Question

Today is 15 May 2022. Hoa has just purchased a Treasury bond with a face value of $100, maturing at par and paying coupon at j2 = 3.4% p.a. The purchase price was $100.065. The maturity date of this bond is 15 May 2024.

For the purposes of this question only, assume that Hoa’s purchase price was not $100.065, and that we don’t know her purchase price. Calculate the duration of this Treasury bond. Assume that Hoa’s pur- chase yield rate was j2 = 5.2% p.a. Give your answer in terms of years, rounded to three decimal places. a. 1.893 0 years b. 1.898 8 years c. 1.946 5 years d. 1.949 4 years e. None of the above+

Answer 1

Bond duration is the weighted average time until a bond's cash flows are received, calculated using discounted cash flows. The duration formula incorporates the present value of cash flows at the given yield rate. A higher discount rate decreases a bond's present value, reflecting the time value of money.

Step-by-step explanation:

To calculate the duration of a Treasury bond, we must understand the bond's cash flows and the present value of these cash flows at the yield rate. The duration measures the bond's sensitivity to interest rate changes and represents the weighted average time until the bond's cash flows are received. In this case, provided there's a Treasury bond maturing on 15 May 2024, with a face value of $100 and paying a coupon at a semi-annual rate of 3.4%. The bond pays out $1.70 every six months until maturity, where it will also pay the final coupon and the face value.

The formula for the present value of a cash flow is: PV = C / (1 + i)^n, where C is the cash flow, i is the discount rate per period, and n is the number of periods until the cash flow is received. We apply this formula to each coupon payment and the face value payment at maturity, then multiply each present value by the time in years until the payment is received, sum these products, and divide by the price of the bond. The yield rate for discounting in this scenario is given as j2 = 5.2% per annum, which we must divide by 2 to adjust for semi-annual compounding, getting a rate of 2.6% per period.

For example, a simple two-year bond with a face value of $3,000 and an interest rate of 8% provides annual interest payments of $240. Using a present value formula, we can discount these payments at the bond's own interest rate or at a different market rate to find the bond's current worth.

If the discount rate is equal to the coupon rate, the bond's present value will be equal to its face value because the bond's payments are exactly compensated by the investment returns of the discount rate. However, if the discount rate is higher (as with the 5.2% yield in the student's question), the present value will be less than the face value, reflecting that the bond's payments are worth less in today’s money. In the case of our hypothetical $3,000 bond, if the discount rate increases to 11%, its present value would decrease accordingly.