Why the Coupon Rate Is Not the Whole Story
A bond’s coupon rate determines its stated interest payments, but it does not tell an investor how much the bond currently costs or what return is implied by buying it at today’s price. If a bond’s price differs from its face value, the bondholder’s cash flows include not only coupon payments but also a gain or loss as the bond approaches repayment at face value.
Three rates or measures are useful to distinguish. The coupon rate describes the contractual coupon relative to face value. The current yield compares the annual coupon with the bond’s current price. The yield to maturity is the annual discount rate that equates the present value of the promised coupon and principal payments to the current price, under a specified payment and compounding convention. They answer different questions.
The examples here assume a conventional, noncallable bond with fixed annual coupons, repayment of face value at maturity, no default, and annual compounding. They use the quoted price as the amount paid, with no accrued interest adjustment. Under these assumptions, the price-yield relationship is straightforward. Different coupon frequencies or market conventions require matching the cash-flow timing and yield convention; do not apply the annual formula unchanged to semiannual payments.
How Price Produces a Premium or Discount
A bond trades at a premium when its price is greater than face value and at a discount when its price is less than face value. For an ordinary fixed-coupon bond, the coupon rate and the market’s required yield help explain which occurs. If the coupon rate exceeds the market yield, the contractual payments are relatively attractive and the bond tends to trade above face value. If the coupon rate is below the market yield, the bond tends to trade below face value.
This conclusion follows from the present value of the promised payments. With annual coupons \(C\), face value \(F\), maturity \(n\), and annual market yield \(y\), the price is
For a bond whose coupon rate is \(c=C/F\), set \(y=c\). The price then equals face value: the coupon stream is the present value of an annuity paying \(Fc\), and the principal is discounted at the same rate. In detail, the present value of the coupons is \(Fc[1-(1+c)^{-n}]/c=F[1-(1+c)^{-n}]\), while the present value of principal is \(F(1+c)^{-n}\). Their sum is \(F\). Also, each positive promised payment becomes smaller in present value when the discount rate rises. Thus, for this standard bond, a market yield above the coupon rate implies a price below face value; a market yield below the coupon rate implies a price above face value.
Worked Example: A Discount Bond and Its Current Yield
A hypothetical bond has face value \(F=\$1{,}000\), a 6% annual coupon rate, and three years to maturity. The market’s annual required yield is 8%. Find its price, current yield, and relationship between the three measures.
Method. Discount each promised annual payment at the market yield. The annual coupon is \(C=0.06(\$1{,}000)=\$60\). The promised payments are \(\$60\) at the end of year 1, \(\$60\) at the end of year 2, and \(\$1{,}060\) at the end of year 3.
The unrounded calculation gives \(P_0=\$948.458059\ldots\), or \$948.46 to the nearest cent. The bond is at a discount: its price is \(\$1{,}000-\$948.458059\ldots=\$51.541941\ldots\), or \(\$51.54\) below face value.
The coupon rate remains 6%, because it is based on the contractual coupon and face value. Current yield is \(C/P_0\), using the unrounded price: \(\$60/\$948.458059\ldots=0.0632606\ldots\), or 6.3261% to four decimal places. The yield to maturity is 8%, the market discount rate used to price the promised cash flows. Thus the coupon rate is 6%, current yield is about 6.3261%, and yield to maturity is 8%.
The discount to face value also explains why current yield is above the coupon rate: the same \(\$60\) coupon is divided by a price below \(\$1{,}000\). The gap between current yield and yield to maturity reflects the additional return implied by paying less than face value and receiving face value at maturity. Common mistake avoided: using the coupon rate as the bond’s current market yield, or rounding the present-value components prematurely. The component \(1{,}060/(1.08^3)\) is \(841.4622\), and the price is \(948.4581\) before cent rounding; the price increase to face value is \(51.5419\) before cent rounding.
Current Yield and Yield to Maturity Answer Different Questions
Current yield is simple to calculate, but it uses only the annual coupon and current price. It does not include the repayment of principal at maturity or the timing of that repayment. Yield to maturity uses all promised cash flows, including principal. It therefore incorporates the price’s movement toward face value in the return implied by the bond’s cash flows.
A yield to maturity is a rate implied by a bond’s price and promised payments, not a guarantee of the investor’s realized return. That interpretation assumes the bond is held to maturity and pays as promised. A realized return can differ if the bond is sold early, defaults, or coupons are reinvested at rates different from the yield convention’s reinvestment assumption. For borrowing-cost analysis, a bond’s yield may provide a market-based indication of financing cost, but the coupon alone does not.
Worked Example: A Premium Bond
A hypothetical two-year bond has face value \(\$1{,}000\), an annual coupon rate of 9%, and an annual market yield of 6%. Find its price and current yield, and compare them with its coupon rate and yield to maturity.
Method. The annual coupon is \(C=0.09(\$1{,}000)=\$90\). The cash flows are \(\$90\) in one year and \(\$1{,}090\) in two years. Discount both at 6%:
The price is \(\$1{,}055.00\) to the nearest cent, a premium of \(\$55.00\) over face value. The current yield, computed from the unrounded price, is \(\$90/\$1{,}055.0017807=0.0853079\ldots\), or 8.5308% to four decimal places. The coupon rate is 9%, while the yield to maturity is 6%, the annual rate used to discount the payments.
