Question 1189743: How much should an investor be willing to pay now for 10%, 50, 000 bond that will mature in 25 years and pays interest semi-annually, if it wants to make 12% nominal interest compounded semi-annually on a bond
investment.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to calculate the bond price:
**Understanding the Problem**
The investor wants to earn a 12% annual return (compounded semi-annually) on their investment. The bond pays a 10% annual coupon (paid semi-annually) and has a face value of $50,000, maturing in 25 years. We need to find the present value of this bond's future cash flows, discounted at the investor's desired rate of return.
**Calculations**
1. **Semi-annual Coupon Payment:**
* Annual coupon = 10% * $50,000 = $5,000
* Semi-annual coupon = $5,000 / 2 = $2,500
2. **Semi-annual Discount Rate:**
* Annual discount rate = 12%
* Semi-annual discount rate = 12% / 2 = 6% = 0.06
3. **Number of Periods:**
* Number of years = 25
* Number of semi-annual periods = 25 * 2 = 50
4. **Present Value of Coupon Payments:**
We use the present value of an annuity formula:
PV of coupons = PMT * [1 - (1 + r)^-n] / r
Where:
* PV = Present Value
* PMT = Periodic payment ($2,500)
* r = Discount rate per period (0.06)
* n = Number of periods (50)
PV of coupons = $2,500 * [1 - (1 + 0.06)^-50] / 0.06
PV of coupons = $2,500 * [1 - 0.0534] / 0.06
PV of coupons ≈ $2,500 * 15.77
PV of coupons ≈ $39,425
5. **Present Value of Face Value:**
We discount the face value back to the present:
PV of face value = FV / (1 + r)^n
Where:
* FV = Face Value ($50,000)
* r = Discount rate per period (0.06)
* n = Number of periods (50)
PV of face value = $50,000 / (1 + 0.06)^50
PV of face value ≈ $50,000 / 18.42
PV of face value ≈ $2,714
6. **Bond Price:**
The bond price is the sum of the present values of the coupon payments and the face value:
Bond Price = PV of coupons + PV of face value
Bond Price ≈ $39,425 + $2,714
Bond Price ≈ $42,139
**Answer:**
The investor should be willing to pay approximately $42,139 for the bond.
|
|
|
