Dahler Corporation has just issued a bond with a maturity of 20 years, coupon rate of 10.25%, and a market price of $1330.25. Dahler makes semiannual coupon payments.
a) what is the YTM expressed as a quoted rate based on semi-annual compounding? And what is the effective annual YTM on this bond?
b)What is the bond's expected price two years from now? What is its capital gains yield for year 2? What is its current yield at the beginning of year 2? What is the expected total return for year 2?
c) What would happen to the price of the bond over time, as the bond gets closer to maturity? (Again, assuming no change in interest rates)
a) To find the YTM (Yield to Maturity), we need to solve for the rate that makes the present value of all the future cash flows equal to the market price of the bond. We can use the following formula:
PV = (C / (1 + r)) + (C / (1 + r)²) + ... + (C / (1 + r)⁴⁰) + (F / (1 + r)⁴⁰)
Where PV is the present value, C is the periodic coupon payment, r is the semi-annual yield, and F is the face value of the bond.
Given:
Coupon Rate (C) = 10.25%
Face Value (F) = $1000
Market Price (PV) = $1330.25
Using this information, we can calculate the YTM:
1330.25 = (51.25 / (1 + r)) + (51.25 / (1 + r)²) + ... + (51.25 / (1 + r)⁴⁰) + (1000 / (1 + r)⁴⁰)
To find the effective annual YTM, we can convert the semi-annual yield to an annual yield by multiplying it by 2.
b) To find the bond's expected price two years from now, we can calculate the present value of the remaining cash flows, which include the 38 remaining coupon payments and the face value:
PV = (C / (1 + r)) + (C / (1 + r)²) + ... + (C / (1 + r)³⁸) + (F / (1 + r)³⁸)
To find the capital gains yield in year 2, we can subtract the expected price at the beginning of year 2 from the price at the beginning of year 1, divide by the price at the beginning of year 1, and multiply by 100%.
To find the current yield at the beginning of year 2, we can divide the coupon payment at the beginning of year 2 by the expected price at the beginning of year 2, and multiply by 100%.
To find the expected total return for year 2, we can sum the capital gains yield and the current yield.
c) Assuming no change in interest rates, the price of the bond over time, as it gets closer to maturity, will approach the face value of the bond. This is because the remaining payments become less significant compared to the face value, and the bond converges towards its par value.
a) To calculate the Yield to Maturity (YTM) as a quoted rate based on semi-annual compounding, we can use the bond pricing formula. The formula is:
Bond Price = (Coupon Payment / (1 + YTM/2)^n) + (Coupon Payment / (1 + YTM/2)^(n-1)) + ... + (Coupon Payment + Face Value / (1 + YTM/2)^(n-1))
Where:
Bond Price = $1330.25 (given)
Coupon Payment = (Coupon Rate / 2) * Face Value
YTM = Yield to Maturity
n = number of periods (2*number of years)
In this case, the bond has a maturity of 20 years, with semiannual payments. Therefore, there are 40 payment periods.
Coupon Payment = (10.25% / 2) * Face Value = 0.05125 * Face Value
Now, we can solve the equation for the quoted YTM. We can do this either manually using trial and error or using financial calculators or spreadsheet software that have built-in functions for calculating Yield to Maturity.
Once the YTM is obtained, the effective annual YTM can be calculated by using the following equation:
Effective Annual YTM = (1 + YTM/2)^2 - 1
b) To calculate the bond's expected price two years from now, we need to calculate the present value of the cash flows. The cash flows include the semiannual coupon payments and the face value received at maturity. The present value can be calculated using the one-period discount factor, which is (1 / (1 + YTM/2)).
The expected price two years from now would be the present value of the remaining cash flows. There will be 18 semiannual coupon payments left (assuming we are calculating the price at the beginning of year 2).
To calculate the capital gains yield for year 2, subtract the expected price at the beginning of year 2 from the current price and divide it by the current price.
Capital Gains Yield for Year 2 = (Expected Price at Year 2 - Current Price) / Current Price
The current yield at the beginning of year 2 can be calculated by dividing the annual coupon payment by the expected price at the beginning of year 2.
Current Yield at Beginning of Year 2 = (Annual Coupon Payment / Expected Price at Year 2) * 100%
The expected total return for year 2 can be calculated by summing up the capital gains yield and the current yield.
Expected Total Return for Year 2 = Capital Gains Yield for Year 2 + Current Yield at Beginning of Year 2
c) Assuming no change in interest rates, the price of the bond would gradually approach its face value as it gets closer to maturity. This is because the remaining cash flows become fewer, making their present value relatively higher in comparison to the price of the bond. The bond's price would converge towards the face value as the bond's maturity date approaches.