the (random variable) amount C N ≡ cY N . Hence,
E[ C N ] = C N = Nc ρ
Var( C N ) = σ 2N = Nc 2 V 2
Suppose that investors are mean-variance maximizers and that the conditions for the Capital
Asset Pricing Model (Section XI) hold. If R = 1 + rate of interest, then the return per dollar
invested in equity of the insurance company is Z N ≡
E[ Z N ] = Z N =
R( NP N + K N ) - C N
KN
, and
R[( NP N ) + K N ] - Ncρ
KN
Var [ Z N ] = σ 2Z N =
σ = Nc 2 V 2
2
K
2
N
2
N
KN
In equilibrium, supply must equal demand, and so, for the equity of the insurance company to be
held, we have that Z N must satisfy the basic equilibrium condition for the CAPM:
supply = K N = Demand for asset; or
KN
= w*Z N = fraction of the market portfolio
NW
*
where W Z N is also the fraction in optimal combination of risky assets given by:
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Finance Theory
n
∑v Z j( Z j - R)
N
j=1
w*Z N =
(A-RC)
Suppose (as is reasonable) that Z N is independent of the returns on all other assets
(i.e., Cov( Z N , Z j ) = 0 for Z N ≠ Z j ), then
n
Cov( Z N , Z M ) = ∑
w*j
Cov( Z N ,Z j ) =
w*Z N Var
( Z N )= (
KN
)(
Nc 2V 2
NW
j=1
K 2N
)=
Nc 2V 2
.
NW K N
From Section XI, we have from the Security Market Line that
⎛ c2 V 2 ⎞
R
=
(
,
)
=
Cov
⎜⎜
⎟⎟
r
Z
Z
r
S
N
M
S
ZN
⎝ WK N ⎠
Substituting for Z N , we have that
R[ NP N + K N ] - Nc ρ
KN
⎛ c 2V 2 ⎞
- R= rS ⎜
⎟ or
⎝ WK N ⎠
⎛ c 2V 2 ⎞
RNP N - Nc ρ = r S ⎜
⎟ or substituting for c = qW
⎝ W ⎠
⎛ q 2WV 2 ⎞
⎟⎟
RP N - q ρW = r S ⎜⎜
N
⎝
⎠
The (equilibrium) premium per policy can be written as
PN =
qWρ
1
cρ 1
2
2
2
+
r S q WV =
r S cq V and therefore ,
R
NR
R NR
2
ZN = R+
r S cq V
, the (equilibrium) expected return on equity in the insurance company.
KN
243
Robert C. Merton
Note: In this formal analysis, we have not taken into account the limited liability feature of the
equity of the insurance company which leads us to the last question to be answered:
What is the appropriate value for K N ?
To answer this question, one must go back and ask what service is the financial intermediary to
provide to the customer? What does he want? The customer wants a certain payment, c, in the
event of death. Now, if the total number of deaths is such that the (ex-post) benefits required to
be paid, c N , is larger than the company's total assets, R( NP N + K N ), then by limited liability on
equity, the customer will not receive the full benefits promised, but only
R
( NP N + K N ) < c.
yN
Obviously, the larger is K N the less likely is default. If K N is "too small", then the probability of
default is higher, and the customer in purchasing the policy does not get the simple security he wanted
which pays $c for sure in the event of his death, but rather has the more complicated security
which pays $c
(r.v.) amount
in the event of his death, conditional on the company being solvent, pays the
R
( P N N + K N ) in the event of death, conditional on the company not being solvent.
yN
Clearly, to assess the probabilities about possible payoffs, the customer would have to know the amount
of capital the firm has; the nature and quantity of policies written for other customers; the probability of
these customers dying; etc. In short, nearly everything about the company that the management knows,
the customer would have to know and analyze. Essentially, the customer takes a (partial) equity position
in the company. Since one purpose of the financial intermediary is to limit the amount of information
required by customers to make a decision and because the service wanted is basically life insurance, the
equity capital should be large enough to (virtually) eliminate the chance of default. In doing so, the
separation between customer and equityholder (or general liability or debtholder) is made as large as
possible. Define: reserves as the amount of assets required to be held by the insurance company
to ensure that payment will be made to customers with some probability.
R N = reserves requires = R (NPN + K N )
Let
So, given the premium, there is a one-to-one
correspondence between reserves and capital (equity). Clearly, the amount of reserves required to ensure
with absolute certainty that all customers will be paid in every state of the world would come by
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Finance Theory
requiring that assets be large enough to payoff everyone in the event that everyone dies.
max
C N = Nc
K
max
N
= Nc(
R max
=
N
or
maximum required reserves
1
cq V 2
.
- ρ ) - rS
R
R
However, for
Nc large and ρ
I.e.,
= Nc = R( NP N + K max
N )
or
reasonably small, the amount of
capital required to meet the maximum reserves could be prohibitively large. Further, if ρ is small,
there is a very small chance that everyone will die (especially since the events of death are independent)
and one would expect that for large N, there would be some diversifying effects. Hence, it may not
be necessary for the company to hold the maximum reserves while still performing the essential
service required.
