Recall from Weeks 1 and 2 that the core idea of finance is deceptively simple: a dollar today is worth more than a dollar tomorrow, and the tools of present value and future value let you compare cash flows across time. We've applied those tools to single lump sums, annuities, and perpetuities. This week, we put all of that machinery to work on one of the most important financial instruments in the world: the bond.
A bond is a loan packaged as a tradeable security. When a corporation or government needs to borrow money, it can issue bonds to investors. The issuer promises to make a series of fixed payments over time and to repay the borrowed amount at the end. If you buy a bond, you're the lender — you're giving money today in exchange for a stream of future cash flows. The value of that bond, therefore, is just the present value of those promised cash flows. Nothing more, nothing less. That's the punchline of this entire week: bond pricing is a time-value-of-money problem.
Before we start calculating, though, you need to know the vocabulary. Bonds come with specific contractual terms, and these terms determine the exact cash flows you'll receive.
The face value (also called par value) is the principal amount the issuer promises to repay at the end of the bond's life. For most corporate and government bonds, the face value is $1,000. This is the amount you get back on the last day — think of it as the original loan amount.
The coupon rate is the annual interest rate the issuer pays, stated as a percentage of face value. A bond with a 6% coupon rate and a $1,000 face value pays 6% × $1,000 = $60 per year in interest. These periodic interest payments are called coupon payments. The name is a relic of the era when bonds were physical certificates with detachable coupons that you'd clip and redeem for cash.
The maturity date is the date on which the issuer repays the face value. The time remaining until maturity is the bond's term to maturity (or just "maturity"). A bond issued in 2026 that matures in 2036 has a 10-year maturity.
Many bonds pay coupons semiannually — twice a year — rather than once. A 6% coupon bond with semiannual payments makes two payments of $30 each, six months apart. This is the default for most U.S. corporate and government bonds, so whenever a problem says "semiannual," you'll need to adjust both the coupon payment and the discount rate to a per-period basis. We'll work through that adjustment shortly.
Beyond these basics, bonds can have additional features that affect their risk and value:
A call provision gives the issuer the right to buy back ("call") the bond before maturity, typically at a specified call price that includes a small premium above face value. Issuers like call provisions because they can refinance if interest rates drop — just like refinancing a mortgage. As a bondholder, though, a call provision works against you: the issuer is most likely to call your bond precisely when it's most valuable to you (i.e., when rates have fallen and your bond's coupon looks generous). For this reason, callable bonds must offer a slightly higher yield to compensate investors for this risk.
Seniority refers to a bond's priority in the event of bankruptcy. Senior bonds get paid before subordinated (junior) bonds, and secured bonds — those backed by specific collateral — get paid before unsecured bonds. Seniority affects the bond's risk and therefore its required yield. We'll see more about how risk maps to yield when we discuss bond ratings later this week.
Now for the mechanics. A bond's cash flows are straightforward: a series of equal coupon payments (an annuity) plus a single lump-sum repayment of face value at maturity. The price of the bond is the present value of all of those cash flows, discounted at the bond's yield to maturity (YTM) — the market's required rate of return for a bond of that risk and maturity.
If you think about it, you already know how to do this. The coupon payments form an ordinary annuity, and the face value is a single future cash flow. Recall from Week 2 that the present value of an annuity is PMT × [(1 − (1 + r)−t) / r], and the present value of a lump sum is FV / (1 + r)t. A bond's price is simply the sum of these two pieces.
The first term is the present value of the coupon annuity. The second term is the present value of the face value repayment. Together, they give you the price a rational investor would pay for this stream of cash flows.
Let's work through this with Blue Acre Industries, a financial services company that's issued a 10-year bond.
Notice that this bond sells for less than its $1,000 face value. When a bond trades below par, we call it a discount bond. Why the discount? Because the bond pays a 6% coupon, but the market demands 8%. Investors aren't willing to pay full price for a bond whose coupon rate is below the going rate — the lower price compensates by boosting the effective return up to 8%.
What happens if market rates are below the coupon rate? Let's find out.
