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$80

$80

$80

$80

$80

$80

$80

$80

$80

$1,080

As shown, the Xanth bond has an annual coupon of $80 and a face, or par, value of $1,000 paid at maturity in 10 years.

As shown, the Xanth bond has an annual coupon of $80 and a face, or par, value of $1,000 paid at maturity in 10 years.

separately and adding the results together. First, at the going rate of 8 percent, the present value of the $1,000 paid in 10 years is:

Present value = $1,000/1.0810 = $1,000/2.1589 = $463.19

Second, the bond offers $80 per year for 10 years; the present value of this annuity stream is:

Annuity present value = $80 X (1 - 1/1.0810)/.08 = $80 X (1 - 1/2.1589)/.08 = $80 X 6.7101 = $536.81

We can now add the values for the two parts together to get the bond's value: Total bond value = $463.19 + 536.81 = $1,000

This bond sells for exactly its face value. This is not a coincidence. The going interest rate in the market is 8 percent. Considered as an interest-only loan, what interest rate does this bond have? With an $80 coupon, this bond pays exactly 8 percent interest only when it sells for $1,000.

To illustrate what happens as interest rates change, suppose that a year has gone by. The Xanth bond now has nine years to maturity. If the interest rate in the market has risen to 10 percent, what will the bond be worth? To find out, we repeat the present value calculations with 9 years instead of 10, and a 10 percent yield instead of an 8 percent yield. First, the present value of the $1,000 paid in nine years at 10 percent is:

Present value = $1,000/1.109 = $1,000/2.3579 = $424.10

Second, the bond now offers $80 per year for nine years; the present value of this annuity stream at 10 percent is:

Annuity present value = $80 X (1 - 1/1.109)/.10 = $80 X (1 - 1/2.3579)/.10

We can now add the values for the two parts together to get the bond's value: Total bond value = $424.10 + 460.72 = $884.82

A good bond site to visit is bonds.yahoo.com, which has loads of useful information.

Ross et al.: Fundamentals of Corporate Finance, Sixth Edition, Alternate Edition

III. Valuation of Future Cash Flows

7. Interest Rates and Bond Valuation

© The McGraw-Hill Companies, 2002

PART THREE Valuation of Future Cash Flows

On-line bond calculators are available at personal.fidelity.com; interest rate information is available at money.cnn.com/markets/ bondcenter/latest rates.html and www.bankrate.com.

Therefore, the bond should sell for about $885. In the vernacular, we say that this bond, with its 8 percent coupon, is priced to yield 10 percent at $885.

The Xanth Co. bond now sells for less than its $1,000 face value. Why? The market interest rate is 10 percent. Considered as an interest-only loan of $1,000, this bond only pays 8 percent, its coupon rate. Because this bond pays less than the going rate, investors are willing to lend only something less than the $1,000 promised repayment. Because the bond sells for less than face value, it is said to be a discount bond.

The only way to get the interest rate up to 10 percent is to lower the price to less than $1,000 so that the purchaser, in effect, has a built-in gain. For the Xanth bond, the price of $885 is $115 less than the face value, so an investor who purchased and kept the bond would get $80 per year and would have a $115 gain at maturity as well. This gain compensates the lender for the below-market coupon rate.

Another way to see why the bond is discounted by $115 is to note that the $80 coupon is $20 below the coupon on a newly issued par value bond, based on current market conditions. The bond would be worth $1,000 only if it had a coupon of $100 per year. In a sense, an investor who buys and keeps the bond gives up $20 per year for nine years. At 10 percent, this annuity stream is worth:

Annuity present value

This is just the amount of the discount.

What would the Xanth bond sell for if interest rates had dropped by 2 percent instead of rising by 2 percent? As you might guess, the bond would sell for more than $1,000. Such a bond is said to sell at a premium and is called a premium bond.

This case is just the opposite of that of a discount bond. The Xanth bond now has a coupon rate of 8 percent when the market rate is only 6 percent. Investors are willing to pay a premium to get this extra coupon amount. In this case, the relevant discount rate is 6 percent, and there are nine years remaining. The present value of the $1,000 face amount is:

Present value = $1,000/1.069 = $1,000/1.6895 = $591.89

The present value of the coupon stream is:

Annuity present value = $80 X (1 - 1/1.069)/.06 = $80 X (1 - 1/1.6895)/.06 = $80 X 6.8017 = $544.14

We can now add the values for the two parts together to get the bond's value:

Total bond value is therefore about $136 in excess of par value. Once again, we can verify this amount by noting that the coupon is now $20 too high, based on current market conditions. The present value of $20 per year for nine years at 6 percent is:

Annuity present value

This is just as we calculated.

Ross et al.: Fundamentals of Corporate Finance, Sixth Edition, Alternate Edition

III. Valuation of Future Cash Flows

7. Interest Rates and Bond Valuation

© The McGraw-Hill Companies, 2002

CHAPTER 7 Interest Rates and Bond Valuation

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