Morningstar Magazine - December 2017/January 2018 - 48

Strategies

q( t ) = exp

Investing Over the Lifecycle, Part II
How life insurance and annuities manage
the uncertainty of the time of death.

1 - exp

()
t
b

exp

( )
a-m
b

In this model, I make use of two pricing formulas
for life insurance. First, I calculate the fair price of
t
a-m
aq(permanent
life insurance
policy
that pays
p( tt )) == exp
q ( t - 1)1 -- exp
q ( t ) b exp b
whenever the insured dies. Per dollar of payment,
the price is analogous to the fair price of an
annuity,
butq(t)
with
exp a b- m instead of
q( t ) =∞exp
1 - death
exp t probabilities
A₀ t =
p(
) =∑ qprobabilities:
( t - 1) -t q( t ) b
survival
t = 0 (1 + r)

()

( )

()

( )

∞
p(t)
L₀ == ∑∞
q(t) t
p(
t
)
=
q
(
t
(1
t
=
0
A₀ = ∑ -+1)r) -t q ( t )
t = 0 (1 + r)

E X H I BI TS 1 & 2 show q( t ) with these parameters
of a 30-year-old.

QUANT U

Paul D. Kaplan

In the last issue of Quant U, I presented a
bare-bones lifecycle model of investing that
focuses on spending and savings. In this
issue, I present a model that shows how to deal
with the uncertainties of death using annuities
and life insurance.
Modeling Mortality

While the old saying about the certainty of death
and taxes is true, what is not certain is when
death will occur. Fortunately, there are very good
models of the probability distribution of when
death occurs. In the February/March 2015 issue of
Quant U, I presented the Gompertz mortality
model. According to the Gompertz model, the
probability of a person surviving for at least t more
years is a function that only has three additional
parameters:
1 The person's current age ( a).
2 The mode of the distribution of the age

of death ( m).
3 The dispersion of the age of death
around the mode ( b). The b parameter is similar
to the standard deviation parameter of a
normal distribution.
Given these parameters, the probability of
surviving for at least t more years is as follows:

q( t ) = exp

1 - exp

()
t
b

exp

( )
a-m
b

In this issue of Quant U, I set m to 86 and b
p( t ) = q( t - 1) - q ( t )
to 10.48.

48
A₀

q(t) December/January 2018
= Morningstar
∑
t
t = 0 (1 + r)
∞

The probability of dyingt in t years
is given by
q( t ) = exp 1 - exp b exp a b- m
the difference in the survival
probabilities for t - 1
and t years:

()

( )

Annuities and Life Insurance
∞
q(t)companies provide two basic
Life
A₀ =insurance
∑
+ r) t to help investors manage the
t = 0 (1
insurance
products
uncertainty
of the time of death: annuities
∞
p(t)
L₀ ==life
and
insurance.
∑
t For purposes of this discussion,
t = 0 (1 + r)
by annuities, I mean single-premium immediatepayout annuities. Here is some detail on each
of these1products:
- q(t + 1|t)
L1₀ =
1+r
Annuities
An annuity provides a guaranteed stream of
+ 1)
income for theq(t
annuitant,
thus providing insurance
q(t + 1|t) =
against runningq(t)
out of money
before death.
t
exp a b- m
q( t ) = exp 1 - exp
In this model, I assumeb that the capital
market
provides
a
single
constant
rate
of
return
r.
L
q(t)y
H₀ == ∑the actuarially
Hence,
fair
price
of
an
annuity
per
t
t = 0 (1 + r)
dollar
of
annual
payments
per
year
in
year
p( t ) = q( t - 1) - q ( t )
0 is:

( )

H₀ + F₀ = A₀c + L₀B
∞
q(t)
A₀ = ∑
t
t = 0 (1 + r)
H₀ + F₀ - L₀B
c =
∞
A₀
p(t)
(In
practice,
insurance
companies charge more
L₀ ==
∑
t
t = 0 (1 + r)
than fair
price to earn profits.)
1+r
(Ht-1 - yt-1 )
Ht =
q(t|t
- 1)
Life Insurance
1 - q(t + 1|t)
L1₀ =insurance is for investors who want to be
Life
1+r
sure to leave
we shall see, life
1 + ar bequest. As
(FAmost
APt-1 -before
AIt-1 )an investor
FAt =
t-1 +sense
insurance
makes
the
q(t|t - 1)
has accumulated
wealth to leave the
q(t +enough
1)
q(t + 1|t) =
- t q(t)
desired(1bequest.
+ )
- 1+ y + AI - c - AP - LP )
FRt = (1 + r) (FRt-1
t-1
t-1
t-1
t-1
L
q(t)y
H₀ == ∑
t
exp
1 - exp
t = 0 (1 + r)

