EDUCATION AND EXAMINATION COMMITTEE
OF THE
SOCIETY OF ACTUARIES
FINANCIAL MATHEMATICS STUDY NOTE
USING DURATION AND CONVEXITY TO APPROXIMATE
CHANGE IN PRESENT VALUE
by
Robert Alps, ASA, MAAA
Copyright 2017 by the Society of Actuaries
The Education and Examination Committee provides study notes to persons preparing
for the examinations of the Society of Actuaries. They are intended to acquaint candidates
with some of the theoretical and practical considerations involved in the various subjects.
While varying opinions are presented where appropriate, limits on the length of the
material and other considerations sometimes prevent the inclusion of all possible
opinions. These study notes do not, however, represent any official opinion,
interpretations or endorsement of the Society of Actuaries or its Education and
Examination Committee. The Society is grateful to the authors for their contributions in
preparing the study notes.
FM-24-17
1
Using Duration and Convexity to Approximate Change in Present Value
Robert Alps
February 1, 2017
Contents
1 Introduction ............................................................................................................................. 2
2 Cash Flow Series and Present Value ....................................................................................... 3
3 Macaulay and Modified Duration ............................................................................................ 4
4 First-Order Approximations of Present Value ......................................................................... 5
5 Modified and Macaulay Convexity ......................................................................................... 6
6 Second-Order Approximations of Present Value .................................................................... 7
Appendix A: Derivation of First-Order Macaulay Approximation ................................................ 9
Appendix B: Comparisons of Approximations............................................................................. 10
Appendix C: Demonstration that the First-Order Macaulay Approximation is More Accurate
than the First-Order Modified Approximation ............................................................................. 13
Appendix D: Derivation of Second-Order Macaulay Approximation .......................................... 17
2
1 Introduction
The study of interest theory includes the concept of duration and how it may be used to
approximate the change in the present value of a cash flow series resulting from a small change
in interest rate. The purpose of this study note is to demonstrate a non-linear approximation using
Macaulay duration that is more accurate than the linear approximation using modified duration,
and that a corresponding second-order approximation using Macaulay duration and convexity is
more accurate than the usual second-order approximation using modified duration and convexity.
These Macaulay approximations are found in formulas (4.2) and (6.2) below.
Most textbooks give the following formula using modified duration to approximate the change in
the present value of a cash flow series due to a change in interest rate:
0 0 mod 0
( ) ( ) 1 ( ) ( )P i P i i i D i
.
This approximation uses only the difference in interest rates and two facts about the cash flow
series based on the initial interest rate,
0
i
, to provide an approximation of the present value at a
new interest rate, i. These two facts are (1) the present value of the cash flow series and (2) the
modified duration of the cash flow series. Furthermore, the approximation of the change in
present value is directly proportional to the change in interest rate, facilitating mental
computations. We will refer to this approximation as the first-order modified approximation.
The following approximation, using Macaulay duration, is, under very general conditions, at
least as accurate as the first-order modified approximation and has other pleasant attributes:
mac 0
()
0
0
1
( ) ( )
1
Di
i
P i P i
i
,
We will refer to this approximation as the first-order Macaulay approximation.
The methods discussed in this note are based on the assumption that the timings and amounts of
the cash flow series are unaffected by a small change in interest rate. This assumption is not
always valid. On one hand, in the case of a callable bond, a change in interest rates may trigger
the calling of the bond, thus stopping the flow of future coupons. On the other hand, non-callable
bonds, or payments to retirees in a pension plan are situations where the assumption is usually
valid.
The developments in this note are also predicated on a flat yield curve, that is to say that cash
flows at all future times are discounted to the present using the same interest rate.
This note is not intended to be a complete discussion of duration. In fact, we assume the reader
already is acquainted with the concept of duration, although it is not absolutely required.
3
2 Cash Flow Series and Present Value
A cash flow is a pair,
( , )at
, where
a
is a real number, and
t
is a non-negative real number. Given
a cash flow
( , )at
, the amount of the cash flow is
a
and the time of the cash flow is
t
. Notice
that we have allowed the amount to be negative, although the time is non-negative. A cash flow
series is a sequence (finite or infinite) of cash flows
( , )
kk
at
defined for
kN
, where N is a
subset of the set of non-negative integers.
