110
S.4
S.4. INSURANCE
Insurance
1. There is no specimen solution for question 1 because it is marked comparatively.
Section 3.3 of Eeckhout, Gollier and Schlesinger includes a proof that an increase in
the cost of insurance (through the loading) does not necessarily reduce the demand
for insurance if the utility function exhibits DARA. There will be some utility
functions where the wealth e↵ect dominates. This means that the increased cost
has reduced their wealth and utility to an extent that the higher risk aversion at
the low wealth levels acts to increase the demand for insurance.
2. (a) Working in units of £1 million, the premium is
1
4
⇥ 8 ⇥ (1 + 0.2) = 2.4.
(b) Denoting the expected utility on choosing coinsurance level
3
1
ln(12 2.4 ) + ln(12 2.4
8(1
4
4
3
2.4
1 5.6
f 0( ) =
+
4 12 2.4
4 4 + 5.6
2
3
2.4
1
5.62
f 00 ( ) =
<0
4 (12 2.4 )2 4 (4 + 5.6 )2
f( ) =
by f ( ), we have
))
So, f is twice di↵erentiable and strictly concave, and so the optimal
solves f 0 ( ⇤ ) = 0 (or is at an endpoint ⇤ = 0 or ⇤ = 1):
3
2.4
1
5.6
+
=0
⇤
4 12 2.4
4 4 + 5.6 ⇤
)7.2(4 + 5.6 ⇤ ) = 5.6(12 2.4 ⇤ )
5.6 ⇥ 12 7.2 ⇥ 4
38.4
) ⇤=
=
= 0.714 (3 s.f.)
7.2 ⇥ 5.6 + 5.6 ⇥ 2.4
53.76
Since this ⇤ 2 [0, 1] and f is concave, the optimal solution will be
or 1. (We will confirm this in the next part of the question.)
⇤
,
⇤
, not 0
(c)
3
1
ln(12) + ln(4) = 2.210 (4 s.f.)
4 ✓
4
◆
✓
◆
3
38.4
1
38.4
⇤
f ( ) = ln 12 2.4
+ ln 4 + 5.6
= 2.268 (4 s.f.)
4
53.76
4
53.76
f (1) = ln(12 2.4) = 2.262 (4 s.f.)
f (0) =
(d) Full insurance (
3. (a) E[x̃] = 0 ⇥
7
10
⇤
= 1) is optimal if loading is zero (Mossin’s Theorem).
+4⇥
1
10
+8⇥
1
10
+ 10 ⇥
1
10
= 2.2
7
1
1
1
(b) E[max(0, x̃ 3)] = 0 ⇥ 10
+ 1 ⇥ 10
+ 5 ⇥ 10
+ 7 ⇥ 10
= 1.3
7
1
1
1
E[max(0, x̃ 6)] = 0 ⇥ 10 + 0 ⇥ 10 + 2 ⇥ 10 + 4 ⇥ 10 = 0.6
The actuarially fair premium is not proportional to the inverse of the deductible and so one would not expect a doubling of the deductible to halve
the premium.
APPENDIX S. SOLUTIONS SB4B · HT 2017
(c) Denoting the coinsurance rate
mium with deductible d, 3 =
s.f.).
111
d which gives
1.3
= 0.591 (3
2.2
the same premium as the pres.f.) and 6 = 0.6
= 0.273 (3
2.2
(d) For loss outcomes (0, 4, 8, 10) net wealth on purchasing insurance with a deductible of 6 is (9.4, 5.4, 3.4, 3.4) and net wealth on purchasing coinsurance
with level of 6 is (9.40, 6.49, 3.58, 2.13). Denoting the cdf of final wealth if
deductible is 6 by F and the cdf of final wealth if coinsurance is 6 by G then
the figure of cdfs is as follows:
cdfs
1
F (w)
G(w)
0.5
0
0
2
4
6
8
Net wealth w
10
Rw
Define S(w) := 0 G(s) F (s)ds. Clearly S(2.13) = 0. Also S(9.4) = 0
since both products have the same mean wealth. Moreover G(s) F (s) 0
for s < 3.4 and G(s) F (s)  0 for s 3.4 and so it must be that S(w) is
weakly increasing from 0 from w = 2.13 to w = 3.4, and weakly decreasing to
0 from w = 3.4 to w = 9.4. By continuity of S, S must always be nonnegative
between 2.13 and 9.4.
