How Does the Value of a Bond Change?
As rates increase or decrease, the discount rate that is used also changes appropriately. Let's change the discount rate in the above example to 10% to see how it affects the value of the bond.
Example: The Value of a Bond when Discount Rates Change
PV of the cash flows is: Year one = 70 / (1.10) to the 1st power = $ 63.63
Year two = 70 / (1.10) to the 2nd power = $ 57.85
Year three = 70 / (1.10) to the 3rd power = $ 52.63
Year four = 70 / (1.10) to the 4th power = $ 47.81
Year five = 1070 / (1.10) to the 5th power = $ 664.60
Value = 63.63 + 57.85 + 52.63 + 47.81 + 664.60 = $ 886.52
- As we can see from the above examples, an important property of PV is that for a given discount rate, the older a cash flow value is, the lower its present value.
- We can also compute the change in value from an increase in the discount rate used in our example. The change = 1,086.59 - 886.52 = 200.07.
- Another property of PV is that the higher the discount rate, the lower the value of a bond and the lower the discount rate the higher the value of the bond.
If the discount rate is higher than the coupon rate the PV will be less than par. If the discount rate is lower than the coupon rate, the PV will be higher than par value.
How Does a Bond's Price Change as it Approaches its Maturity Date?
As a bond moves closer to its maturity date, its price will move closer to par. The break down on the three scenarios is as follows:
1. If a bond is at a premium, the price will decline over time towards its par value.
2. If a bond is at a discount, the price will increase over time towards its par value
3. If a bond is at par, its price will remain the same.
To show how this works lets use our original example of the 7% bond, but now let's assume a year has passed and a discount rate remains the same at 5%.
Example: Price Changes Over Time
Let's compute the new value to see how the price moves closer to par. You should also be able to see how the amount by which the bond price changes is attributed to it being closer to it's maturity date.
PV of the cash flows is: Year one = 70 / (1.05) to the 1st power = $66.67
Year two = 70 / (1.05) to the 2nd power = $ 63.49
Year three = 70 / (1.05) to the 3rd power = $ 60.47
Year four = 1070 / (1.05) to the 4th power = $880.29
Value = 66.67 + 63.49 + 60.47 + 880.29 = 1,070.92
As the price of the bond decreases, it moves closer to its par value. Theamount of change attributed to the year's difference is 15.67.
An individual can also decompose the change that results when a bond approaches its maturity date and the discount rate changes. This is accomplished by first taking the net change in the price that reflects the change in maturity, and then adding it to the change in the discount rate. The two figures should equal the overall change in the bond's price.
Computing the Value of a Zero-coupon Bond
This may be the easiest of securities to value because there is only one cash flow - the maturity value.
Value of a zero coupon bond that matures N years from now is:
Zero coupon bond value = Maturity value / (1 + I) to the power of the number of years x 2
Where I is the semi-annual discount rate.
Example: The Value of a Zero-Coupon Bond
For illustration purposes, let's look at a zero coupon with a maturity of three years and a maturity value of $1,000 discounted at 7%
I = 0.035 (.07 / 2)
N = 3
Value of a Zero = 1,000 / (1.035) to the 6th power (3 x 2)
= 1,000 / 1.229255
Arbitrage-free Valuation Approach
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