# Essay

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LECTURE 10: DURATION
BONDS III

FIN300 (Matt Marcinkowski, Fall '13)

DURATION
• Consider two bonds with 10 years to maturity and \$1,000 face value (assume annual coupons/compounding): • Bond A: Coupon rate = 10% • Bond B: Coupon rate = 0% (discount paper)
Yield Bond 8% A B \$1,134.20 (+13.4%) \$463.19 (+20%) 10% \$1,000 \$385.54 12% 887.00 (-11.3%) \$321.97 (-16.5%)

FIN300 (Matt Marcinkowski, Fall '13)

DURATION
• Now, consider two bonds with 10 percent coupon rate and \$1,000 face value (assume annual coupons/compounding): • Bond C: Time to maturity = 5 years • Bond D: Time to maturity = 25 years
Yield Bond 8% C D \$1,079.85 (+8%) \$1,213.50 (+21.4%) 10% \$1,000 \$1,000 12% \$927.90 (-7.2%) \$843.14 (-15.7%)

FIN300 (Matt Marcinkowski, Fall '13)

DURATION
• We have observed the following:
• The price of Bond A is less sensitive (in relative terms) to interest rate changes than the price of Bond B. • This is due to the fact that Bond A has a higher coupon rate (10%) than Bond B (0%)

• We have also observed that:
• The price of Bond C is less sensitive (in relative terms) to interest rate changes than the price of Bond D. • This is due to the fact that Bond D has a longer time to maturity than Bond C.

FIN300 (Matt Marcinkowski, Fall '13)

DURATION
• Bond prices are more sensitive in relative terms to interest rate changes if the coupon rate is lower and if the time to maturity is longer. • To compare interest rate sensitivity of bonds with different coupon rates and times to maturity we need to compute duration.

FIN300 (Matt Marcinkowski, Fall '13)

DURATION
• Consider the following stream of cash flows
CF1 CF2 CF3 1 2 3 …

CFT T

• If the discount rate is r, then the duration of these cash flows is given by:
CF  t (1  rt )t Duration  t T1  CFt  (1  r )t t 1
T

 t (1  r ) t 1

T

CFt

t

P

CF2 CFT CF1 2 (1  r

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