For the purpose of calculating present values and durations, we introduce a periodic effective
interest rate, i, where the period of time is the same time unit used to measure the times of the
cash flows. For example, if the times are measured in months, then the interest rate, i, is a
monthly effective interest rate. We define P to represent the present value of the cash flow series
as a function of the interest rate as follows.
( ) (1 )
k
t
k
kN
P i a i
(2.1)
If the cash flow series is infinite, the sum in (2.1) may not converge or be finite. In what follows,
we implicitly make the assumption that any sums so represented converge. In the case that N is a
finite set of the form
{1,..., }n
, we may choose to write the sum as
1
(1 )
k
n
t
k
k
ai
.
The following examples show the present value of a 10-year annuity immediate calculated at an
annual effective interest rate of 7.0% and at an annual effective interest rate of interest of 6.5%.
We will use this same cash flow series as an example throughout this note.
Suppose
( , ) (1000, )
kk
a t k
and
1,...,10N
. Then,
10 0.07
(0.07) 1000 7023.5815Pa
(2.2)
and
10 0.065
(0.065) 1000 7188.8302.Pa
(2.3)
We would like to approximate the change in the present value of a cash flow series resulting
from a small change in the interest rate. This is a valuable technique for several reasons. First,
much of actuarial science involves the use of mathematical models of various levels of
complexity and sophistication. To be able to use a model effectively, one needs to understand the
dynamics of the model, i.e., how one variable changes based on a change to a different variable.
The present value formula is such a mathematical model. An actuary should understand how
present value changes when the amounts change, when the times change, and when the interest
rate changes.
4
A second reason is that as a practical matter, actuaries are required sometimes to approximate
changes in present value without being able to use the computer power needed for a complete
calculation. For example, consider an investment actuary meeting with the president of a large
insurance company with a substantial bond portfolio. The president is concerned that interest
rates will increase, which will decrease the value of the bond portfolio. The investment actuary
has recently calculated the value of the bond portfolio using an interest rate of 6.5%. The
president wants to know the value of the bond portfolio if interest rates increase to 6.75% or even
7.0%. Since the value of the bond portfolio is merely the present value of future cash flows,
using the concepts of duration defined below, such approximations can be done quickly using
nothing more than a handheld calculator.
Even when full computing power is available, approximations like the ones in this note are
essential. For example, when doing multi-year projections using Monte Carlo techniques for
interest rate scenarios, thousands of present value calculations may be needed. It is not feasible
to do full calculations and approximations make it possible for such projections to be done.
3 Macaulay and Modified Duration
The definition of Macaulay duration is
mac
(1 ) (1 )
( ) .
()
(1 )
kk
k
tt
k k k k
k N k N
t
k
kN
t a i t a i
Di
Pi
ai
(3.1)
The definition of modified duration is
1
mod
(1 )
()
( ) .
( ) ( )
k
t
kk
kN
t a i
Pi
Di
P i P i
(3.2)
Macaulay duration is the weighted average of the times of the cash flows, where the weights are
the present values of the cash flows. Modified duration is the negative derivative of the present-
value function with respect to the effective interest rate, and expressed as a fraction of the
present value. Therefore it is expected that modified duration gives us information about the rate
of change of the present-value function as the interest rate changes. We note the following
relation between the two notions of duration:
mac
mod
()
( ) .
1
Di
Di
i
(3.3)
Because both definitions of duration involve division by P(i), we will assume for the remainder
of this note that
( ) 0.Pi
(3.4)
5
As an example of Macaulay and modified duration, we first consider a cash flow series that
consists of a single flow,
11
( , )at
. For this situation, we have
1
mac 1 mod
( ) and ( ) .
1
t
D i t D i
i
(3.5)
Next, using the 10-year immediate annuity and setting
0.07i
we have
10
1
mac
10
1
( 1000 1.07 )
34739.1332
(0.07) 4.9460710.