4. For the standard portfolio problem let the utility function be u, satisfying u0 > 0,
u00 < 0, and initial wealth be w0 . Since the risk free (risky) investment earns return
zero (x̃) an investment in the risky asset of ↵ leads to realised wealth of w0 + ↵x̃
and expected utility, f (↵), of:
f (↵) := E [u(w0 + ↵x̃)]
f is strictly concave and so the unique optimal ↵⇤ is the solution to the first order
condition
f 0 (↵⇤ ) = E [x̃u0 (w0 + ↵⇤ x̃)] = 0.
(a) Now, let ỹ := (x̃, q; 0, 1 q), and let denote the optimal amount of (new)
asset ỹ to purchase. The new first order condition is
Eỹ [ỹu0 (w0 +
⇤
ỹ)] = 0
Applying the law of total expectation (EY (Y ) = EX (EY (Y |X)) we find
Eỹ [ỹu0 (w0 +
⇤
ỹ)] = Ex̃ [Eỹ [ỹu0 (w0 + ⇤ ỹ)|x̃]]
= Ex̃ [qx̃u0 (w0 + ⇤ x̃) + (1 q) ⇥ 0]
= qEx̃ [x̃u0 (w0 + ⇤ x̃)]
112
S.4. INSURANCE
Now the unique solution to this is ⇤ = ↵⇤ , that is you purchase the same
amount of the risky asset as you would have in the original problem. Note
that this comes from the Independence Axiom: you prefer w0 + ↵⇤ x̃ to any
other w0 + ↵x̃ and therefore prefer any stochastic mix of w0 + ↵⇤ x̃ and the
risk-free asset to any stochastic mix of w0 + ↵x̃ and the risk-free asset, and so
↵⇤ is the optimal amount to purchase in both problems.
(b) Let z̃ := qx̃, and let denote the optimal amount of (new) asset z̃ to purchase.
The new first order condition is
Ez̃ [z̃u0 (w0 + ⇤ z̃)] = 0
)Ex̃ [qx̃u0 (w0 + ⇤ qx̃)] = 0
From the definition of ↵⇤ , we have
⇤
q = ↵⇤ , and therefore
⇤
=
↵⇤
.
q
5. The question did not specify whether ↵ can be negative, or whether the expected
excess return from the risky asset was positive. We don’t impose any restrictions
in this solution.
(a) Let w0 denote initial wealth and ↵ denote the demand for the risky asset. The
optimisation problem is
max p ln(w0 (1 + r)
↵
↵(r
a)) + (1
p) ln(w0 (1 + r) + ↵(b
r))
This is strictly concave in ↵ and so the optimal choice of ↵, ↵⇤ satisfies first
order condition
p(r a)
(1 p)(b r)
+
=0
w0 (1 + r) ↵⇤ (r a) w0 (1 + r) + ↵⇤ (b r)
)p(r a)(w0 (1 + r) + ↵⇤ (b r)) = (1 p)(b r)(w0 (1 + r) ↵⇤ (r a))
)↵⇤ [p(r a)(b r) + (1 p)(r a)(b r)] = w0 (1 + r)[(1 p)(b r) p(r

1 p
p
⇤
)↵ = w0 (1 + r)
r a b r

(1 p)(b r) + p(a r)
= w0 (1 + r)
(r a)(b r)
(Note that ↵⇤ is the same sign as the excess expected return of the risky asset
over the risk-free asset.)
(b) Yes, for a < r < b.
