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Chapter 20: Duration
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Measure sensitivity of bond price to small changes in interest rate
Average time it takes the bond to return its price
Often used as a risk measure for a bond
Associated with well-known names in economics/finance:
(1882-1970)
(1904-1989, Nobel prize 1972)
(1915-2009, Nobel prize 1970)
Notice the two definitions of duration (highlighted):
Weighted average term to maturity of bond cash flows
Price sensitivity of bond to change in interest rates
Calculating Duration
P = bond price =
Ct = bond payment at time t
r = bond yield to maturity
Example: Bond A
10-year bond with 7% coupon with a face value $1,000
Issued at par
Meaning: price of bond = face value = $1,000
Meaning yield to maturity (IRR) = bond coupon = 7%
Bond cash flows:
Years 1-9: payment of coupon only (7%*1,000)
Year 10: payment of coupon + face value (7% * $1000 + $1,000)
Calculating the duration of Bond A
Example: Bond B
10-year bond with 13% coupon with a face value $1,000
Current bond price = $1,421.41 =
Meaning yield to maturity (IRR) of bond is 7% ( bond coupon)
Years 1-9: payment of coupon only = 13% * $1,000
Year 10: payment of coupon + face value = $13% * $1,000 + $1,000
Calculating the duration of Bond B
Functions in Excel
Excel has a function called Duration
Financial Modeling defines an alternative (VBA) function called DDuration
Excel’s Duration function
Duration and Dduration functions applied to bond A
Duration applied to Bond A: Screens
Bond A has 10-year maturity. I’ve chosen the Settlement (starting) date and the Maturity dates to be 10 years apart. The Frequency refers to the number of payments per year (in our example: 1 payment/year).
Dduration applied to Bond A
Dduration is much simpler to use. TimeFirst refers to the time between today and the first payment.
Dduration is a “home-made” VBA function that has to be inserted into the spreadsheet. See Chapter 20 or Chapter 0 for instructions.
What does duration mean?
Duration can be used to approximate the change in bond price when the discount factor (1 + r) changes by a small amount
Using duration to approximate bond price changes
The discount rate increases, from 7% to 7.7%.
This reduces the bond price from 1,000 to 952.39 (DP = 49.17)
The change in price is approximated by
-Duration*Dr/(1+r)
(cell B17 )
Duration as a function of coupon
Larger coupons mean that the bond returns its price earlier. Since Duration measures the average time that the bondholder has to wait to get his/her money back, larger coupons mean lower duration.
101,0705.4393
Bond price1,000.00<-- =NPV(B2,B5:B14)
Duration7.5152<-- =SUM(C5:C14)
BOND A DURATION
11300.0855
21300.1598
31300.2240
41300.2791
51300.3260
61300.3657
71300.3987
81300.4258
91300.4477
101,1304.0413
Bond price1,421.41<-- =NPV(B2,B5:B14)
Duration6.7535<-- =SUM(C5:C14)
BOND B DURATION
101,0705.4393
Bond price1,000.00<-- =NPV(B2,B5:B14)
Duration7.5152<-- =SUM(C5:C14)
Using the Excel function
and the "home-made" function
Bond A7.5152<-- =DURATION(DATE(1996,12,3),DATE(2006,12,3),7%,B2,1)
7.5152<-- =dduration(A14,7%,B2,1)
BOND A DURATION
Discount rate 7%
Coupon rate 7%
Face value 1,000
Maturity 10
Original price 1,000.00<-- =PV($B$2,B7,-B5*B6)+B6/(1+$B$2)^B7
Duration 7.5152<-- =DURATION(DATE(1996,1,1),DATE(2006,1,1),B5,B2,1)
New discount rate 7.70%
New price 952.39<-- =PV($B$12,B7,-B5*B6)+B6/(1+$B$12)^B7
Change in price
Actual 47.61<-- =B9-B13
Using duration as approximation
- Duration *Price*
49.17<-- =-B10*B9*($B$2-$B$12)/(1+$B$2)
DURATION AS PRICE ELASTICITY
The change in the bond price can be approximated by
- Duration*Price*
Current date21-05-96<-- =DATE(1996,5,21)
Maturity, in years 21
Maturity date21-05-17<-- =DATE(1996+B3,5,21)
YTM 15%Yield to maturity (i.e., discount rate)
Face value 1,000
Duration 9.0110<-- =DURATION(B2,B4,B6,B5,1)
Data table: Effect of coupon on duration
9.0110<-- =B9 , data table header
Bond coupon --> 2%10.7865
EFFECT OF COUPON ON DURATION
579111315171921230%5%10%15%DurationCoupon rateEffect of Coupon on Duration Maturity = 21, YTM = 15.00%
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