# Fixed Income: Asset Liability Management

Submitted By seemamaloo0308
Words 7983
Pages 32
Financial Mathematics for Actuaries
Chapter 8
Bond Management

Learning Objectives
1. Macaulay duration and modiﬁed duration
2. Duration and interest-rate sensitivity
3. Convexity
4. Some rules for duration calculation
5. Asset-liability matching and immunization strategies
6. Target-date immunization and duration matching
7. Redington immunization and full immunization
8. Cases of nonﬂat term structure
2

8.1 Macaulay Duration and Modiﬁed Duration
• Suppose an investor purchases a n-year semiannual coupon bond for
P0 at time 0 and holds it until maturity.
• As the amounts of the payments she receives are diﬀerent at diﬀerent times, one way to summarize the horizon is to consider the weighted average of the time of the cash ﬂows.
• We use the present values of the cash ﬂows (not their nominal values) to compute the weights.
• Consider an investment that generates cash ﬂows of amount Ct at time t = 1, · · · , n, measured in payment periods. Suppose the rate of interest is i per payment period and the initial investment is P .

3

• We denote the present value of Ct by PV(Ct ), which is given by
Ct
.
PV(Ct ) = t (1 + i)

and we have
P =

n
X

(8.1)

PV(Ct ).

(8.2)

t=1

• Using PV(Ct ) as the factor of proportion, we deﬁne the weighted average of the time of the cash ﬂows, denoted by D, as
D =
=

n
X
t=1 n X

t

"

PV(Ct )
P

twt ,

#

(8.3)

t=1

where

PV(Ct ) wt =
.
P
4

(8.4)

P

• As wt ≥ 0 for all t and n wt = 1, wt are properly deﬁned weights t=1 and D is the weighted average of t = 1, · · · , n.
• We call D the Macaulay duration, which measures the average period of the investment.
• The value computed from (8.3) gives the Macaulay duration in terms of the number of payment periods.
• If there are k payments per year and we desire to express the duration
in…...

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