Question

Suppose we see the following prices for zero coupon bonds with maturities ranging from one to six years:

Maturity in Years Bond Price

1 $98.04 2 $95.18 3 $92.18 4 $89.28 5 $86.52 6 $83.90

Note: Each bond has a face value of $100

a) What do you expect the five-year spot rate to be one year from now? Please report the annual rate.

b) What is the yield-to-maturity of a six-year coupon bond that has a face value of $1,000 and an annual coupon rate of 8%? The coupons are paid annually.

c) What is the (Macaulay’s) duration of the bond introduced in part (b)? What’s the economic meaning of duration? How can you interpret its weights?

d) How much would the price of the bond change if the yield increased by 1%?

Answer 1

A.

Let's find the Zero Coupon yield curve from the given information

(For Zero Coupon Bond)

Using above equation we can find Yield Curve

Time to Maturity	Spot Rates
1	2.00%
2	2.50%
3	2.75%
4	2.88%
5	2.94%
6	2.97%

Now from bootstrapping process we can use,

(1+r₅)⁵ = (1+r₁)*(1+₁f₅)

where ₁f₅ is 5 year spot rate 1 year from now

(1+2.94%)⁵ = (1+2%) * (1+x)

x = 13.32%

B.

To calculate YTM we need to first calculate price of the bond from zero coupon rates

Price = 80/(1+2%) + 80/(1+2.5%)^2 + 80/(1+2.75%)^3 + 80/(1+2.88%)^4 + 80/(1+2.94%)^5 + 1080/(1+2.97%)^6

Price = $1275.081

Now to Calculate YTM use the above equation but discount it by a single YTM rate

Therefore, YTM = 2.93%..................Answer

C.

By plugging in all the values we get,

Duration = 5.113

Macaulay Duration tells us average amount of years an investor must invest in bond to break-even that is he/she will get back his/her invested amount.

D.

New yield =2.93%+1% = 3.93%

Now again use above equation to calculate New price of the bond

Price = $1213.842

Threfore, change in price =1275.081 - 1213.842

= $61.24....................Answer