You own a corporate bond that carries a 5.8 percent coupon rate and pays $10000 at maturity in exactly 2 years. The current market yield on the bond is 6.1 percent. Coupon interest is paid semiannually and the market price is $9944.32. Calculate Macaulay's duration and modified duration.
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To calculate Macaulay's duration and modified duration of a bond, you need the following information:
1. Coupon rate: The annual interest rate the bond pays.
2. Face value: The amount the bond pays at maturity.
3. Market yield: The current market interest rate.
4. Coupon payments: The frequency at which the bond makes coupon payments.
5. Time to maturity: The number of years until the bond matures.
6. Market price: The current price of the bond.
Using this information, you can calculate Macaulay's duration and modified duration as follows:
Step 1: Calculate the present value of each cash flow.
- Divide the coupon rate by the coupon payments to get the periodic coupon rate. In this case, the coupon rate is 5.8% and the bond makes semiannual coupon payments, so the periodic coupon rate would be 5.8% / 2 = 2.9%.
- Calculate the number of coupon payments by multiplying the time to maturity (2 years) by the coupon payments per year (2). In this case, you would have 2 x 2 = 4 coupon payments.
- Using the market yield, calculate the present value of each coupon payment by discounting them back to the present using the formula: Present value = Coupon payment / (1 + Market yield / Coupon payments)^n, where n is the number of periods until the cash flow.
- Calculate the present value of the face value payment at maturity using the same formula.
Step 2: Calculate the weighted average time until the bond's cash flows are received.
- For each cash flow, multiply the present value of the cash flow by the period until the cash flow is received.
- Sum up the weighted cash flows.
Step 3: Divide the weighted average time by the market price to calculate Macaulay's duration.
- Macaulay's duration = weighted average time / market price.
Step 4: Calculate modified duration.
- Modified duration = Macaulay's duration / (1 + Market yield / Coupon payments).
Plugging the given values into these formulas, we can calculate the Macaulay's duration and modified duration.
Coupon rate = 5.8%
Face value = $10000
Market yield = 6.1%
Coupon payments = Semiannual (2 times per year)
Time to maturity = 2 years
Market price = $9944.32
Step 1: Calculate present value of cash flows.
- First, calculate the periodic coupon payment: $10000 x 5.8% / 2 = $290.
- Using the market yield of 6.1% and the number of coupon payments of 4, calculate the present value of each coupon payment:
- Present value of first coupon payment = $290 / (1 + 6.1% / 2)^1 = $283.07
- Present value of second coupon payment = $290 / (1 + 6.1% / 2)^2 = $277.02
- Present value of third coupon payment = $290 / (1 + 6.1% / 2)^3 = $271.07
- Present value of fourth coupon payment = $290 / (1 + 6.1% / 2)^4 = $265.22
- Calculate the present value of the face value payment at maturity:
- Present value of face value payment = $10000 / (1 + 6.1% / 2)^4 = $8751.95
Step 2: Calculate the weighted average time until cash flows.
- Calculate the weighted cash flows by multiplying the present value of each cash flow by the period until the cash flow is received:
- Weighted cash flow for first coupon payment = $283.07 x 1 = $283.07
- Weighted cash flow for second coupon payment = $277.02 x 2 = $554.04
- Weighted cash flow for third coupon payment = $271.07 x 3 = $813.21
- Weighted cash flow for fourth coupon payment = $265.22 x 4 = $1060.88
- Weighted cash flow for face value payment = $8751.95 x 4 = $35007.80
- Sum up the weighted cash flows: $283.07 + $554.04 + $813.21 + $1060.88 + $35007.80 = $37418.00
Step 3: Calculate Macaulay's duration.
- Macaulay's duration = $37418.00 / $9944.32 = 3.7632
Step 4: Calculate modified duration.
- Modified duration = 3.7632 / (1 + 6.1% / 2) = 3.6645
Therefore, Macaulay's duration is approximately 3.7632 and the modified duration is approximately 3.6645.