7023.5815
(1000 1.07 )
k
k
k
k
k
D
(3.6)
Alternatively, for this example, we can see that
mac
10 0.07 10 0.07
10 0.07 10 0.07
1000 ( ) ( )
34.7391332
(0.07) 4.9460710.
1000 7.0235815
Ia Ia
D
aa
(3.7)
Also, for this example, we have
mac
mod
(0.07)
4.9460710
(0.07) 4.6224963.
1.07 1.07
D
D
(3.8)
4 First-Order Approximations of Present Value
The first-order modified approximation of the present-value function is
0 0 mod 0
( ) ( ) (1 ( ) ( )).P i P i i i D i
(4.1)
This approximation is presented on Page 369 in [1], on Page 396 in [2], on Page 455 in [3], and
on Page 216 in [4]. It is derived using the first-order Taylor approximation for
()Pi
about
0
i
.
The first-order Macaulay approximation of the present-value function is
mac 0
()
0
0
1
( ) ( ) .
1
Di
i
P i P i
i
(4.2)
The derivation of this approximation is given in Appendix A.
Using the 10-year annuity immediate, we calculate the first-order modified approximation for
P(0.065) and compare it to the true present value. The result is
mod
(0.065) (0.07) 1 (0.065 0.07) (0.07)
7023.5815 (1 0.005 4.6224963) 7185.9139.
P P D
(4.3)
Because P(0.065) = 7188.8302, the percent error is –0.0406%.
Next we calculate the corresponding values for the first-order Macaulay approximation:
6
4.9460710
1.07
(0.065) 7023.5815 7188.1938.
1.065
P
(4.4)
The percent error is –0.0089%.
Thus, the error from Macaulay approximation is about 22% of the error from the modified
approximation.
It is worthwhile noting that in the case where the cash flow series consists of a single cash flow,
the first-order Macaulay approximation gives the exact present value, while the first-order
modified approximation does not.
In Appendix B, we have compared the two approximations over 180 scenarios. At worst, the
error from the first-order Macaulay approximation is 39% of the error from the first-order
modified approximation. At best, the error from the first-order Macaulay approximation is 14%
of the error from the first-order modified approximation.
In Appendix C, it is shown that the first-order Macaulay approximation is more accurate than the
first-order modified approximation whenever the cash flow amounts are positive. When this
condition is not met, it is possible for the first-order modified approximation to be more accurate
than the first-order Macaulay approximation.
5 Modified and Macaulay Convexity
The definition of modified convexity is:
2
mod
( 1) (1 )
( ) ( ) / ( ) .
()
k
t
k k k
kN
t t a i
C i P i P i
Pi
(5.1)
The definition of Macaulay convexity is:
22
mac
(1 ) (1 )
( ) .
()
(1 )
kk
k
tt
k k k k
k N k N
t
k
kN
t a i t a i
Ci
Pi
ai
(5.2)
Thus, Macaulay convexity is the weighted average of the squares of the times of the cash flows,
where the weights are the present values of the cash flows. The following relationship is easily
derived:
mac mac
mod
2
( ) ( )
( ) .
(1 )
C i D i
Ci
i
(5.3)
As an example of Macaulay and modified convexity, we first consider a cash flow series that
consists of a single cash flow,
11
( , )at
. For this situation, we have
7
2
11
mac 1 mod
2
( 1)
( ) and ( ) .
(1 )
tt
C i t C i
i
(5.4)
Using the 10-year annuity example from Sections 2 and 3, we can see that
10
2
1
mac
10
1
(1000 1.07 )
228,451.20
(0.07) 32.526311
7,023.5815
(1000 1.07 )
k
k
k
k
k
C
(5.5)
and
mac mac
mod
2
(.07) (0.07)
(0.07) 32.729830.
1.07
CD
C
(5.6)
6 Second-Order Approximations of Present Value
The second-order modified approximation of the present value function is:
2
0
0 0 mod 0 mod 0
()
( ) ( ) 1 ( ) ( ) ( ) .
2
ii
P i P i i i D i C i
(6.1)
This approximation can be found in most of the texts.
Letting
2
mac 0 mac 0
( ) and ( )T D i Q C i T
, the second-order Macaulay approximation of
the present value function is:
2
00
0
0
1
( ) ( ) 1 .
