6.2 Bond Valuation

Bond valuation lets you figure out the fair price of a bond based on its future cash flows. Since bond prices shift constantly with interest rates and market conditions, knowing how to value them is a core skill for any finance course and for real-world investing.

Bond Pricing Calculations

Time Value of Money in Bond Valuation

The price of a bond equals the present value of all its future cash flows: the periodic coupon payments plus the face value (par value) you get back at maturity. Because a dollar received in the future is worth less than a dollar today, you discount each cash flow back to the present.

The bond pricing formula:

$P = \sum_{t=1}^{n} \frac{C}{(1 + r)^t} + \frac{FV}{(1 + r)^n}$

Where:

• $P$ = bond price
• $C$ = coupon payment per period
• $r$ = yield to maturity per period
• $t$ = each time period
• $n$ = total number of periods until maturity
• $FV$ = face value (typically $\$1{,}000$)

Example walkthrough: Price a bond with a $\$1{,}000$ face value, a 5% annual coupon rate, 10 years to maturity, and a YTM of 6%.

1. Calculate the annual coupon payment: $C = 0.05 \times \$1{,}000 = \$50$

2. Find the present value of the coupon stream: discount each $\$50$ payment at 6% for periods 1 through 10. Using the annuity formula: $PV_{\text{coupons}} = \$50 \times \frac{1 - (1.06)^{-10}}{0.06} = \$50 \times 7.3601 = \$368.00$

3. Find the present value of the face value: $PV_{\text{face}} = \frac{\$1{,}000}{(1.06)^{10}} = \$558.39$

4. Add them together: $P = \$368.00 + \$558.39 \approx \$926.40$

Notice the price is below $\$1{,}000$. That's because the bond's 5% coupon is less than the 6% yield the market demands, so the bond sells at a discount.

(Note: Slight differences in rounding can produce values like $\$922.78$ or $\$926.40$ depending on how intermediate steps are rounded. On exams, use the method your professor specifies.)

Yield to Maturity and Coupon Rate

Yield to maturity (YTM) is the discount rate that makes the present value of a bond's future cash flows equal to its current market price. It represents the total annualized return an investor earns if the bond is held to maturity and all coupons are reinvested at that same rate.

Coupon rate is the annual interest rate the issuer pays, expressed as a percentage of face value. It's fixed at issuance and doesn't change. A bond with a $\$1{,}000$ face value and $\$50$ annual coupon payments has a coupon rate of $\frac{\$50}{\$1{,}000} = 5\%$.

The distinction matters: the coupon rate determines the cash flows you receive, while the YTM reflects the market's required return. These two rates drive whether a bond trades at a premium, discount, or par.

Bond Prices and Interest Rates

Inverse Relationship between Bond Prices and Market Interest Rates

Bond prices and market interest rates move in opposite directions. When rates rise, existing bond prices fall. When rates drop, existing bond prices rise.

Why? A bond's coupon rate is locked in at issuance. If market rates climb above that coupon rate, your bond's fixed payments become less attractive compared to newly issued bonds offering higher yields. Buyers will only purchase your bond at a lower price, which effectively raises its yield to match the market. The reverse happens when market rates fall: your bond's relatively higher coupon becomes more valuable, pushing its price up.

Example: You hold a bond paying a 5% coupon. If market rates rise to 6%, new bonds pay more than yours. Your bond's price drops so that a buyer purchasing it at the lower price would earn a yield close to 6%. If market rates fell to 4% instead, your 5% coupon looks generous, and the price rises above par.

Bond Price Premiums, Discounts, and Par

The relationship between the coupon rate and the market interest rate determines whether a bond trades above, below, or at face value:

• Premium: Coupon rate > market rate → price > face value. Investors pay extra for the higher-than-market coupon.
• Discount: Coupon rate < market rate → price < face value. The price drops to compensate for the below-market coupon.
• Par: Coupon rate = market rate → price = face value. The bond's payments exactly match what the market demands.