Suppose that instead it was required that reserves be such that the probability of default is
*
*
less than some assigned level, p , i.e., Prob{C N > R N } ≤ p , and define the associated required
max
capital as K N ( p ). Note: K N (0) = K N .
*
n
Prob{(∑ yi)c
1
Prob{C N > R N } =
R
> R K N(p ) + Ncρ + rscqV } = Prob{X N > ( )
2
*
( *)
KN p
+
rs cqV
V c N
N
For large N,
X N will be
N
where X N ≡
∑ yi-Nρ
1
and E(X ) = 0; Var(X ) = 1 .
N
N
V N
distributed approximately standard normal (Gaussian). Hence,
Prob{ X N > (
*
*
R K N ( p ) r S qV
R K ( p ) r S qV
)
+
} ≈ 1 - Φ [( ) N
+
] = p*
Vc
Vc
N
N
N
N
where
Φ[ ]
is the
cumulative density function for the normal distribution. For this distribution, there is a one-to-one
*
correspondence between p
and the number of standard deviations to the right of the mean. I.e., let
µ = number of standard deviations, then
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Robert C. Merton
*
*
µ(p )
p
0
1.0
2.0
2.33
3.10
3.70
4.00
.5000
.1600
.0230
.0100
.0010
.0001 (1 chance in 10,000)
.00004 (1 chance in 25,000)
*
Vc
R K N ( p ) r S qV
*
+
)µ ( p* )N - r SqV for large N. Thus,
or K N ( p ) = (
Hence, µ ( p ) = ( )
R
Vc
N
N
*
*
*
for a given p (or µ ( p )) , we have an expression for the required equity capital for large N.
1
*
Asymptotic Results for large N(N → ∞) ( > p > 0)
2
*
KN( p ) ≈ (
Vc
)µ( p* ) N
R
limit ( P N ) =
N →∞
limit
N →∞
(
cρ
;
R
2
lim ( Z N ) = R;
N →∞
R
> 0.
2
µ ( p* )
limit ( σ 2Z N ) =
N →∞
*
Required Equity
K (p )
) = limit ( N
) = 0;
Total Premiums
N PN
N →∞
lim (
N →∞
*
KN( p )
) = 0;
NW
*
Required Equity
KN( p )
=
(
)= 0
limit
max
Maximum Equity limit
N →∞
N →∞
KN
*
Suppose Suppose ρ = .0025; c = $30,000 p = .00004 =
2
V = .0025 R = 1
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1
25,000
[ µ ( p* ) = 4]
Finance Theory
NPN
K max
= K N (0)
N
K N (.00004)
K N (.00004)
NPN
$750,000
$299,250,000
$600,000
.8000
$6,750,000
$2,693,250,000
($2.7 billion)
$1,800,000
.2667
$75,000,000
$30,000,000,000
($30 billion)
$6,000,000
.0800
$675,000,000
$270 billion
$18,000,000
.0267
$7,500,000,000
($7.5 billion)
$3 trillion
12
(3x10 )
$60,000,000
(60 million)
7
(6x10 )
.0080
N
10,000
4
(1x10 )
90,000
4
(9x10 )
1,000,000
6
(1x10 )
one million
9,000,000
6
(9x10 )
8
1x10
one hundred
million
Thus, we observe a characteristic property of (many) financial intermediaries: namely, that net
worth is a small fraction of total assets (in the example, less than 1%); further total (potential)
liabilities are many orders of magnitude larger than total assets or reserves. Of course, sales and
other operating expenses would have to be added to the premium and other assets (buildings,
etc.) have been excluded.
The benefits of the financial intermediary in this case are obvious: if each insurance
policy were written by one person for one other person (and if
r S ≈ 2 and q ≈ 1), then the
minimum premium for a $30,000 policy would be $225 versus $75 charged by the company.
max
Further, the reserves required would be $30,000 (or K 1 ) per policy versus $.60 per policy for
the intermediary! (50,000 times as much!)
Note despite the tremendous diversifying power of many policies, if there were no equity
capital market to raise the funds, it is doubtful if such an organization could occur without
substantially higher premiums. Suppose one (wealthy) individual provided all the capital ($60
million): (at the derived rates with R = 1) the expected rate of return is zero and there is a .16
probability of one standard deviation to the right which translates into a $15 million loss! Few
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Robert C. Merton
risk-averse utility maximizers would accept such an investment. But, by diversifying the risk by
issuing equity in the capital market and if individuals hold well-diversified equity portfolios (as
they should), then the loss would be around 15¢ per investor which is trivial for an investor with
initial wealth of $30,000. Thus, through the combined use of the capital market and the financial
intermediary, the individual investor can get the service or asset he wants (virtually no-default
life insurance) to eliminate a substantial non-systematic risk, at minimum cost.
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