Question

A 2 -year bond with \(\$ 100\) face value pays a \(\$ 6\) coupon each quarter and has \(11 \%\) yield. Compute the duration.

Short answer

The duration is 6.740 quarters.

Step-by-step solution

- Determine the Cash Flows

Identify the cash flows from the bond. This bond pays a \(6 coupon each quarter and has a face value of \)100. Therefore, the cash flows for each quarter are: \(6, \)6, \(6, \)6, \(6, \)6, \(6, \)106.

- Convert Yield to Quarterly Rate

The bond’s yield is 11% per year, so the quarterly yield rate is \[ r = \frac{11\text{ \% }}{4} = 2.75\text{ \% } = 0.0275 \]

- Calculate the Present Value of Each Cash Flow

We need to discount each cash flow to the present value using the formula for present value (PV): \( PV = \frac{CF}{(1 + r)^t} \), where CF is the cash flow, r is the quarterly yield rate, and t is the time period. Calculate this for each cash flow: \[ PV_1 = \frac{6}{(1 + 0.0275)^1} \approx 5.84 \ PV_2 = \frac{6}{(1 + 0.0275)^2} \approx 5.68 \ PV_3 = \frac{6}{(1 + 0.0275)^3} \approx 5.53 \ PV_4 = \frac{6}{(1 + 0.0275)^4} \approx 5.38 \ PV_5 = \frac{6}{(1 + 0.0275)^5} \approx 5.24 \ PV_6 = \frac{6}{(1 + 0.0275)^6} \approx 5.10 \ PV_7 = \frac{6}{(1 + 0.0275)^7} \approx 4.97 \ PV_8 = \frac{106}{(1 + 0.0275)^8} \approx 84.66 \]

- Compute the Weight of Each Cash Flow

The weight of each cash flow is the present value divided by the bond's price. The price of the bond is the sum of the present values: \[ P = PV_1 + PV_2 + PV_3 + PV_4 + PV_5 + PV_6 + PV_7 + PV_8 = 5.84 + 5.68 + 5.53 + 5.38 + 5.24 + 5.10 + 4.97 + 84.66 = 122.40 \ w_1 = \frac{5.84}{122.40} \approx 0.048 \ w_2 = \frac{5.68}{122.40} \approx 0.046 \ w_3 = \frac{5.53}{122.40} \approx 0.045 \ w_4 = \frac{5.38}{122.40} \approx 0.044 \ w_5 = \frac{5.24}{122.40} \approx 0.043 \ w_6 = \frac{5.10}{122.40} \approx 0.042 \ w_7 = \frac{4.97}{122.40} \approx 0.041 \ w_8 = \frac{84.66}{122.40} \approx 0.692 \]

- Compute the Duration

Multiply each weight by the time period for the corresponding cash flow, and sum these products: \[ D = \sum_{i=1}^n t_i \times w_i = (1 \times 0.048) + (2 \times 0.046) + (3 \times 0.045) + (4 \times 0.044) + (5 \times 0.043) + (6 \times 0.042) + (7 \times 0.041) + (8 \times 0.692) = 0.048 + 0.092 + 0.135 + 0.176 + 0.215 + 0.252 + 0.287 + 5.536 = 6.740 \]

Key concepts

cash flow

When dealing with bonds, 'cash flow' refers to the payments made to the bondholder over the bond's life. These payments typically include periodic interest payments, known as coupons, and the principal amount repaid at the end of the bond's term, known as the face value or par amount. For a bond that pays a coupon and has a face value, the cash flow is structured periodic payments followed by a lump sum at the end.
The example bond provided pays a \(6 coupon each quarter. Additionally, at the end of the 2-year term, it repays the bond's face value of \)100. This results in the following cash flows over eight quarters: \(6, \)6, \(6, \)6, \(6, \)6, \(6, and \)106 (which includes the final coupon plus the face value). Understanding these cash flows is essential, as they are used to calculate the bond's present value and duration.

yield rate

'Yield rate' is the rate of return that investors expect to earn if they hold the bond until maturity. It is an essential factor in determining the present value of future cash flows. Given that the bond in the problem has an annual yield of 11%, this needs to be converted into a quarterly rate because the bond pays quarterly coupons.
To do this, divide the annual yield by the number of quarters in a year (4 in this case), resulting in a quarterly yield rate of 2.75%. This rate will be used to discount each of the bond's future cash flows to their present values. Mathematically, this can be expressed as: \( r = \frac{11\%}{4} = 0.0275 \) or 2.75%.

present value

The 'present value' (PV) of a cash flow is the amount of money needed today to satisfy future payments, discounted at the bond's yield rate. The formula for present value is given by: \( PV = \frac{CF}{(1 + r)^t} \), where CF is the cash flow, r is the yield rate, and t is the time period.
For each quarterly cash flow from the bond, we apply this formula using the quarterly yield rate of 2.75%. For instance, the PV of the first quarter's \(6 coupon payment is approximately \( PV_1 = \frac{6}{(1 + 0.0275)^1} = 5.84 \). We repeat this calculation for each of the subsequent cash flows (6 quarters of \)6 and the final payment of \(106), resulting in respective present values. Adding all these present values gives us the bond price, which is \)122.40.

duration

Bond 'duration' measures the sensitivity of the bond's price to changes in interest rates. It is the weighted average time until the bond's cash flows are received. The duration calculation involves two main steps: computing the weights of each cash flow and then summing the products of each cash flow's weight and its time period.
First, we find the weight of each cash flow by dividing its present value by the total price of the bond, e.g., \( w_1 = \frac{5.84}{122.40} \approx 0.048 \). We repeat this for each cash flow present value.
Next, we calculate the duration, \( D \), as the sum of the products of each cash flow’s time period and its weight: \( D = \sum_{i=1}^n t_i \times w_i = 6.740 \). This result indicates that the bond's average weighted timing of cash flows is around 6.74 quarters.