Bond A pays ,000 in 20 years. Bond B pays ,000 in 40 years. (To keep things simple, assume these

step 1 Okay, guys, this is Chapter 27 Problem 3. So in this question, we're given that there are two bo

Bond A pays ,000 in 20 years. Bond B pays ,000 in 40 years....

Bond A pays $\$$8,000 in 20 years. Bond B pays $\$$8,000 in 40 years. (To keep things simple, assume these are zero-coupon bonds, which means the $\$$8,000 is the only payment the bondholder receives.) a. If the interest rate is 3.5 percent, what is the value of each bond today? Which bond is worth more? Why? ($Hint$: You can use a calculator, but the rule of 70 should make the calculation easy.) b. If the interest rate increases to 7 percent, what is the value of each bond? Which bond has a larger $percentage$ change in value? c. Based on the example above, complete the two blanks in this sentence: "The value of a bond [rises/falls] when the interest rate increases, and bonds with a longer time to maturity are [more/less] sensitive to changes in the interest rate."

Bond A pays $\$$8,000 in 20 years. Bond B pays $\$$8,000 in 40 years. (To keep things simple, assume these are zero-coupon bonds, which means the $\$$8,000 is the only payment the bondholder receives.)

a. If the interest rate is 3.5 percent, what is the value of each bond today? Which bond is worth more? Why? ($Hint$: You can use a calculator, but the rule of 70 should make the calculation easy.)

b. If the interest rate increases to 7 percent, what is the value of each bond? Which bond has a larger $percentage$ change in value?

c. Based on the example above, complete the two blanks in this sentence: "The value of a bond
[rises/falls] when the interest rate increases, and bonds with a longer time to maturity are
[more/less] sensitive to changes in the interest rate."

Key Concepts

Present Value

Present value is the concept of determining how much a future amount of money is worth today, given a specific interest rate. It reflects the time value of money, which means that a dollar received in the future is less valuable than a dollar received today, because money available now can be invested to earn returns.

Zero-Coupon Bonds

Zero-coupon bonds are securities that do not offer periodic interest payments. Instead, they are issued at a discount to their face value and pay the full face value at maturity. Their valuation relies entirely on the present value of a single future cash flow.

Discounting and Exponential Decay

Discounting is the process of calculating the present value of a future payment by applying a discount rate that reflects the cost of capital or interest rate. This process uses an exponential decay formula, where the longer the time until payment, the greater the reduction in value, illustrating how future cash flows diminish in present value terms.

Interest Rate Risk and Duration

Interest rate risk refers to the inverse relationship between bond prices and interest rates; as rates rise, bond prices fall and vice versa. Duration, a measure of a bond's sensitivity to interest rate changes, indicates that bonds with longer maturities are generally more sensitive to interest rate fluctuations compared to those with shorter maturities.

Transcript

00:01 Okay, guys, this is chapter 27 problem 3.

00:06 So in this question, we're given that there are two bonds.

00:08 Bond a pays $8 ,000 in 20 years, and bond b pays $8 ,000 in 40 years.

00:15 So the first part of the question is asking us that the interest rate is 3%, what is the current value of the bond? so this is a net present value question.

00:29 And the formula for net present value can be found on page 565.

00:35 Of the textbook.

00:37 The formula is x, which is the payment you get over one plus the interest rate to the power of n, which is the number of years in the future that you're going to receive the payment, it's equal to the net present value.

00:50 So for the first bond, we're going to receive $8 ,000 on an interest rate of 3 % in 20 years, and this is going to equal $4 ,020.

01:03 So you can do this with the rule of 70, like the question, the hint in the question says, but it's very easy to do this with a formula.

01:12 So for bond b, it's going to be 8 ,000 over one plus the interest rate to the power of 40, and this is 2020, right? this is bond b, and this is bond a.

01:29 So obviously bond b is worth less, and that's because you have to wait longer to receive the payment, which means because of inflation and price changes, the value of a dollar in 40 years is worth less than the value of the dollar in 20 years, which is worth less than the value of a dollar today.

01:49 So part b then asks that if the interest rate changes to 7%, what's the present value of each bond, which bonds change? change the most.

01:59 So we're going to be using the same formulas.

02:01 And for bond a, it's $8 ,000 over one plus 0 .07 now to the power of 20.

02:11 And this equals $2 ,067.

02:14 And for bond b, it's $8 ,000 over one plus 0 .07 to the power of 40...

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