Question

4. (Bond yield) A firm issues a bond today with a $1000 face value, an 8% coupon rate, and 25-year maturity.

a. An investor purchases the bond for $900, what is the YTM?

b. Suppose the investor buys the bond for $1,200 instead, what is the YTM?

c. Suppose the investor buys the bond for $1,000, what is the YTM?

d. Suppose the bond in fact pays coupons semi-annually and has a price of $820, what is the YTM?

5. (Interest Rate risk) Bond S has 3 years to maturity. Bond T has 20 years to maturity. Both have 8% coupons paid semi-annually, and are priced at par value.

a. If interest rate suddenly rises by 2%, what is the percentage change in the price of Bond S? Of Bond T?

b. If interest rate suddenly falls by 2%, what is the percentage change in the price of Bond S? Of Bond T.

c. What does this problem tell you about the interest rate risk of longer-term bonds?

Answer 1

And 4: $ 1000 face value bond with 8% coupon and 25 year maturity. The YTM of the bond is simply the discount rate at which the future cash flows of the bond equal the current market price.

(a) If the investor pays $900 for the bond which is paying 8% coupon ($80) for 25 years and maturity value of $1000, then if the YTM is r%, then we have:

900 = 8/(1+r) + 8/(1+r)² + .... (1000+8)/(1+r)²⁵; using excel to solve for r, we get r = 9.02%

Hence YTM = 9.02%

(b) If now the purchase price is $1200, then if the YTM is given by r%, we have:

1200 = 8/(1+r) + 8/(1+r)² + .... (1000+8)/(1+r)²⁵; using excel to solve for r, we get r = 6.38%

Hence YTM = 6.38%

(c) If the investor purchases at $1000, then the YTM will be equal to coupon rate itself i.e. 8%

(d) Now if the bond pays semi annual coupons then it will pay $ 40 per 6 month period (number of periods will be 50 now) and at purchase price of $820, the YTM will be r, as below:

820 = 40/(1+r) + 40/(1+r)² + .... + (1000+40)/(1+r)⁵⁰; solving for r using excel we get r = 4.98% which is the semi annual rate; in annualised terms YTM will be (4.98%*2) = 9.96%

Ans. 5 We have two bonds both priced at par value hence their respective YTM is equal to their coupons (8% paid semi annually). To calculate the sensitivity to change in the interest rates, we will calculate the modified durarion for each of the bonds. The formula for modified duration is = Macaulay Duration / (1 + (YTM/k)) where k are the number of coupons paid per year. In this case k = 2 (since we have semi annual payments)

Macaulay duration = (PV of future cash flows * time period )/market price of bond

The calculations for both the bonds are below:

3 year bond, 8% annual coupon paid semi annually ($40 semi annually) and price = $ 1000. Since they are priced at par value the YTM = coupon rate = 4% semi annually or 8% annually.

We have the Macaulay duration above at 5.4518 which is in semi annual periods and in annual periods it will be (5.4518/2) = 2.7259 years. Now the modified duration = 2.7259/(1+(YTM/k))= 2.7259 / (1+(8%/2)) = 2.6211

So now if the interest rates rise by 2%, the value of this bond will change -(2*2.6211)% = -5.2422% times i.e the bond price will fall by -5.2422%

20 year bond, 8% annual coupon paid semi annually ($40 semi annually) and price = $ 1000. Since they are priced at par value the YTM = coupon rate = 4% semi annually or 8% annually.

The Macaulayduration is 14.1339 in semi annual period terms and in annuallised periods terms it will be = (14.1339/2) = 7.0670

Modified duration = 7.0670 / (1+(8%/2) = 6.7952

Hence for an interest rate increase of 2%, the price of this bond will decline by (6.7952*2)% = - 13.5904%

Part (b) If the interest rates fall by 2%, the price of Bond S will increase now by 5.2422% and Bond T by 13.5904%

Part (c) Thus we see that the interest rate risk is magnified for longer maturity bonds .