Now the bond trades above par — it's a premium bond. The logic is the mirror image: the bond's 6% coupon is more generous than what the market requires (4%), so investors bid the price above face value. The premium you pay offsets the "extra" coupon income, bringing your effective return back down to 4%.
There's a clean pattern here. Compare Examples 1 and 2 with one more scenario — what if the YTM equals the coupon rate exactly?
This is one of the most important relationships in fixed-income finance: bond prices and interest rates move in opposite directions. When rates rise, bond prices fall. When rates fall, bond prices rise. The reason is mechanical — you're discounting fixed cash flows at a different rate. A higher discount rate means each future dollar is worth less today, so the price drops. A lower discount rate means each future dollar is worth more today, so the price rises.
This inverse relationship isn't just an academic curiosity. It explains why bond investors watch Federal Reserve announcements so carefully — a rate hike means the bonds they already hold lose value. It also explains why long-duration bonds are more volatile than short-duration ones, a topic we'll return to shortly.
A 6% coupon bond with 10 years to maturity. When YTM equals the coupon rate (6%), the bond trades at par. When YTM is below the coupon rate, the bond trades at a premium; when above, at a discount. The curve is convex — price drops decelerate as yields rise.
Most bonds in practice pay coupons twice a year. The adjustment is straightforward but easy to get wrong if you're not careful: divide the annual coupon payment by 2, divide the annual YTM by 2, and multiply the number of years by 2. Everything converts to a per-period (semiannual) basis.
Recall from Week 2 that dividing the annual rate by the number of compounding periods gives you the stated rate per period. This is the convention for bond markets — the quoted YTM is always a stated annual rate (an APR), not an effective annual rate. If you wanted the EAR, you'd compute (1 + r/2)2 − 1, but for bond pricing you use the semiannual rate directly.
The key pitfall with semiannual bonds is forgetting to adjust all three inputs — the coupon payment, the discount rate, and the number of periods. If you halve the coupon but forget to halve the rate (or vice versa), your answer will be wrong. Develop the habit of converting everything to per-period terms as your first step.
So far we've been given a YTM and asked to find the price. But in practice, you often observe a bond's market price and need to figure out the implied yield. The yield to maturity (YTM) is the discount rate that makes the present value of a bond's cash flows equal to its current market price. It's the return you'd earn if you bought the bond today and held it until maturity, assuming all coupons are reinvested at the same rate.
Solving for YTM is the inverse of the pricing problem. You know the price, the coupons, the face value, and the maturity — you need to find the rate r that makes the pricing equation hold:
Unlike solving for PV or FV, solving for r in the bond pricing equation can't be done with simple algebra — the rate appears in multiple places (both in the annuity factor and the lump-sum discount factor), so you need either a financial calculator, a spreadsheet solver, or trial-and-error. On a financial calculator, you enter N (periods), PMT (coupon), FV (face value), and PV (negative of the price), then compute I/Y.
The trial-and-error approach works too and builds good intuition. You start by guessing a rate, computing the implied price, and then adjusting: if your computed price is too high, the rate must be too low (remember the inverse relationship), so you try a higher rate.
Notice the logic in step 2: because the bond trades at a premium, the YTM must be below the coupon rate. This is the inverse of the pricing relationship we established — if price > par, yield < coupon rate. Use this as a quick reasonableness check on your answer.
Students sometimes confuse three different measures of bond return. They measure different things, and it's important to know which is which:
The coupon rate is simply the annual coupon divided by face value. It's set at issuance and never changes. It tells you nothing about the return you'll actually earn at today's market price — it's just a contractual rate.
The current yield is the annual coupon divided by the bond's current market price. It measures the income return — how much cash you receive each year relative to what you paid. But it ignores any capital gain or loss you'll realize if you hold the bond to maturity. A discount bond will generate a capital gain (you paid less than par, but receive par at maturity), so the current yield understates the total return. A premium bond generates a capital loss, so the current yield overstates the total return.
The yield to maturity (YTM) captures everything — coupon income plus the capital gain or loss from the price converging to face value at maturity. It's the most complete single measure of a bond's return, and it's the one the market quotes and investors compare.
