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Sessio n 5: Duration

A. Duration is the weighted average time for investors to recover the cost of theirinvestment from the cash flows of the investment. The weight attached to eachcash flow is defined as the present value of the cash flow divided by the cost. The

time attached to each cash flow is defined as the time period when the cash flowis received.

B. Answers:

Coupon rate Duration Years to maturity Duration YTM Duration10% 1.8684 2 1.8684 5 1.868412% 1.8473 4 3.4483 7 1.865714% 1.8276 6 4.8175 9 1.863016% 1.8091 8 6.0257 11 1.860218% 1.7918 10 7.1066 13 1.857520% 1.7754 12 8.0833 15 1.8547

Relationship betw een Duration & Coupon Rate: 2-year bond w ith yield of 5%

1.760 0

1.780 0

1.800 0

1.820 0

1.840 0

1.860 0

1.880 0

8 % 10% 12% 14 % 16% 18% 2 0% 2 2%

Coupon Rate

D u r a t i o n

Relat ionship b etw een Durat ion & Matur i ty - 10% coupon bond w ith yie ld of 5%

0.0000

2.0000

4.0000

6.0000

8.0000

10.0000

0 2 4 6 8 10 12 14

Years to Matur i ty

D u r a t i o n

Relations hip bet w een Duration & Yield - 2-year 10% coupo n bon d

1.8500

1.8550

1.8600

1.8650

1.8700

3 5 7 9 11 13 15 17

Yield to Matur i ty

D u r a t i o n

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The results and diagrams show that duration isi) inversely related to coupon rate. The higher the coupon rate, the sooner the

investor will recover the cost of investment and hence the smaller duration.ii) directly related to maturity. The longer the maturity, the longer it takes to

recover the cost of investment and hence the larger duration.iii) inversely related to yield to maturity. If you plot a diagram to illustrate the

relationship between the yield and price of a bond, you should realise that thebond price decreases at a decreasing rate with yield. This relationship impliesthat the larger the yield, the lower the price sensitivity of a bond to interestrate changes. But smaller duration also corresponds to lower bond pricesensitivity. Thus the larger the yield, the smaller the duration.

C. From the session on bond pricing:As Coupon rate and maturity , price sensitivity .

From the previous question:As Coupon rate and maturity , duration .

Thus a direct relationship between duration and price sensitivity is expected.

D. Since cash flows in the form of coupon interests are received prior to maturity, theinvestor does not need to wait until maturity to recover the cost of investment.Hence the duration of a coupon bond is generally smaller than its time tomaturity.

E. The duration is 0.5 years since the remaining cash flows (FV and final coupon

interest) are all received six months later when the bond matures and the presentvalue of the cash flows is equal to the cost of investment. Thus t he duration of acoupon bond is not always smaller than its time to maturity. It can be the same as or lessthan the time to maturity.

F. Yes, this investment is risk-free since you are certain of the cost and terminal value of theinvestment.

G. Assumption 1: The yield curve is flat at the time of bond investment. Thisensures that the expected return from the bond is the same return as its initial yieldto maturity.

Assumption 2: Any unexpected change in rates applies to the entire spectrum ofmaturities, i.e., the yield curve experiences a parallel upward/downward shift.This ensures that the unexpected rise (fall) in coupon reinvestment income offsetsthe unexpected fall (rise) in the selling price of the bond completely.

Assumption 3: Prior to any unexpected change in rates, the bonds durationmatches the intended holding period. Otherwise, price risks will not offsetincome risks completely.