(c)
d↵⇤
↵⇤
=
dw0
w0
So a small increase in initial wealth increases the magnitude of ↵⇤ ; if ↵⇤ is
positive it becomes more positive, if ↵⇤ is negative it becomes more negative.
a)]
APPENDIX S. SOLUTIONS SB4B · HT 2017
(d)
d↵⇤
↵⇤
=
dr
1+r
113
w0 (1 + r)

1
(r
p
p
+
2
a)
(b r)2
An increase in the risk free rate of return has two e↵ects on the optimal choice
↵⇤
of ↵. First there is a ‘wealth’ e↵ect of 1+r
, whereby by increasing the riskfree rate of return e↵ectively increases the ex-post wealth of the individual,
increasing the magnitude of ↵. Second by making the risky asset relatively
less attractive it reduces ↵.
(e)
d↵⇤
1 p
= w0 (1 + r)
da
(r a)2
⇤
d↵
p
= w0 (1 + r)
db
(b r)2
0
0
An increase (decrease) in a or b leads to a first order stochastic dominant
improvement (deterioration) in the returns from the risky asset, thereby increasing (decreasing) optimal demand.
(f) Let c := (1
p)(b
r) + p(a
r) and so r
↵⇤ = w0 (1 + r)
"
a=
(1 p)(b r) c
.
p
p
b
r
p
c
1 p
b
r
Then
#
Di↵erentiating ↵⇤ with respect to b whilst holding expected excess return c
constant gives
"
#
@↵⇤
1
1
= w0 (1 + r)p
+
@b
(b r 1 c p )2 (b r)2
⇤
Since b r > 0 and r a > 0, and therefore b r 1 c p > 0 we have @↵
has
@b
⇤
the opposite sign to c, and therefore the opposite sign to ↵ . So an increase in
b whilst holding the expected excess return of the risky asset constant reduces
the magnitude of ↵⇤ .
6. (a) The utilities of Angela and Bertie before and after the trade, to 3 s.f., are
E[uA (!A )] = ln(10) = 2.30
1
1
E[uB (!B )] = ln(15) + ln(5) = 2.16
2
2
1
1
1
1
E[uA ((13, ; 8, )] = ln(13) + ln(8) = 2.32
2
2
2
2
1
1
1
1
E[uB ((12, ; 7, )] = ln(12) + ln(7) = 2.22
2
2
2
2
2.30 < 2.32 and 2.16 < 2.22 and so the trade is mutually beneficial
114
S.4. INSURANCE
(b) A necessary condition for the allocation to be Pareto efficient is that it satisfies
the Borch Rule for states 1 and 2, and so
u0A (cA (1))
u0B (cB (1))
=
u0A (cA (2))
u0B (cB (2))
1/13
1/12
However
6=
1/8
1/7
and so the Borch Rule does not hold. The suggested allocation is not Pareto
efficient.
(c) The following three equations must hold (the first is the Borch Rule, and the
second two are the budget constraint):
u0A (cA (1))
u0B (cB (1))
=
u0A (cA (2))
u0B (cB (2))
cA (1) + cB (1) = 25
cA (2) + cB (2) = 15
So we have three equations in four unknowns. Substituting the second two into
the first we may write the efficient allocation in terms of cA (1) = 2 (0, 25)
cA (2)
15 cA (2)
=
cA (1)
25 cA (1)
) cA (2)(25
) = (15 cA (2))
15
) cA (2) =
25
cB (1) = 25
15
cB (2) = 15
25
(d) The set of Pareto efficient allocations wouldn’t change.
7. From risk aversion, agent n + 1 would be willing to sell a fraction of the risky
endowment "c̃ to any other agent for a price lower than the expected value E["c̃].
For small enough " any of the other agents would be willing to accept this trade.
This is the same as Mossin’s Theorem, whereby a risk averse agent will not purchase
full insurance if the premium is actuarially unfair, or equivalently that a risk averse
agent will always purchase a positive amount of the risky asset in the static portfolio
choice problem. Intuitively, this arises because risk aversion is a second order e↵ect.
Although not requested in the question, the following is a proof of the statement.
Suppose that the more risk averse agent is bearing all the aggregate risk with
consumption profile c̃, and that the other i = 1, . . . , n agents have constant consumption ci . Let µc and c2 denote the mean and variance of c̃ respectively. Now,
suppose that the more risk averse agent was to pay "⌘ to agent 1 to take on the
zero mean risk "(c̃ µc ).