The ordering makes intuitive sense. The current yield is higher than the coupon rate because you paid less than face value — your $50 coupon is a bigger fraction of the $920 price than of the $1,000 face value. The YTM is higher still because it also accounts for the capital gain: you paid $920 but will receive $1,000 at maturity, and that $80 gain boosts your total return beyond just the coupon income.
We've established that bond prices move inversely with interest rates. But how much a bond's price moves depends on the bond's characteristics. This sensitivity is called interest rate risk — the risk that changes in market interest rates will cause a bond's price to fluctuate. Not all bonds are equally exposed.
Two factors determine how much interest rate risk a bond carries: maturity and coupon rate. Understanding why each matters builds real intuition about how bonds behave.
Longer-maturity bonds are more sensitive to rate changes than shorter-maturity bonds. The reason is that longer bonds have more cash flows in the distant future, and distant cash flows are more sensitive to the discount rate. Think about it this way: if you discount a payment arriving in 2 years, a 1% change in the rate barely moves the present value. But if you discount a payment arriving in 20 years, that same 1% change compounds over 20 periods and moves the present value significantly.
The 20-year bond lost nearly two and a half times as much value as the 5-year bond for the same 2-percentage-point rate increase. This is why investors who are nervous about rising rates tend to shorten the maturity of their bond holdings — shorter bonds are simply less volatile.
The second factor is less obvious: lower-coupon bonds have more interest rate risk than higher-coupon bonds, all else equal. Here's the intuition. A high-coupon bond returns more of your investment sooner (through the larger coupon payments), so its value is less dependent on the distant face-value payment. A low-coupon bond, by contrast, has more of its value concentrated in the face-value repayment at maturity — that distant, lump-sum cash flow is exactly the part most sensitive to rate changes.
To summarize: interest rate risk is highest for bonds that are long-maturity and low-coupon. These are the bonds where the most value is concentrated in distant cash flows. Conversely, short-maturity, high-coupon bonds have the least interest rate risk. This is useful when constructing a portfolio: if you expect rates to rise, you'd prefer shorter, higher-coupon bonds. If you expect rates to fall, longer, lower-coupon bonds will give you the most price appreciation.
So far, we've treated the bond's promised cash flows as certain. But in reality, there's always a chance the issuer can't make its payments — the company might hit financial trouble, or in the extreme case, file for bankruptcy. This possibility is called default risk (or credit risk), and it's a central concern for bond investors.
To help investors assess default risk, independent agencies — most notably Moody's, Standard & Poor's (S&P), and Fitch — assign bond ratings that reflect the issuer's creditworthiness. The rating scales differ slightly across agencies, but the logic is the same: higher ratings mean lower default risk.
The highest-quality bonds are rated AAA (S&P) or Aaa (Moody's). These are issuers with extremely strong capacity to meet their financial obligations — think of the U.S. government (though even sovereign ratings can change, as the U.S. discovered when S&P downgraded it in 2011). Investment-grade bonds range from AAA down to BBB− (or Baa3). Below that threshold, bonds are classified as high-yield bonds (sometimes called "junk bonds") — rated BB+ and below. High-yield issuers have a materially higher probability of default, which is precisely why they must offer higher yields to attract investors.
The relationship between ratings and yields is intuitive: investors demand compensation for bearing default risk. A bond rated BBB will offer a higher yield than an otherwise identical bond rated AAA, because the BBB bond has a greater chance of not paying as promised. The difference in yields between a risky bond and a risk-free benchmark (typically a Treasury bond of similar maturity) is called the credit spread or default risk premium.
A basis point is 1/100th of a percentage point (0.01%), so 160 basis points = 1.60%. Bond traders quote spreads in basis points because rate differences are often small, and it avoids the awkwardness of saying "one point six zero percent" when "160 basis points" is cleaner.
Credit spreads aren't fixed — they widen during economic downturns (when investors become more worried about defaults) and narrow during expansions (when default risk seems lower). This dynamic is part of why bond markets are sensitive barometers of economic sentiment. We'll formalize the relationship between risk and return much more carefully when we study the Capital Asset Pricing Model in Week 9.