The premium makes the current yield lower than the coupon rate: the annual \(\$90\) coupon is divided by more than \(\$1{,}000\). The yield to maturity is lower still because the investor pays more than face value and receives only \(\$1{,}000\) of principal at maturity. Common mistake avoided: inferring that a 9% coupon means the bond offers a 9% yield. The coupon is fixed by the contract; the market price determines the yields available to a new purchaser.
Comparing Bonds at Different Prices
A bond’s coupon rate can stay unchanged while its market yield and current yield change. This is why comparing coupon rates alone can mislead a borrower or investor. The following example isolates the effect of price using one-year bonds, where the yield calculation is especially transparent.
Worked Example: Same Coupon, Different Market Yields
Two hypothetical one-year bonds each have face value \(\$1{,}000\) and an annual coupon rate of 8%, so each pays \(\$80\) in one year in addition to repayment of principal. Bond A trades at par. Bond B is priced using a 10% annual market yield. Compare their prices, current yields, and yields to maturity.
Bond A. At par, its price is \(\$1{,}000\). Its current yield is \(\$80/\$1{,}000=8\%\). Because the only payment in one year is \(\$1{,}080\), its yield to maturity satisfies \(\$1{,}000=\$1{,}080/(1+y)\), giving \(y=1{,}080/1{,}000-1=8\%\).
Bond B. Discount its \(\$1{,}080\) payment at the 10% market yield:
Its price is \(\$981.82\), a discount to face value. Its current yield is \(\$80/\$981.8182\ldots=0.0814815\ldots\), or 8.1481% to four decimal places. Its yield to maturity is 10%, the rate used to discount the one-year cash flow. Both bonds have the same 8% coupon rate, but their current yields and yields to maturity differ because their prices differ. Common mistake avoided: assuming identical coupon rates imply identical market returns. Bond B’s discount adds a gain when it repays \(\$1{,}000\) after the purchaser paid less than that amount.
Exam Traps and Practical Interpretation
The most frequent error is to use face value as the denominator when calculating current yield. Face value belongs in the coupon-rate calculation; the current market price belongs in the current-yield calculation. For example, a \(\$60\) annual coupon on a \(\$1{,}000\) face-value bond gives a 6% coupon rate. If that bond currently sells for \(\$950\), its current yield is \(\$60/\$950\), not \(\$60/\$1{,}000\).
Another trap is to treat current yield as though it were yield to maturity. Current yield omits the gain or loss between purchase price and principal repayment. It is a useful snapshot of annual coupon income relative to current price, but it is not a complete return measure. The yield to maturity incorporates the full promised cash-flow pattern under its stated assumptions.
For the conventional fixed-coupon bond considered here, the following comparison is useful:
| Measure | Numerator or basis | What it describes |
|---|---|---|
| Coupon rate | Annual coupon divided by face value | Contractual coupon relative to principal |
| Current yield | Annual coupon divided by current price | Annual coupon income relative to the amount paid now |
| Yield to maturity | Discount rate equating all promised cash flows to current price | Return implied by price, coupons, and principal repayment, subject to assumptions |
These relationships are not a substitute for checking the bond’s actual terms. Embedded options, default risk, changing coupons, payment-frequency conventions, transaction costs, and taxes can affect the appropriate valuation or the investor’s realized return. When comparing financing alternatives, match cash-flow timing and yield conventions, and keep contractual coupon payments separate from the market-based return implied by price. This distinction is aligned with the SOA VEE Accounting and Finance syllabus.
Check Your Understanding
For each problem, distinguish what the coupon rate, current yield, and market price tell you. Round rates to four decimal places when needed.
- A hypothetical bond has face value \(\$2{,}000\), an annual coupon rate of 5%, and a current price of \(\$1{,}900\). Calculate its annual coupon and current yield, and identify whether it trades at a premium or discount. (Answer: Annual coupon \(=\$100\); current yield \(=5.2632\%\); discount.)
- A hypothetical bond has face value \(\$1{,}000\), annual coupons of \(\$75\), and a current price of \(\$1{,}050\). Calculate its coupon rate and current yield. (Answer: Coupon rate \(=7.5\%\); current yield \(=7.1429\%\); premium.)
- A hypothetical one-year bond has face value \(\$1{,}000\), an annual coupon rate of 4%, and a market yield of 5%. Find its price and current yield. (Answer: Price \(=\$1{,}040/1.05=\$990.48\); current yield \(=\$40/\$990.4762\ldots=4.0385\%\).)
- A hypothetical two-year bond has face value \(\$1{,}000\), annual coupons of \(\$60\), and an annual market yield of 5%. Is it priced above or below face value? State its yield to maturity and explain how the coupon rate compares with that yield. (Answer: Above face value, because the 6% coupon rate exceeds the 5% market yield; yield to maturity is 5% under the stated pricing assumption.)
- A hypothetical bond’s annual coupon is \(\$80\), its face value is \(\$1{,}000\), and its current price is \(\$960\). A candidate reports its current yield as 8%. Identify the calculation error and give the correct current yield. (Answer: The candidate divided by face value instead of current price; correct current yield \(=\$80/\$960=8.3333\%\).)