APPENDIX S. SOLUTIONS SB4B · HT 2017
115
The more risk averse agent would strictly prefer c̃ + "(µc c̃) to c̃ since the former
strictly second order stochastically dominates the latter. From continuity of the
utility function, the more risk averse agent would be willing to pay some positive
amount "⌘ > 0.
It is then sufficient to use the Arrow-Pratt approximation to show that for any
positive ⌘, there is a small enough " such that agent 1 would accept this o↵er.
Agent 1 starts with certain consumption of c1 . From the definition of the risk
premium ⇡(c1 + ⇡, u, "(c̃ µc )) at wealth level c1 + ⇡(c1 + ⇡, u, "(c̃ µc )) we know
that
E[u(c1 + ⇡ + "(c̃
µc ))] = u(c1 )
For sufficiently small ", the risk premium is approximately ⇡ = 12 "2 c2 A(c1 + ⇡).
⇡/" tends to zero as " ! 0 and so for any ⌘ > 0 there is some " such that ⇡ < "⌘
and therefore agent 1 would accept the o↵er.
8. (a) Shashi and Vibhuti would be willing to pay up to the same insurance premium
Pa , satisfying
1
1
ln(100) + ln(50) = ln(100 Pa )
2
2
1
1
) Pa = 100 exp
ln(100) + ln(50)
2
2
p
p
= 100
100 ⇥ 50
= 29.3 (3 s.f.)
(b) This is a Pareto efficient allocation since utility functions and beliefs are the
same (and they are risk averse). In this case the group would now be willing
to pay up to an amount Pb which satisfies
1
1
1
ln(100) + ln(75) + ln(50) = ln(100
4
2
4
) Pb = 2 ⇥ 100 1001/4 ⇥ 751/2 ⇥ 501/4
= 54.4 (3 s.f.)
Pb /2)
(c) In this case the group would now be willing to pay up to an amount Pc which
satisfies
1
1
1
1
3
ln(100) + ln(75) + ln(50) = ln(100 Pc /2) + ln(75
4
2
4
4
4
3
2
)(100 Pc /2) ⇥ (75 Pc /2)
100 ⇥ 75 ⇥ 50 = 0
Solving by numerical methods we find Pc = 15.4 (3 s.f.).
Pc /2)
116
S.4. INSURANCE
(d) The implied loading for the insurance products described in parts (a), (b) and
(c) are
a
=
b
=
c
Pa
50/2
1 = 17.2% (3 s.f.)
Pb
1 = 8.70% (3 s.f.)
100/4 + 50/2
Pc
=
1 = 23.3% (3 s.f.)
50/4
We would expect b < a since pooling reduces the risk faced by each brother
(in the sense of SSD) and so the risk premium for the less risky risk will be
lower, and therefore the maximum loading that the brothers are willing to pay
for full insurance will be lower.
We would also expect c > b since, once the brothers have pooled their
risk, the policy in (c) protects only against the worst possible outcome (where
the marginal utility of consumption is the highest) whereas the policy in (b)
protects against all bad outcomes.
It is not immediately obvious that c > a , and I suspect that this will not
necessarily hold for any risk averse utility function.
9. The indi↵erence curves for the risk averse agent must be concave with respect to
the corner where the risk averse agent has 0 wealth in each state. By symmetry,
they must have a gradient of 1 at c1 =c2 for this agent. The indi↵erence curves
for the risk neutral agent are just straight lines with a gradient of 1 - since a
move of consumption from one state to the other does not change expected utility.
Therefore the curves meet at a tangent where the risk averse agent’s utility curves
have gradient -1.
The contract curve is then a line with gradient 1, starting at the corner where the
risk averse agent has 0 wealth in each state. The risk neutral agent will always o↵er
full insurance to the risk averse agent.
𝑧(1)
Consumption of risk neutral agent in state 1
0
𝑧(2)
Consumption of risk averse
agent in state 2
Consumption of risk neutral
agent in state 2
0
0
𝑧(2)
0
Consumption of risk averse agent in state 1
𝑧(1)