Post on 31-Dec-2015
description
6-3
Long Horizon Investors
P0 C C + PAR
Maturity
Time
Reinvest
Value at some distant date n is important.
10 2
C
n
6-7
Duration as an Approximation of Price Change
Price
Interest rate
Slope of tangent equals numerator of durationActual price change equals P0 P1
Duration approximation of price changeequals P0 P´1
Price
Tangent
P0
P1
P´1
y0 y1
6-8
tangent of Slope dy
dP
Move along tangent to approximate price change.From calculus
Divide both sides by price
ydy
dPP
yP
dy/dP
P
PP%
P
dy/dP= a measure of sensitivity of bond
prices to changes in yields
= a measure of risk
6-10
Macaulay’s Duration (DUR)
Often used by short horizon investors as a measure of price sensitivity.
DUR= % change in price as yield changes
DUR = .-[dP /dy](1 + y)
Price
6-11
This expression may be interpreted as the weighted average maturity of a bond.
DUR = .
1c/(1 + y)1 + 2c/(1 + y)2 + … + n(c + PAR)/(1 + y)n
Price
6-12
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)/y
)PVA)(y1( y,n
6-14
Duration for Various Coupons and Maturities YTM of 8%
Coupon
Maturity 0 0.04 0.06 0.08 0.10 0.121 1 1 1 1 1 1 5 5 4.59 4.44 4.31 4.20 4.11
10 10 8.12 7.62 7.25 6.97 6.7415 15 10.62 9.79 9.24 8.86 8.5720 20 12.26 11.23 10.60 10.18 9.8825 25 13.25 12.15 11.53 11.12 10.84 30 30 13.77 12.73 12.16 11.80 11.55
Note: Perpetual bond has duration of 1.08/0.08 = 13.50.
6-17
(Risk)
Feasible
Low Risk
High Risk
30
Duration versus MaturityDuration
Zero-c
oupon
Discount
ParPremium
1
1Maturity
1 + yy
1 + yy
.
6-18
Duration Gap
Bank Balance Sheet
Assets Liabilities & Equity
Cash Deposits
Loan Bonds
Buildings Equity
DURA DURL
GAP = DURA – DURL
6-19
Immunization at a Horizon Date
Points in Time
0 n
The zero coupon strategy
Buy zero coupon bond-$P
Receive par value+$X
6-20
Points in Time0
Maturity strategy
Receive par + 1 coupon
2 n
Buy coupon-bearingbond
1
-$P +c +c c + Par
Reinvest coupons
. . .
Receive coupons
. . .