1 1 2
T
i i i
Q
P i P i
ii
(6.2)
A derivation of this formula can be found in Appendix D.
To illustrate these two approximations, we will apply them to the 10-year annuity example.
Using the convexity values from Section 5 and the duration values from Section 3, we can
calculate the two second-order approximations of P(0.065), with the following results.
First, for the second-order modified approximation, we get
2
mod mod
2
(0.065 0.07)
(0.065) (0.07) 1 (0.065 0.07) (0.07) (0.07)
2
(0.005)
7023.5815 1 0.005 4.6224963 32.729830
2
7023.5815 1 0.0231125 0.0004091
7188.7874.
P P D C
(6.3)
8
Because P(0.065) = 7188.8302, the percent error is –0.00060%.
Next we calculate the second-order Macaulay approximation:
4.9460719
2
2
1 0.07
(0.065) (0.07)
1 .065
0.065 0.07 32.526311 4.9460710
1
1 .07 2
7023.5815 1.02343708 1.0000881
7188.8266.
PP
(6.4)
Here the percent error is –0.00005%. For this example, the error for the second-order Macaulay
approximation is less than 10% of the error of the second-order modified approximation.
Table (B.3) of Appendix B shows that the error from the second-order Macaulay approximation
is less than 20% of the error from the second-order modified approximation over 180 different
scenarios.
As a final observation about the second-order methods, we note that the Macaulay approximation
gives the exact present value at the new interest rate in the case of a single cash flow, because in
this case, using (3.5) and (5.4),
22
11
0Q t t
.
9
Appendix A: Derivation of First-Order Macaulay Approximation
To derive this approximation of P(i) we reason as follows. For each time T, we define a function
T
V
to represent the current value of the given cash flow series at time T:
( ) ( ) (1 ) .
T
T
V i P i i
(A.1)
Note that if we set
0T
in (A.1), we obtain the present-value function. It is important to
understand that each function
T
V
is a function of a single real variable, which we think of as
representing an effective rate of interest. Below, when we take the derivative of one of these
functions, it is with respect to that variable.
For the moment, let us consider a specific interest rate,
0
i
, and consider current-value functions
for various values of T. If T is small enough, for example before the time of the first payment,
then a small increase in the interest rate will decrease the current value, i.e.,
0
( ) 0
T
Vi
.
However, if T is large enough, then a small increase in the interest rate will increase the current
value, i.e.,
0
( ) 0
T
Vi
. This suggests that there is some value of T such that the function
T
V
is
neither increasing nor decreasing at
0
i
. That is, for this value of T, we would have
0
( ) 0
T
Vi
.
We solve for this value:
1
0 0 0 0 0
0 ( ) ( ) (1 ) ( ) (1 )
TT
T
V i P i T i P i i
.
Thus,
0 0 0 0
mod 0 0 mac 0
1
0
00
( ) (1 ) ( ) (1 )
( ) (1 ) ( )
()
( ) (1 )
T
T
P i i P i i
T D i i D i
Pi
P i i
.
It is easily checked that, in fact,
0
0 ( )
T
Vi
if
mac 0
()T D i
. Let us now define the function V,
with no subscript, as
T
V
with
mac 0
()T D i
. Thus,
mac 0
()
( ) ( ) (1 )
Di
V i P i i
and
0
( ) 0Vi
. By applying the first-order Taylor approximation to V(i) about
0
i
we see
mac 0 mac 0
mac 0
0 0 0 0
( ) ( )
00
()
0
0
( ) ( ) ( ) ( ) ( )
( ) (1 ) ( ) (1 )
1
( ) ( ) .
1
D i D i
Di
V i V i i i V i V i
P i i P i i
i
P i P i
i
(A.2)
10
Appendix B: Comparisons of Approximations
The percent error has been analyzed for both the modified duration approximation and the
Macaulay duration approximation under a variety of scenarios. We have considered nine
different cash flow series, each with up to 25 cash flows at times 1 through 25. The series are
defined as follows.
