Morningstar - Q3 2023 - 50

Strategies
TG
EXHIBIT 3
Annuity Prices for Males and Females at Various Ages With Various
Discount Rates
Gompertz Annuity Pricing Model ($)
Age
r = 2%
55
60
65
70
75
80
85
Male Female
21.12
18.62
16.09
13.57
11.15
8.89
6.87
22.90
20.33
17.64
14.91
12.20
9.63
7.28
Source: Author's calculations.
r = 4%
Male Female
16.33
14.81
13.16
11.41
9.63
7.88
6.23
17.45
15.94
14.26
12.41
10.46
8.49
6.59
Male
13.12
12.15
11.03
9.78
8.44
7.05
5.70
r = 6%
Female
13.85
12.93
11.83
10.54
9.10
7.56
6.01
Male
10.88
10.23
9.44
8.52
7.49
6.38
5.24
r = 8%
Female
11.37
10.79
10.04
9.12
8.03
6.81
5.52
gAnnuities for women are subject to more
interest-rate risk than for men.
A(a, TG
, r ) =
∑
t=1
A(a, TG, r ) =
(1 ) t + ∑ 1 ) t
1
+r
1- exp( -rTG
gThe higher the discount rate, the less the
interest-rate risk. The same is true for bonds.
exp
( m - a )
b -rb, exp
r - exp
(
(
(
EXHIBIT 4
Duration of Annuities for Males and Females at Various Ages With Various
Discount Rates (Years)
r = 2%
Age
55
60
70
75
80
TG
Male Female
14.26
12.66
A(a, TG, r ) =∑ 11.08(1 ) t + ∑ 1 ) t
65
1
t=1
+r
A(a, TG, r ) =
85
1- exp( -rTG 6.95 )
5.46
9.55
8.09
6.72
Source: Author's calculations.
of a bond per a small change in the bond's
interest rate.1
exp
( m - a )
b -rb, exp
r - exp
(
(
(
( y^x ) EXP(-y)
*
, r ) =
)
a - m
b
)
Milevsky generalizes duration to apply to fixed
annuities. Here, I rewrite his equation for duration
using my notation and with the rearrangement
of some terms. Duration is given by:
( x,y ) = GAMMA( x+1 )*
1 - GAMMADIST( y, x+1, 1, TRUE )
/ x
D(a, TG
A(a, TG
∑ t=1
TG
t
(1 + r )t
amax
+ ∑ t=TG
-a
+1
(1+r) A(a, TG, r )
1 Strictly speaking, this is the definition of modified duration and is what the next equation shows. Here, I follow Milevsky and simply call it duration.
, r+ r) - A(a, TG
A(a, TG, r )
50
=
c
q
$12,000A( 65, 0, r ) = $300,000
Morningstar Q3 2023
c (y)
, r )
≈ -D(a, TG, r ) r
)
/ x -
tqt
a
(1 + r )t
(
a - m + TG
b
)
o
Duration is usually measured in years
because it is basically the weighted average
of times to future possible payments.
A key difference between the calculation
of duration for a bond and for an annuity is
that for an annuity, the weights include
survival probabilities.
EXHIBIT 4 presents the duration of each annuity
in EXHIBIT 1 . Here are some observations.
All else equal:
5.55
15.08
13.40
11.72
10.06
8.46
amax
qt
a
t=TG+1 ( +r
r = 4%
-a
+
Male Female
12.20
10.99
9.76
8.53
7.32
6.15
5.07
12.87
11.61
10.31
8.98
7.66
6.37
5.16
Male
10.50
9.59
8.64
7.65
6.65
5.66
4.71
r = 6%
Female
11.03
10.10
9.10
8.05
6.96
5.86
4.80
Male
9.11
8.43
7.69
6.89
6.06
5.22
4.39
r = 8%
Female
9.52
8.85
8.08
7.24
6.34
5.41
4.48
Milevsky's Equation 7 in Life Annuities shows
how to use duration to estimate how
different the price of an annuity would be if
interest rates were to change:
1 - GAMMADIST( y, x+1, 1, TRUE )
/ x
( y^x ) EXP(-y)
*
D(a, TG
, r ) =
A(a, TG
)
∑ t=1
TG
t
(1 + r )t
amax
+ ∑ t=TG
-a
+1
(1+r) A(a, TG, r )
, r+ r) - A(a, TG
A(a, TG, r )
, r )
≈ -D(a, TG, r ) r
$12,000A( 65, 0, r ) = $300,000
tqt
a
(1 + r )t
a - m
b
)
EXHIBIT 5 illustrates how annuity prices
for women vary with age and discount rates.
( x,y ) = GAMMA( x+1 )*
)
/ x -
(
a - m + TG
b
)
qt
a
t=TG+1 ( +r
gThe higher the age, the lower the duration.
This is because of survival probabilities-
the higher the age, the fewer the number of
likely future payments, just as the shorter
term a bond is, the lower its interest-rate risk.
)
+
amax
-a
where Δr is the change in the discount rate.
The expression on the left side of the equation
is the percentage change in the annuity
price if the discount rate changes by Δr. For
example, consider the annuity in EXHIBIT 1
for 55-year-old men. If the discount rate
were 6%, from EXHIBIT 1, we have that its price
is $12.63 and, from EXHIBIT 4 , its duration
is 10.50 years. Now suppose that the discount
rate were to increase by half a percent
(0.50%). From the right side of the above equation,
we have that the percentage change in
the annuity price is -10.50 × 0.50% = -5.25%.
So, the price of the annuity would fall by
about 5.25%. Applying this to the price
of $12.63, we would predict that the price would
go to (1-0.0525) × $12.63 = $11.97.
f = n- qT
n = qT
n
P =
c
q
∑
j=1
n
P(y; c, T) = ∑(1+
j=1
c
q
D(y; c, T) =
∑ n
j=1
j -f
q
y
q
)
f-j
+ 1+
(
( ) (1+
c
q
Take a Pension or a Lump Sum?
Insurance companies are not the only entities
that issue fixed annuities. A defined
pension plan offers retirees a lifetime income,
which is in effect a fixed annuity. However,
-j
P(y; c, T) = P(y; c,
c
q
(
P(y; c, T)
1+
y
q
)
(
P =
D ( )
y
1+ q
P(y; c,
(
1
y
1
(
f-j
=
)
n
q
)
1+ y
q
y
)
f
)
f
c
q
n
q
)(1+
(
P(y; c,
y
q
)
1+
n
q
y
q
)
f-j
)(1+
y
q
)
f
y
q
)
f
P(y; c, T)
y
q
f-j
)
- qT
-j
+ 1+
(
y
q
)
-n
(
1+
y
q
)
f
+ 1+T
(
y
q
)
- qT
Morningstar - Q3 2023

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