Finance

Bond Valuation

Pricing a bond, current yield, and the inverse link between price and rates.

A bond's price is the present value of its cash flows: annual coupons $C$ for $n$ years plus the face value $F$ at maturity, discounted at the market yield $r$ : $P = \sum_{t = 1}^{n} \frac{C}{( 1 + r ) ^{t}} + \frac{F}{( 1 + r ) ^{n}}$ . Because $r$ is in the denominator, price and yield move in opposite directions.

1. Pricing a two-year bond
easy
A bond has face value $\$ 1{,}000 $, p a y s anann u a l co u p o n o f$ $50 $(a$ 5% $co u p o n), an d ma t u r es in$ 2 $y e a r s . T h e ma r k e t y i e l d i s$ 6%$. What is its price?
Show solution
$P = \dfrac{50}{1.06} + \dfrac{50 + 1000}{1.06^2} = 47.17 + \dfrac{1050}{1.1236} = 47.17 + 934.50 = \$ 981.67$.

The bond trades at a discount to par ( $<\$ 1{,}000 $) b ec a u se i t s$ 5% $co u p o ni s b e l o w t h e$ 6%$ market yield.
2. Current yield
medium
A bond with a $\$ 60 $ann u a l co u p o n c u r r e n tl y t r a d es a t a p r i ceo f$ $1{,}200$. Compute its current yield and explain how it differs from the coupon rate.
Show solution
Current yield $= \frac{annual coupon}{price} = \frac{60}{1 , 200} = 5%$ .

The coupon rate ( $60/1000 = 6%$ of face value) is fixed, but the current yield uses the market price. Since the bond trades at a premium ( $\$ 1{,}200 > $1{,}000 $), t h ec u r r e n t y i e l d ($ 5%$) is below the coupon rate — you pay more, so your income return is lower.
3. Price when rates rise
hard
Reprice the bond from the first problem (face $\$ 1{,}000 $,$ $50 $ann u a l co u p o n,$ 2 $y e a r s t o ma t u r i t y) i f t h e ma r k e t y i e l d j u m p s f r o m$ 6% $t o$ 8%$. By how much does its price change, and why?
Show solution
$P = \dfrac{50}{1.08} + \dfrac{1050}{1.08^2} = 46.30 + \dfrac{1050}{1.1664} = 46.30 + 900.21 = \$ 946.51$.

The price falls from $\$ 981.67 $t o$ $946.51 $, a d r o p o f ab o u t$ $35.16 $($ -3.6%$).

Higher market yields discount the same fixed cash flows more heavily, so the price falls. This inverse relationship is the core risk of holding bonds — rising rates lower the value of existing, lower-coupon bonds.

Bond Valuation

1. Pricing a two-year bond

2. Current yield

3. Price when rates rise

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