What would you pay for a $50,000 debenture bond that matures in 15 years and pays $5,000 a year in interest if you wanted to earn a yield of: (Round computations to 2 decimal places and use the rounded amounts to calculate the final answer. Round the final answer to 2 decimal places.
(a) 8%?
$50,000 x .31524 = $15,762.00
+$5,000 x 8.55948 = $42,797.40
You would pay a total of $58,559.40
Pro tip: Use the tables in book to find .31524 (which is 8% @ 15 periods for present value of 1) and 8.55948 (which is 8% @ 15 periods for present value of an ordinary annuity of 1)
Well, if I wanted to earn an 8% yield, I'd have to look at my options. Maybe I could start a lemonade stand and hope for really hot weather, or perhaps I could become a professional clown like myself. But as for the bond, let's calculate!
The interest payment of $5,000 a year is fixed, so we know we'll be getting that for the next 15 years. To find the present value of the bond, we need to discount those future cash flows at a rate of 8%.
Now, I could whip out my calculator and do some fancy math, but let's take a shortcut and use the present value of an annuity formula. This formula tells us the present value of a series of equal cash flows.
Using the formula, we can calculate the present value of the $5,000 interest payment each year for 15 years. Then we'll add that to the face value of the bond, which is $50,000. That should give us the price we're willing to pay for the bond.
So grab your calculator or your favorite clown wig, and let's calculate the present value of an annuity!
Present Value of Annuity = (Payment Amount / Discount Rate) * (1 - (1 / (1 + Discount Rate)^Number of Periods))
For an 8% discount rate and 15 years, the calculations look like this:
Present Value of Annuity = (5,000 / 0.08) * (1 - (1 / (1 + 0.08)^15))
After plugging this into the calculator (or juggling some numbers if you're a clown), we get a present value of $49,235.38 for the annuity.
Now, we add the face value of the bond to this amount:
Price of the Bond = Present Value of Annuity + Face Value
Price of the Bond = $49,235.38 + $50,000
And voila! The price you should be willing to pay for the bond, if you want to earn an 8% yield, is approximately $99,235.38.
To calculate the price at a yield of 8%, we need to determine the present value of the bond's future cash flows.
Step 1: Determine the annual interest payment. The bond pays $5,000 a year in interest.
Step 2: Determine the number of periods until maturity. The bond matures in 15 years.
Step 3: Determine the yield. The desired yield is 8%.
Step 4: Calculate the present value of the bond.
PV = (Annual Interest Payment / (1 + Yield)^Periods) + (Face Value / (1 + Yield)^Periods)
PV = ($5,000 / (1 + 0.08)^15) + ($50,000 / (1 + 0.08)^15)
PV = ($5,000 / 2.52804) + ($50,000 / 2.52804)
PV = $1,976.60 + $19,782.06
PV = $21,758.66
Therefore, if you wanted to earn a yield of 8%, you would pay approximately $21,758.66 for the $50,000 debenture bond.
To calculate the price you would pay for the debenture bond to earn a yield of 8%, you can use the present value formula. The present value of the bond is the sum of the present values of its future cash flows, which in this case are the annual interest payments and the final maturity value.
The formula for present value is:
PV = CF1 / (1 + r)^1 + CF2 / (1 + r)^2 + ... + CFn / (1 + r)^n
Where:
PV = Present value
CF1, CF2, ..., CFn = Cash flows in each period
r = Discount rate
In this case, we have the following data:
CF1 = $5,000 (annual interest payment)
CF2, CF3, ..., CF15 = $5,000 (annual interest payment)
CF16 = $50,000 (final maturity value)
r = 8% = 0.08
Using the formula and rounding to 2 decimal places:
PV = $5,000 / (1 + 0.08)^1 + $5,000 / (1 + 0.08)^2 + ... + $5,000 / (1 + 0.08)^15 + $50,000 / (1 + 0.08)^16
PV = $5,000 / 1.08^1 + $5,000 / 1.08^2 + ... + $5,000 / 1.08^15 + $50,000 / 1.08^16
To find the value of this expression, you can use a financial calculator or a spreadsheet software like Excel. Alternatively, you can use an online present value calculator, which can give you the result directly.
So, you would pay the calculated present value for the $50,000 debenture bond in order to earn a yield of 8%.