Bond Valuation: How to Price Bonds Using Present Value
A bond is a fixed-income security where the issuer borrows money from investors and promises to make regular interest payments plus return the principal at maturity. The price of a bond is the present value of those future cash flows. If you understand discounting, you already have everything you need to price a bond.
Key takeaway: A bond's price equals the present value of its coupon payments plus the present value of its face value. When market interest rates rise, bond prices fall, and vice versa.
What a Bond Pays
Every bond has three defining features:
- Face value (par value): The principal amount the issuer repays at maturity, typically $1,000.
- Coupon rate: The annual interest rate applied to the face value. A 6% coupon on a $1,000 bond pays $60 per year.
- Maturity: The date when the issuer returns the face value. A 5-year bond makes coupon payments for 5 years, then returns $1,000 at the end.
The investor's cash flows are straightforward: receive a fixed coupon payment each period, then receive the face value at maturity.
The Bond Pricing Formula
The price of a bond is the sum of two present values:
Bond Price = PV of Coupons + PV of Face Value
Written out fully:
Price = C/(1+r)^1 + C/(1+r)^2 + ... + C/(1+r)^n + F/(1+r)^n
- C: Annual coupon payment (coupon rate x face value)
- F: Face value
- r: Market interest rate (the required yield for bonds of similar risk)
- n: Years to maturity
The market interest rate is not the same as the coupon rate. The coupon rate is fixed when the bond is issued. The market rate changes constantly based on economic conditions, credit risk, and supply and demand. The gap between these two rates determines whether the bond trades above or below its face value.
Pricing a Bond Step by Step
Consider a bond with the following terms:
- Face value: $1,000
- Coupon rate: 6% ($60 per year)
- Maturity: 5 years
- Market interest rate: 8%
The bond pays $60 at the end of each year for 5 years, plus the $1,000 face value at the end of year 5. Discount each cash flow at the 8% market rate:
| Year | Cash Flow | Discount Factor (8%) | Present Value |
|---|---|---|---|
| 1 | $60 | 0.926 | $56 |
| 2 | $60 | 0.857 | $51 |
| 3 | $60 | 0.794 | $48 |
| 4 | $60 | 0.735 | $44 |
| 5 | $1,060 | 0.681 | $721 |
| Bond Price | $920 |
The year 5 cash flow is $1,060 because the investor receives both the final coupon ($60) and the face value ($1,000).
The bond is worth $920, which is less than its $1,000 face value. This makes sense: the bond pays a 6% coupon, but the market demands 8%. To compensate for the below-market coupon, the bond must sell at a discount.
Premium, Par, and Discount
The relationship between the coupon rate and the market rate determines whether a bond trades above, at, or below face value:
| Scenario | Coupon vs Market Rate | Price vs Face Value |
|---|---|---|
| Premium | Coupon rate > market rate | Price > $1,000 |
| Par | Coupon rate = market rate | Price = $1,000 |
| Discount | Coupon rate < market rate | Price < $1,000 |
Using the same 6% coupon bond with 5 years to maturity:
| Market Rate | Bond Price | Trades At |
|---|---|---|
| 4% | $1,089 | Premium |
| 6% | $1,000 | Par |
| 8% | $920 | Discount |
When the coupon rate exceeds the market rate, the bond's fixed payments are more generous than what the market currently offers. Investors bid the price above face value. When the coupon rate falls short, the price drops below face value to bring the effective yield up to the market rate.
At par, the coupon rate and market rate are equal. The present value of all future cash flows works out to exactly the face value.
How Interest Rates Affect Bond Prices
Bond prices and interest rates move in opposite directions. When rates rise, existing bonds become less attractive because new bonds offer higher coupons. Prices fall to compensate. When rates drop, existing bonds with higher coupons become more valuable, and prices rise.
Using the same 6% coupon, 5-year bond:
| Market Rate | Bond Price |
|---|---|
| 4% | $1,089 |
| 5% | $1,043 |
| 6% | $1,000 |
| 7% | $959 |
| 8% | $920 |
| 10% | $848 |
A 1-percentage-point increase from 6% to 7% drops the price by $41. A move from 6% to 10% drops it by $152. The further rates move from the coupon rate, the larger the price change.
This inverse relationship is one of the most important concepts in fixed income. It explains why bond portfolios lose value when central banks raise interest rates and gain value when rates are cut.
Yield to Maturity
Yield to maturity (YTM) is the total annual return an investor earns if they buy the bond at its current market price and hold it until maturity. Mathematically, YTM is the discount rate that makes the present value of all future cash flows equal to the bond's price. It is the bond equivalent of the internal rate of return (IRR).
For the bond priced at $920, the YTM is 8%. That is the rate that makes:
60/(1.08)^1 + 60/(1.08)^2 + 60/(1.08)^3 + 60/(1.08)^4 + 1060/(1.08)^5 = $920
If the bond were trading at $1,000, the YTM would equal the coupon rate (6%). If it were trading at $1,089, the YTM would be 4%. In each case, YTM adjusts to reflect the actual return based on the price paid.
Like IRR, there is no algebraic shortcut for YTM. It is solved by trial and error or using Excel's =YIELD() or =IRR() function.
Why Bond Valuation Matters
Bond pricing shows up across finance:
- Fixed income investing: Portfolio managers price bonds to decide whether they are cheap or expensive relative to their risk.
- Corporate finance: When a company issues bonds, the market rate at issuance determines the cost of debt, which feeds into WACC.
- Interest rate risk: The inverse price-rate relationship means rising rates create losses for existing bondholders. Understanding this helps manage portfolio risk.
- Valuation foundation: Bond pricing is discounting in its cleanest form. The same logic extends to valuing any asset that produces predictable cash flows.
Conclusion
A bond's price is the present value of its coupons plus the present value of its face value. When market rates exceed the coupon rate, the bond sells at a discount. When rates fall below the coupon rate, it sells at a premium. YTM captures the total return at the current price.
For the fundamentals behind discounting, see our guide on time value of money. For a deeper look at how present value works, see our guide on present value vs. future value. To understand how YTM relates to IRR, see our guide on NPV and IRR.
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