Duration

1
Duration

• Measuring Interest Rate Sensitivity

2
Measuring Interest Rate Risk

• We know
• An increase in interest rates causes bond prices
  to fall, and a decrease in interest rates causes
  bond prices to rise.
• We also know that longer maturity debt securities
  tend to be more volatile in price.
• For a given change in interest rates, the price
  of a longer term bond generally changes more than
  the price of a shorter term bond.

3
Measuring Interest Rate Risk

• Two bonds with the same term to maturity do not
  have the same interest-rate risk.
• A 10 year zero coupon bond makes all of its
  payments at the end of the term.
• A 10 year coupon bond makes payments before the
  maturity date.
• Which bond has the highest interest-rate risk?

4
Interest Rate Risk Problem

• Calculate the rate of capital gain or loss on a
  ten year zero coupon bond for which the interest
  rate has increased from 10 to 20. The bond has
  a face value of 1000.
• Capital gain (Pt1 - Pt) / P t
• - 49.7 (193.81 - 385.54)/385.54

5
Interest Rate Risk Problem

• The rate of capital gain or loss on a ten year
  coupon bond that has a face value of 1000 for
  which the interest rate has increased from 10 to
  20 is -40.3.
• The interest rate risk on a ten year coupon bond
  is less than the interest rate risk on a 10 year
  zero coupon bond.
• Why?

6
Varying Coupon Rates Coupon Effect

• A security promising lower annual coupon payments
  behaves as though it has a longer maturity even
  if it is due to mature on the same date as a
  security carrying a higher coupon rate.
• Investors must wait longer to realize a
  substantial return.
• The farther in the future cash payments are to be
  received, the more sensitive the present value of
  the stream of payments to changes in interest
  rates.

7
Coupon Effect Definition

• When interest rates rise, the prices of low
  coupon securities tend to fall faster than the
  prices of high coupon securities.
• Similarly, when interest rates decline, the
  prices of low coupon rate securities tend to rise
  faster than the prices of high coupon rate
  securities.
• Therefore, the potential for capital gains and
  capital losses is greater for low coupon
  securities.

8
Duration Introduction

• Knowledge of the impact of varying coupon rates
  on security price volatility led to the
  development of a new index of maturity other than
  straight calendar time.
• The new measure permits analysts to construct a
  linear relationship between term to maturity and
  security price volatility, regardless of
  differing coupon rates.

9
Duration
Present value of interest and principal payments
from a security weighted by the timing of those
payments
n
CPt (1 i)t
S
t
t1
D


n
CPt (1 i)t
Present value of the securitys promised stream
of interest and principal payments
S
t1
10
Duration
CP represents the expected payment of principal
and interest income. t represents the time
period in which each payment is to
be received. And i is the securitys yield to
maturity.
n
CPt (1 i)t
S
t
t1
D

n
CPt (1 i)t
S
t1
11
Duration Example
Assume there is an investor who is interested in
buying a 1,000 par value bond that has a term
to maturity of 10 years, a 10 percent annual
coupon rate, and a 10 percent yield to maturity
based on its current price.
100(1) 100(2) 100(10)
1000(10) (1.10) (1.10)2
(1.10)10 (1.10)10
...




6758.9 1000
D



100 100 100
1000 (1.10) (1.10)2 (1.10)10
(1.10)10
...




6.758 years
12
Duration Zero Coupon Bonds Example

• To get the effective maturity of a set of zero
  coupon bonds we must
• Sum the effective maturity of each zero coupon
  bond, weighting it by the percentage of the total
  value of all the bonds that it represents.
• The duration of the set is the weighted average
  of the effective maturities of the individual
  zero coupon bonds, with the weights equaling the
  proportion of the total value represented bye
  each zero coupon bond.

13
Duration ExampleYield 10
Year Cash Payments Present Value
Weights Weighted of Cash
Payments of Total PV Maturity 1
100 90.01 0.09001 0.09091 2
100 82.64 0.08264 0.16528 3
100 75.13 0.07513 0.22539 4
100 68.30 0.06830 0.27320 5
100 62.09 0.06209 0.31045 6
100 56.44 0.0.644 0.33864 7
100 51.32 0.05132 0.35924 8
100 46.65 0.04665 0.37320 9
100 42.41 0.04241 0.38550 10
100 38.55 0.03855 0.38550 10 1000
385.54 0.38554
3.85500 Total 1000.00
1.00 6.75850
14
Zero Coupon Bond Example Steps

• Calculate the present value of each of the zero
  coupon bonds when the interest rate is 10
  (column 3).
• Divide each of these present values by 1000 (the
  total present value of the set of zero-coupon
  bonds) to get the percentage of the total value
  of all the bonds that each bond represents. Note
  that the sum equals 1 (column 4).
• Calculate the weighted maturities (column 5) by
  multiplying column 1 by column 4.
• Get the effective maturity of the set of bonds by
  adding column 5.

15
Duration Another Example Yield 20
Year Cash Payments Present Value
Weights Weighted
of Cash Payments of Total
PV Maturity 1 100 83.33
0.14348 0.14348 2 100 69.44
0.11957 0.23914 3 100 57.87
0.09650 0.29895 4 100 48.23
0.08305 0.33220 5 100 40.19
0.06920 0.34600 6 100 33.49
0.05767 0.34602 7 100 27.91
0.04806 0.33642 8 100 23.26
0.04005 0.32040 9 100 19.38
0.03337 0.30033 10 100 16.15
0.02781 0.27810 10 1000 161.15
0.27808 2.78100 Total
580.76 1.00 5.72204
16
Things to Notice

• When the yield to maturity rises, the duration of
  the coupon bond falls.
• The higher the coupon rate on the bond, the
  shorter the duration of the bond.
• When the maturity of a bond lengthens, the
  duration rises as well.
• Duration is additive the duration of a portfolio
  of securities is the weighted-average of the
  durations of the individual securities, with the
  weights equaling the proportion of the portfolio
  invested in each.

17
Duration is Additive

• The duration of a portfolio of securities is the
  weighted average of the durations of the
  individual securities with the weights reflecting
  the proportion invested in each.
• Example Let 25 of a portfolio be invested in a
  bond with a duration of 5 and let 75 of the
  portfolio be invested in a bond with a duration
  of 10.
• Dp (0.25 x 5) (0.75 x 10) 8.75 years

18
Duration and Interest Rate Risk

• Because duration is related in linear fashion to
  the price volatility of a security, there is an
  approximate relationship between changes in
  interest rates and percentage changes in security
  prices.

19
Duration and Interest Rate Risk
Change in the price of a debt security
/\ i 1 i

-D x
x
100
D duration /\ i change in interest rates
Change in the price of a debt security
0.02 1 0.10
x
100

-11.91

-6.55 x
An increase in interest rates of 2 causes a
decline in the bonds price of approximately 12.