(B.1) Table of Cash Flow Series Scenarios
Time
Level-5
Level-10
Level-15
Level-20
Level-25
Increasing
Decreasing
Inc/Dec
Dec/Inc
1
1,000
1,000
1,000
1,000
1,000
1,000
26,000
1,000
26,000
2
1,000
1,000
1,000
1,000
1,000
2,000
25,000
2,000
25,000
3
1,000
1,000
1,000
1,000
1,000
3,000
24,000
3,000
24,000
4
1,000
1,000
1,000
1,000
1,000
4,000
23,000
4,000
23,000
5
1,000
1,000
1,000
1,000
1,000
5,000
22,000
5,000
22,000
6
0
1,000
1,000
1,000
1,000
6,000
21,000
6,000
21,000
7
0
1,000
1,000
1,000
1,000
7,000
20,000
7,000
20,000
8
0
1,000
1,000
1,000
1,000
8,000
19,000
8,000
19,000
9
0
1,000
1,000
1,000
1,000
9,000
18,000
9,000
18,000
10
0
1,000
1,000
1,000
1,000
10,000
17,000
10,000
17,000
11
0
0
1,000
1,000
1,000
11,000
16,000
11,000
16,000
12
0
0
1,000
1,000
1,000
12,000
15,000
12,000
15,000
13
0
0
1,000
1,000
1,000
13,000
14,000
13,000
14,000
14
0
0
1,000
1,000
1,000
14,000
13,000
12,000
15,000
15
0
0
1,000
1,000
1,000
15,000
12,000
11,000
16,000
16
0
0
0
1,000
1,000
16,000
11,000
10,000
17,000
17
0
0
0
1,000
1,000
17,000
10,000
9,000
18,000
18
0
0
0
1,000
1,000
18,000
9,000
8,000
19,000
19
0
0
0
1,000
1,000
19,000
8,000
7,000
20,000
20
0
0
0
1,000
1,000
20,000
7,000
6,000
21,000
21
0
0
0
0
1,000
21,000
6,000
5,000
22,000
22
0
0
0
0
1,000
22,000
5,000
4,000
23,000
23
0
0
0
0
1,000
23,000
4,000
3,000
24,000
24
0
0
0
0
1,000
24,000
3,000
2,000
25,000
25
0
0
0
0
1,000
25,000
2,000
1,000
26,000
11
For each cash flow series, the present value was approximated at 20 interest rates that differed
from the initial interest rate of 7.0% by multiples of 0.2% between 5.0% and 9.0%. The percent
errors were averaged using a subjectively selected weighting of
0
e
ii
to give greater value to
rates nearer the initial rate.
(B.2) Table of average weighted percent errors for first-order approximations
Cash Flow
Series
1st-order
modified
1st-order
Macaulay
Macaulay err/
modified err
Level-5
0.0820%
0.0125%
15.24%
Level-10
0.2351%
0.0506%
21.52%
Level-15
0.4402%
0.1112%
25.26%
Level-20
0.6765%
0.1905%
28.16%
Level-25
0.9266%
0.2837%
30.62%
Increasing
1.6473%
0.2601%
15.79%
Decreasing
0.5313%
0.1776%
33.43%
Inc/Dec
1.0181%
0.1689%
16.59%
Dec/Inc
0.8984%
0.3138%
34.93%
Table (B.2) shows that the first-order Macaulay approximation is consistently markedly better
than the first-order modified approximation. Overall, the error from the Macaulay approximation
is about 1/3 or less of the error from the modified approximation.
(B.3) Table of weighted-average percent errors for second-order approximations
Cash Flow
Series
2
nd
-Order
modified
2
nd
-Order
Macaulay
Macaulay err/
modified err
Level-5
0.0023%
0.0002%
8.70%
Level-10
0.0107%
0.0009%
8.41%
Level-15
0.0272%
0.0024%
8.82%
Level-20
0.0522%
0.0051%
9.77%
Level-25
0.0851%
0.0095%
11.16%
Increasing
0.1666%
0.0028%
1.68%
Decreasing
0.0405%
0.0071%
17.53%
Inc/Dec
0.0844%
0.0034%
4.03%
Dec/Inc
0.0853%
0.0122%
14.30%
12
Table (B.3) shows that the second-order Macaulay approximation is consistently markedly better
than the second-order modified approximation. Overall, the error from the Macaulay
approximation is about 1/5 or less of the error from the modified approximation.
