Chapter 6
The Risk of Changing
Interest Rates
6-1
Short Horizon Investors
0
1
Maturity
n
Time
P0
P1
y0
y1
P1, the price at Time 1, is important.
6-2
Long Horizon Investors
0
1
2
Maturity
n
Time
P0
C
C
C + PAR
Reinvest
Value at some distant date n is important.
6-3
Bond
Price
Interest Rates
c
c
c  Par
P


2
1  y (1  y)
(1  y)n
6-4
Bond
Price
P0
Actual
Price
Change
P1
y0
y1
Interest Rates
6-5
dP
dy
= derivative of bond price as
yield to maturity changes
= slope of tangent of price
curve
6-6
Duration as an Approximation of
Price Change
Price
Price
Slope of tangent equals
numerator of duration
Actual price change
equals P0  P1
Duration approximation
of price change
equals P0  P´1
P0
P1
P´1
Interest rate
y0
y1
6-7
dP
 Slope of tangent
dy
Move along tangent to approximate price change.
From calculus
 dP 
P    y
 dy 
Divide both sides by price
P  dP/dy 
%P 

y

P  P 
dP/dy
= a measure of sensitivity of bond
P
prices to changes in yields
= a measure of risk
6-8
 dP/dy
is called “modified” duration.
P
Percent
Price  [Duration][Yield Change].
Change
6-9
Macaulay’s Duration (DUR)
Often used by short horizon investors as
a measure of price sensitivity.
DUR= % change in price as yield changes
-[dP /dy](1 + y)
DUR =
.
Price
6-10
DUR =
1c/(1 + y)1 + 2c/(1 + y)2 + … + n(c + PAR)/(1 + y)n
Price
This expression may be interpreted as
the weighted average maturity of a
bond.
6-11
.
Macaulay’s Duration for
Special Types of Bonds
Bond Price Volatilities for Special Types of Bonds
Type of bond
Duration
Zero-coupon
n
Par
Perpetual
(1 y)(PVAn,y )
(1 + y)/y
6-12
Simplified Way of Computing
Macaulay’s Duration
c / P 
DUR  n  n  DUR Par 

y


DUR Par  (1  y)(PVA n,y ).
6-13
Duration for Various Coupons
and Maturities YTM of 8%
Maturity
1
5
10
15
20
25
30
0
1
5
10
15
20
25
30
0.04
1
4.59
8.12
10.62
12.26
13.25
13.77
Coupon
0.06 0.08
1
1
4.44 4.31
7.62 7.25
9.79 9.24
11.23 10.60
12.15 11.53
12.73 12.16
0.10
1
4.20
6.97
8.86
10.18
11.12
11.80
0.12
1
4.11
6.74
8.57
9.88
10.84
11.55
Note: Perpetual bond has duration of 1.08/0.08 = 13.50.
6-14
Bond
Price
High
Risk
Bond
PH,2

PL,2
P0
PL,1
PH,1

y2
y0
y1
Low
Risk Bond
Interest Rates
6-15
Duration versus Maturity
Duration
Discount
1+y
y
1+y
y
Par
Premium
1
.
1
Maturity
6-16
Duration versus Maturity
Duration
(Risk)
Feasible
High
Risk
1+y
y
Discount
1+y
y
Low
Risk
1
Par
Premium
.
1
30
Maturity
6-17
Duration Gap
Bank Balance Sheet
Assets
Liabilities & Equity
Cash
Deposits
Loan
Bonds
Buildings
Equity
DURA
DURL
GAP = DURA – DURL
6-18
Immunization at a Horizon Date
Points in Time
0
n
Buy zero coupon bond
-$P
Receive par value
+$X
The zero coupon strategy
6-19
Points in Time
0
Buy couponbearing
bond
-$P
1
2
...
Receive
coupons
+c
n
Receive par
+ 1 coupon
+c
...
c + Par
Reinvest coupons
Maturity strategy
6-20
Points in Time
0
Buy couponbearing
bond
-$P
1
2
...
Receive
coupons
+ reinvest
+c
+c
...
n
m
Sell original Maturity of
bond +
bond
reinvested
coupons
c
c + Par
Reinvest coupons
Duration strategy
6-21