Answer-first summary for fast verification
Answer: 1.77 years
The duration is the present-value weighted average time of the cash flows. Using the weights given: \[ D = 0.5(0.080) + 1.0(0.074) + 1.5(0.069) + 2.0(0.776) \] \[ D = 0.040 + 0.074 + 0.1035 + 1.552 = 1.7695 \] So the bond’s duration is approximately **1.77 years**. Because the yield is compounded continuously, this is also the modified duration in this special case.
Author: Manit Arora
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Question 714.1. Duration, modified duration and dollar duration
Learning objectives: Calculate the duration, modified duration, and dollar duration of a bond. Evaluate the limitations of duration and explain how convexity addresses some of them.
$100.00 pays a semi-annual coupon of 18.0% and has a yield (yield to maturity) of 15.0% with continuous compounding. Its price is therefore $104.032 as shown below. Please note that we could infer the price by translating the continuous rate into its semi-annual equivalent which is given by $2 * [\exp(0.15/2) - 1] = 15.577%104.03`2. The final column displays the weight of each cash flow's present value as a proportion of the bond's price.| Face value | $100.00 |
|---|---|
| Semi-annual coupon | 18.0% |
| Yield (continuously comp, CC) | 15.0% |
| Semi-Annual Period (t) | d.f. | Cash flow | |
|---|---|---|---|
| FV | PV | ||
| FV(t) | PV(t) |
| 0.5 | 0.928 | $9.00 | $8.35 | 0.080 |
| 1.0 | 0.861 | $9.00 | $7.75 | 0.074 |
| 1.5 | 0.799 | $9.00 | $7.19 | 0.069 |
| 2.0 | 0.741 | $109.00 | $80.75 | 0.776 |
| Total | | $136.00 | $104.032 | 1.000 |
Which is nearest to the bond's duration (please note that because the yield is continuously compounded, this is the special case where Macaulay duration is equal to modified duration so we don't really need to specify which!)?
A
1.77 years
B
1.85 years
C
1.93 years
D
2.00 years
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