We can use the second-order results to measure the success of the Macaulay first-order
approximation. For the Level-5 cash flow series, the difference between the first-order modified
average error and the second-order modified average error is 0.0820% - 0.0023%, or 0.0797%.
The difference between the first-order modified average error and the first-order Macaulay
average error is 0.0820% – 0.0125%, or 0.0695%. Thus the first-order Macaulay approximation
takes you 87% of the way from the first-order modified to the second-order modified
approximation. This percentage varies between 72% and 94% over the nine different cash flow
series studied.
13
Appendix C: Demonstration that the First-Order Macaulay
Approximation is More Accurate than the First-Order Modified
Approximation
We assume in this appendix that the cash flow amounts are positive. We first establish some
notation. We are given an initial periodic effective interest rate,
0
i
. For our given cash flow
series, we set
mac 0
0
1 0 0 mod 0 0
0
0
2 0 0
0
()
( ) ( ) (1 ( ) ( )) ( ) 1
1
1
1
( ) ( ) ( )
11
T
T
T D i
ii
F i P i i i D i P i T
i
i
i
F i P i P i
ii
so that
1
()Fi
is the first-order modified approximation to P(i), and
2
()Fi
is the first-order
Macaulay approximation to P(i).
In Theorem (C.5) below, we show that, the first-order modified approximation is less than or
equal to the first-order Macaulay approximation which is less than or equal to the actual present
value. Thus the first-order Macaulay approximation is always a better approximation.
We begin by showing that the first-order modified approximation is less than or equal to the
first-order Macaulay approximation.
(C.1) Theorem:
12
( ) ( )F i F i
Proof:
We have
1
20
00
2
20
2
0
0
11
( ) ( )
11
11
( ) ( 1) ( ) 0.
1
1
T
T
i
F i T P i
ii
i
F i T T P i
i
i
14
By Taylor’s Theorem with remainder there is j between
0
i
and i such that
2
0
2 2 0 0 2 0 2
2 0 0 2 0
0
00
0
1
()
( ) ( ) ( ) ( ) ( )
2
( ) ( ) ( )
()
( ) ( )
1
( ).
ii
F i F i i i F i F j
F i i i F i
T P i
P i i i
i
Fi
Theorems (C.2) through (C.5) are devoted to showing that the first-order Macaulay
approximation is less than or equal to the present value. While Theorems (C.2) through (C.4) are
important in their own right, the reader may wish to think of these as Lemmas. For these
theorems, our argument is simplified by using a continuously compounded rate of interest,
, as
the independent variable. Thus we will define the present value function, Macaulay duration,
Macaulay convexity, and the first-order Macaulay approximation in terms of this variable. We
begin with an initial
00
ln(1 )i
and make the following definitions.
0
mac
2
mac
()
20
( ) e 1 e ;
e
( ) e 1 ;
()
e
( ) e 1 ; and
()
( ) e 1 ( ) e .
k
k
k
t
k
kN
t
kk
kN
t
kk
kN
T
P P a
ta
DD
P
ta
CC
P
F F P
(C.2) Theorem: If
()DD
and
()CC
then
2
0CD
.
Proof:
For each
kN
, set
e
()
k
t
k
k
a
q
P
, and note that
0
k
q
and
1
k
kN
q
and
()
kk
kN
D t q
and
2
()
kk
kN
C t q
. Then
15
22
22
22
2
21
2
2
0.
k k k k k
k N k N k N
k k k
kN
kk
kN
C D C D D D
t q D t q D q
t D t D q
t D q
(C.3) Theorem:
( ) 0D
Proof: We first note that
( ) e
()
()
()
()
( ) .
()
k
t
kk
kN
P t a
P
D
P
P
C
P
We can now see that
2
( ) ( ) ( ) ( )
()
()
( ) ( ) ( )
0.
P P P P
D
P
C D D
Theorem (C.3) shows that Macaulay duration decreases as the interest rate increases.