Example: A bond with a 6% coupon trades at a premium when market rates are 5% (price above $\$1{,}000$) and at a discount when market rates are 7% (price below $\$1{,}000$). As a bond approaches maturity, its price gradually converges toward par regardless of whether it was trading at a premium or discount. This is called "pull to par."

Factors Influencing Bond Prices

Market Interest Rates, Credit Risk, and Time to Maturity

Three primary factors drive bond price movements:

Interest rate risk is the risk that changing market rates cause your bond's price to fluctuate. Two characteristics make a bond more sensitive to rate changes:

• Longer maturity: A 30-year bond's price swings far more than a 5-year bond's price for the same rate change, because more distant cash flows are discounted more heavily.
• Lower coupon rate: A bond with a 2% coupon is more rate-sensitive than one with an 8% coupon, because a larger share of its value comes from the distant face value payment rather than near-term coupons.

Credit risk (or default risk) is the chance the issuer fails to make coupon or principal payments. Credit rating agencies (Moody's, S&P, Fitch) assign ratings to bonds. Higher credit risk means investors demand a higher yield, which pushes the bond's price down. A downgrade in credit rating typically causes an immediate price drop.

Time to maturity affects price volatility directly. All else equal, longer-maturity bonds carry more uncertainty and therefore more risk.

Liquidity, Inflation, and Call Risk

Beyond the big three, several other risks affect bond prices:

Liquidity risk arises when a bond can't be sold quickly at a fair price. U.S. Treasury bonds are highly liquid; a small corporate issue might not be. Less liquid bonds typically trade at slightly lower prices (higher yields) to compensate.

Inflation risk is the danger that rising prices erode the purchasing power of your fixed coupon payments. If inflation runs at 4% and your bond pays a 3% coupon, you're losing ground in real terms. This is why investors watch inflation expectations closely when pricing bonds.

Call risk applies to callable bonds, where the issuer has the right to redeem the bond before maturity. Issuers typically call bonds when market rates have fallen significantly, because they can refinance at a lower rate. The problem for you as an investor: you get your principal back early and then have to reinvest at the new, lower rates. Callable bonds usually offer slightly higher yields to compensate for this risk.

Bond Price Quotes and Investment Decisions

Understanding Bond Price Quotes

Bond prices are quoted as a percentage of face value, not in dollar terms. A quote of 98.5 means the bond is priced at 98.5% of its face value.

Three price concepts to know:

• Clean price: The quoted price, which excludes any interest that has accumulated since the last coupon payment. This is the number you see on screens and in the financial press.
• Accrued interest: The interest earned by the seller between the last coupon date and the settlement date. The buyer compensates the seller for this amount.
• Dirty price (invoice price): The actual amount the buyer pays, calculated as clean price + accrued interest.

Example: A $\$1{,}000$ face value bond quoted at 98.5 with $\$20$ in accrued interest:

• Clean price = $0.985 \times \$1{,}000 = \$985$
• Dirty price = $\$985 + \$20 = \$1{,}005$

The buyer pays $\$1{,}005$, but $\$20$ of that is reimbursing the seller for interest already earned.

Making Informed Investment Decisions

When evaluating a bond, compare its YTM to your required rate of return. If the YTM exceeds what you need given the bond's risk profile, it may be a good buy.

Duration is a key metric here. It measures how sensitive a bond's price is to a 1% change in interest rates. A bond with a duration of 7 years will see roughly a 7% price change for every 1% move in rates. Higher duration means more interest rate exposure.

A few practical guidelines for building a bond portfolio:

• Match your time horizon to bond maturities. If you need the money in 3 years, a 30-year bond exposes you to unnecessary price risk.
• Diversify across issuers, maturities, and credit qualities to spread risk.
• Conservative investors often lean toward shorter-term, higher-rated bonds (government or investment-grade corporate). Investors willing to accept more risk might pursue longer-term or lower-rated bonds for the higher yields they offer.