This example brings together several ideas: seniority, security, credit risk, and the principle that risk and return go together. A bond isn't just defined by its coupon and maturity — its position in the capital structure and the issuer's overall creditworthiness determine whether those promised cash flows will actually arrive.
Let's step back and see the full picture. Bond valuation is fundamentally a time-value-of-money exercise — you discount a known stream of cash flows at a rate that reflects the bond's risk. Everything we've covered this week follows from that core idea:
The bond pricing formula combines the annuity formula (for coupons) and the lump-sum formula (for face value). If the coupon rate exceeds the YTM, the bond trades at a premium; if the coupon rate is below the YTM, it trades at a discount; if they're equal, it trades at par. Semiannual bonds require halving the coupon, halving the rate, and doubling the periods.
Yield to maturity is the discount rate implied by the market price — it's the total return an investor earns from coupons plus any capital gain or loss. It captures more than the coupon rate or the current yield alone.
Interest rate risk varies across bonds: longer maturity and lower coupons mean greater price sensitivity. These relationships help you understand why different bonds respond differently to the same rate change.
Credit risk adds another dimension: not all promised cash flows are equally certain. Ratings, seniority, and security features determine how much additional yield investors demand beyond the risk-free rate.
Next week, we'll turn from debt to equity and ask the same fundamental question: what is a share of stock worth? We'll formalize this idea when we study dividend discount models in Week 4. Spoiler: the logic is the same — price equals the present value of expected future cash flows — but the cash flows from stocks are less predictable than the contractual payments from bonds, which makes the valuation both more powerful and more uncertain.
| Formula | Expression |
|---|---|
| Bond Price (annual coupon) | C × [1 − (1 + r)−t] / r + F / (1 + r)t |
| Bond Price (semiannual) | (C/2) × [1 − (1 + r/2)−2t] / (r/2) + F / (1 + r/2)2t |
| Coupon Rate | C / F |
| Current Yield | C / Price |
| Yield to Maturity | Rate r that solves: Price = C × [1 − (1+r)−t] / r + F / (1+r)t |
Yield measure ordering: For a discount bond, Coupon Rate < Current Yield < YTM. For a premium bond, Coupon Rate > Current Yield > YTM. For a par bond, all three are equal.
Green Acre Agriculture has a bond with a 7% annual coupon, a $1,000 face value, and 12 years to maturity. If bonds of similar risk are currently yielding 9%, what is the price of this bond? Is it trading at a premium or a discount?
A bond has a 5.5% coupon rate with semiannual payments, a $1,000 face value, and 20 years to maturity. The yield to maturity is 6%. What is the bond's price?
Blue Acre Industries has a bond with an 8% annual coupon, a $1,000 face value, and 6 years to maturity. The bond is currently trading at $1,052. What is its yield to maturity?
A bond has a 4.5% annual coupon, a $1,000 face value, and 15 years to maturity. It currently trades at $850. Compute the coupon rate, current yield, and yield to maturity. Verify that the ordering relationship for a discount bond holds.
Two bonds both have 4% annual coupons and $1,000 face values. Bond A has 5 years to maturity; Bond B has 25 years. Both currently yield 6%. If market interest rates rise to 7%, compute the new price and percentage change for each bond. Which bond is riskier, and why?
Explain, in two to three sentences, why a bond must sell at exactly par value when its coupon rate equals the yield to maturity. Your explanation should reference the relationship between the coupon payments and the discount rate, not just restate the rule.
You're considering two bonds, both with $1,000 face values, annual coupons, and 10 years to maturity, both currently yielding 7%. Bond X has a 9% coupon; Bond Y has a 5% coupon. Suppose you expect interest rates to fall to 5% over the next year. After one year (with 9 years remaining), which bond would have experienced a greater percentage price increase? Compute both and recommend which to buy if you believe rates will fall.
Red Acre Industries (rated A) and Orange Acre Logistics (rated BB) are both issuing 10-year bonds. Red Acre offers a 5.2% coupon; Orange Acre offers a 7.8% coupon. Both are unsecured. Explain why Orange Acre must offer the higher coupon. If the economy enters a recession and default risk increases across all firms, what would you expect to happen to the credit spread between these two bonds? Why?