()
t
b

exp

∞
p(t)
L₀ == ∑ 1 - q(t +t 1|t)
+ 1)
L1₀ +=t1|t)
= 0 (1 + q(t
q(t
=1 +r)r
q(t)

n=
f=n

Where q1 (-t +q(t1 | t+) 1|t)
means the probability of surviving
-t
L1₀ = (1
) +q(t
+ 1)the investor survived into year
q(t)y
year+t1|t)
+L 1,+ given
that
1
r
q(t
=
-1
H₀ == ∑
t
q(t)
= 0 (1 +
t. It is tgiven
by:r)

p( t ) = q( t - 1) - q ( t )

()

Second,
∞ in each year t, I calculate the fair
- p(t)
q(t
+ 1|t)
L₀
q(t)
∑∞1life
price
insurance
for year t + 1 alone. This is
L1₀====of
A₀
++r)rtt
t∑
= 0 (1 1
= 0 (1 + r)
given tby:

( )
a-m
b

()

+ 1)bt
exp
1 q(t
- exp
L
q(t
+ F₀
1|t)
=q(t)y
H₀
+
=
A₀c
+
L₀B
H₀ == ∑
q(t)
t
t = 0 (1 + r)

exp

( )

P=

P ( y;

a-m
b

Hence,H₀L1 +- qF₀( t-+L₀B
1 | t ) is the probability that the
c =
q(t)y
H₀ +==F₀∑ =dies
A₀in +year
H₀
A₀c
L₀Bt + 1, given that he
investor
t
t = 0 (1 + r)
survived to year t. This is the probability that the
insurance
+F₀ r- L₀B will need to make a
H₀ +1company
-
(H
cHt =+= F₀ in
payout
year
1. t-1 yt-1 )
H₀
=
q(t|tA₀c
A₀- 1)+t +L₀B

D ( y;

P ( y;

The Model and Its Solution

H₀ +11+F₀+r-r L₀B
cFA
Unlike
presented
of)this series,
- AIIt-1
-+
(FA
AP)int-1Part
yt-1
Ht =t == the model
t-1t-1
A₀ I(H
q(t|t
I do notq(t|t
solve--a1)1)
utility maximization problem.
Instead, I assume a constant level of consumption
11+the
(so
long as
I solve for
+r r investor
(H+t-1y -is+
) which
yalive),
Ht =
t-1AI
+ r)-(FR
AP
FR
) t-1 - LPt-1 )
FA
t = (1
t-1(FA
t-1 +AP
t-1
t-1 - cAI-t-1
q(t|t
1)
using the
intertemporal
budget
constraint.
q(t|t- -t 1)
(1 + )
-1
As in Part 1I, +torform the intertemporal budget
- cAI-t-1 ) -
(FAt-1 ++AP
FA
+ r)
AIt-1
AP
FRtt == (1q(t|t
constraint,
I first
value
t-1 + yto
t-1calculate
t-1 - the
t-1 ofLPt-1 )
-(FR
1)need
t
a-m
expcapital
1 -inexp
human
yearb 0. exp
Let: b

()

( )

+ ywhich
- c - APt-1 - LPt-1 )
r) (FR
LFRt== (1
the+last
year
the
t-1 in
t-1 + AI
t-1 investor
earns income
y = the amount of labor income the investor
earns in years 0, 1, 2,... L
n = qTa person can only earn income when
Because
alive, the income level needs to be multiplied by
f = n- qT
survival probabilities in the calculation of human
n
- n 0, we have:
-j
capital. For
human capital
iny year
c
y f
+ 1+ q
P= q
1+ q

) (

(

∑
j =1

n

P ( y; c, T ) =

c
q

f- j

)

- qT

∑ (1 + ) + (1 + )
y
q

j =1

y
q

c
q

P(
n=
f=n
= qc
P=
Pc ( y )
P ( y;
Dc ( y )

D ( y;
Dm (y;

P ( y;(y;
D
m
∆P
P P(

c
q

DV01
= qc



Table of Contents for the Digital Edition of Morningstar Magazine - December 2017/January 2018

Contents
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