(C.4) Theorem:
( ) ( )FP
Proof: Set
( ) ( ) e
T
VP
. Then
( ) ( ) e ( ) e ( ) e ( ) .
T T T
V P T P P T D
Using Taylor’s Theorem with Remainder, there is j between
and
0
such that
00
( ) ( ) ( ) ( )V V V j
and hence
0
0
00
0 0 0
( ) e ( ) e ( ) ( ) e ( ( ))
( ) e ( ) ( ) e ( ( ) ( )).
T
T T j
T
Tj
P P P j T D j
P P j D D j
16
If
0
then
0
j
, and because of (C.3),
0
( ) ( )D j D
, and
00
( ( ) ( )) 0.D D j
Similarly, if
0
, then
00
( ( ) ( )) 0D D j
.
Thus
0
0
0
()
0
( ) e ( ) e
( ) ( ) e
( ).
T
T
T
PP
PP
F
(C.5) Theorem:
12
( ) ( ) ( )F i F i P i
Proof:
12
( ) ( )
(ln(1 ))
(ln(1 ))
( ).
F i F i
Fi
Pi
Pi
17
Appendix D: Derivation of Second-Order Macaulay Approximation
As in Appendix A, we let
( ) ( ) (1 )
T
V i P i i
where
mac 0
()T D i
, and we remember that
0
( ) 0Vi
. We will use a second-order Taylor approximation for V, and therefore we compute
the first and second derivatives of V:
1
( ) ( ) (1 ) ( ) (1 )
TT
V i P i T i P i i
(D.1)
and
21
22
22
mod mod
2
( ) ( ) ( 1) (1 ) 2 ( ) (1 ) ( ) (1 )
( ) ( )
( ) (1 ) ( 1) 2 (1 ) (1 )
( ) ( )
( ) (1 ) ( 1) 2 ( ) (1 ) ( ) (1 )
( ) (1 ) (
T T T
T
T
T
V i P i T T i P i T i P i i
P i P i
P i i T T T i i
P i P i
P i i T T D i T i C i i
P i i T
mac mac mac
1) 2 ( ) ( ) ( ) .T D i T C i D i
(D.2)
In particular, for
0
ii
, we have
2
0 0 0 mac 0 mac 0 mac 0
2
0 0 mac 0
22
0 0 mac 0
( ) ( ) (1 ) ( 1) 2 ( ) ( ) ( )
( ) (1 ) ( 1) 2 ( )
( ) (1 ) ( ) .
T
T
T
V i P i i T T D i T C i D i
P i i T T T T C i T
P i i C i T
(D.3)
We now use the second-order Taylor approximation for V(i) about
0
i
:
2
0
0 0 0 0
()
( ) ( ) ( ) ( ) ( )
2
ii
V i V i i i V i V i
This translates to
2
22
0
0 0 0 0 mac 0
22
0 mac 0
00
2
0
()
( ) (1 ) ( ) (1 ) 0 ( ) (1 ) ( )
2
( ) ( )
( ) (1 ) 1
(1 ) 2
T T T
T
ii
P i i P i i P i i C i T
i i C i T
P i i
i
from which we obtain the second-order Macaulay approximation:
22
0 0 mac 0
0
2
0
1 ( ) ( )
( ) ( ) 1 .
1 (1 ) 2
T
i i i C i T
P i P i
ii
(D.4)
18
Acknowledgements
The author thanks Steve Kossman, David Cummings, and Stephen Meskin for their suggestions
during the preparation of this note and for their review of drafts containing various errors. Of
course, any errors that remain in the note are the responsibility of the author.
References
[1] Broverman, Samuel A., Mathematics of Investment and Credit, Sixth Edition, Actex
Publications, Inc., 2015
[2] Vaaler, Leslie Jane Federer and Daniel, James W., Mathematical Interest Theory, Second
Edition, Pearson Prentice Hall, 2009
[3] Kellison, Stephen G., The Theory of Interest, Third Edition, McGraw Hill Irwin, 2009
[4] Ruckman, Chris and Francis, Joe, Financial Mathematics, Second Edition, BPP Professional
Education, Inc., 2005
