six years ago singleton company issued 20 year bonds with a 14% annual coupon rate at their 1000 par value . the bonds had a 9% call premium, with 5 years of call protection. today singleton called the bonds. compute the realized rate of rate of return for an ivestor who purchased the bonds when they were issued and held them until they were called.
14.87
To compute the realized rate of return for an investor who purchased the bonds when they were issued and held them until they were called, we need to calculate the total cash flows received by the investor and then determine the rate of return that equates the present value of those cash flows with the initial investment.
Step 1: Determine the cash flows received by the investor over the holding period.
Annual coupon payment = Coupon rate * Par value = 14% * $1000 = $140
The investor receives this coupon payment for the first 14 years because the bonds were called after 20 years and there were 5 years of call protection.
Total coupon payments received = Annual coupon payment * Number of years = $140 * 14 = $1960
Additionally, the investor will receive the call price, which is $1000 * (1 + Call premium%) = $1000 * (1 + 9%) = $1000 * 1.09 = $1090.
Therefore, the total cash flow received by the investor is $1960 + $1090 = $3050.
Step 2: Determine the initial investment.
The initial investment is the purchase price of the bonds, which is not given in the question. Therefore, we need this information to calculate the realized rate of return.
Step 3: Calculate the realized rate of return.
To calculate the realized rate of return, we need to find the discount rate that equates the present value of the cash flows received ($3050) with the initial investment.
The formula to calculate the present value (PV) of the cash flows is:
PV = Cash flow / (1 + Rate)^n
where Rate is the discount rate and n is the number of periods.
Since we don't have the initial investment value, we cannot directly calculate the realized rate of return. You would need to know the purchase price of the bonds to determine the realized rate of return correctly.
To compute the realized rate of return for an investor who purchased the bonds when they were issued and held them until they were called, we need to consider the cash flows associated with the investment.
1. Determine the cash flows:
- The bonds were issued with a 20-year maturity and a 14% annual coupon rate.
- The par value of each bond is $1000.
- Therefore, the annual coupon payment is $1000 * 0.14 = $140.
2. Calculate the present value of the cash flows:
- To calculate the present value of the coupon payments, use the formula for the present value of an ordinary annuity. In this case, the annuity is the coupon payments over the 20-year bond maturity period, with a discount rate equal to the investor's required rate of return.
- The discount rate is not provided in the question, so we will assume it is the annual market interest rate at the time of purchase. Let's assume it was 10% per year.
- Using a financial calculator or spreadsheet, the present value of the coupon payments can be calculated as follows:
PV(Coupon Payments) = $140 * [1 - (1 + 0.10)^(-20)] / 0.10
3. Calculate the present value of the call price:
- The bonds can be called after 5 years with a 9% call premium.
- The call premium is 9% of the par value, which is $1000 * 0.09 = $90.
- The present value of the call price can be calculated using the formula for the present value of a single amount:
PV(Call Price) = $90 / (1 + 0.10)^5
4. Calculate the total present value of cash flows:
- The total present value of cash flows is the sum of the present values of the coupon payments and the present value of the call price:
Total PV = PV(Coupon Payments) + PV(Call Price)
5. Calculate the realized rate of return:
- The realized rate of return is the discount rate that equates the total present value of cash flows to the initial cost of the investment, which is the purchase price of the bond.
- In this case, the initial cost is the bond's par value of $1000.
- The realized rate of return can be calculated using trial-and-error or an iterative method to find the discount rate that makes the total present value of cash flows equal to $1000.
Once you have the total present value of cash flows and the initial cost, you can calculate the realized rate of return.