Reading the Three Bond Rates
A bond quote can display several rates that describe different things. The coupon rate sets the bond’s scheduled interest payment relative to face value. The current yield compares that annual coupon payment with the bond’s market price. The yield to maturity is the rate that equates the present value of the bond’s promised cash flows to its price. These rates are related, but they are not interchangeable.
The relation among them is especially informative when a bond trades above or below face value. For a standard positive-coupon bond redeemable at face value, the current yield lies between the coupon rate and the yield to maturity. Thus, a premium bond has YTM below current yield, which is below its coupon rate; a discount bond has coupon rate below current yield, which is below its YTM.
The results below use annual coupons and an annual effective YTM, so the rate period matches the cash-flow period. Assume a fixed positive coupon, repayment at face value, no default, and a price measured on a coupon date. A bond is at a premium if \(P>F\), at a discount if \(P<F\), and at par if \(P=F\). The ordering results concern the promised cash flows under this model; they do not guarantee the return an investor will actually earn.
Price, Coupon Rate, and YTM
For a bond with the terms just defined, its price is the present value of its remaining coupons and principal. Write \(v=1/(1+y)\), and let \(a_{\overline{n}|}\) denote the present value of \(n\) annual payments of one at rate \(y\). Then the price equation is:
The expression \(a_{\overline{n}|}=(1-v^n)/y\) when \(y\ne 0\); equivalently, \(1-v^n=y a_{\overline{n}|}\). This identity also holds at \(y=0\), because then \(v=1\) and both sides are zero.
This result follows directly by subtracting face value from the price equation. The steps are useful because they show exactly why the coupon rate and YTM determine whether the price is above or below par.
Since \(C=cF\), \(P=F(ca_{\overline{n}|}+v^n)\).
\(P-F=F[ca_{\overline{n}|}-(1-v^n)]\).
Because \(1-v^n=ya_{\overline{n}|}\), \(P-F=F(c-y)a_{\overline{n}|}\).
\(F\) and \(a_{\overline{n}|}\) are positive, so the sign of \(P-F\) is the sign of \(c-y\).
Why Current Yield Falls Between the Other Rates
Current yield uses the market price as its denominator, whereas coupon rate uses face value. To compare current yield with YTM, start with \(CY-y=C/P-y\) and substitute the price equation:
For a positive bond price, the sign of \(CY-y\) is therefore the same as the sign of \(c-y\). If the bond is at a premium, then \(c>y\), so \(CY>y\). Also, \(P>F\) implies \(C/P<C/F\), or \(CY<c\). Together these give \(y<CY<c\). If the bond is at a discount, \(c<y\), so \(CY<y\); and \(P<F\) implies \(CY>c\). Thus \(c<CY<y\). At par, all three rates are equal.
| Price relative to face value | Rate ordering | Interpretation |
|---|---|---|
| Premium: \(P>F\) | \(y<CY<c\) | Coupon rate exceeds current yield, which exceeds YTM. |
| At par: \(P=F\) | \(y=CY=c\) | All three rates are equal. |
| Discount: \(P<F\) | \(c<CY<y\) | Coupon rate is below current yield, which is below YTM. |
This ranking explains the intuition behind the premium and discount cases. A premium buyer pays more than the amount repaid at maturity, so the coupon income alone, measured against the purchase price, is lower than the coupon rate measured against face value. The YTM is lower still because it also accounts for the price moving toward face value by maturity. For a discount bond, the direction reverses: the coupon is large relative to the purchase price, and the gain as the bond moves toward face value raises YTM above current yield.
Worked Applications
Worked Example: A Premium Bond’s Three Rates
A two-year bond has face value \(\$1{,}000\), an annual coupon of \(\$80\), and an annual effective YTM of \(5\%\). Find its price and current yield, then order the coupon rate, current yield, and YTM. Use annual coupon periods and round the final price to cents and rates to four decimal places.
Method. Discount each promised cash flow at the annual YTM. The coupon rate is \(c=80/1{,}000=0.08=8\%\). The cash flows are \(\$80\) at the end of year 1 and \(\$1{,}080\) at the end of year 2.
Rounded to cents, \(P=\$1{,}055.78\). Since this exceeds \(\$1{,}000\), the bond is at a premium. Its current yield, calculated using the unrounded price, is:
The ordering is \(5.0000\%<7.5773\%<8.0000\%\), or \(y<CY<c\). The result is consistent with the premium-price test: the coupon rate exceeds YTM. Common mistake avoided: dividing the coupon by face value again gives the coupon rate, not current yield; current yield uses the bond’s price.
Worked Example: A Discount Bond’s Three Rates
A two-year bond has face value \(\$1{,}000\), an annual coupon of \(\$40\), and an annual effective YTM of \(8\%\). Calculate its price and current yield, then identify the ordering. Round the final price to cents and rates to four decimal places.
Method. Discount the \(\$40\) coupon at the end of year 1 and the \(\$1{,}040\) coupon-plus-principal payment at the end of year 2. The coupon rate is \(40/1{,}000=4\%\).
The price rounded to cents is \(\$928.67\), below face value, so the bond is at a discount. Using the unrounded price:
Thus \(4.0000\%<4.3072\%<8.0000\%\), or \(c<CY<y\). Common mistake avoided: the discount does not make current yield equal to YTM. The current yield counts coupon income relative to price, while YTM also reflects the maturity payment and the price’s movement toward face value.
Worked Example: Reading Yields From a Market Price
A three-year bond has face value \(\$1{,}000\), pays an annual coupon of \(\$60\), and is quoted at \(\$960\) on a coupon date. Calculate its coupon rate and current yield, classify its price, and state what can be concluded about its YTM. Report rates to four decimal places.
Method. First calculate each rate using its own denominator. The coupon rate is \(c=60/1{,}000=6.0000\%\). The current yield is:
Because \(\$960<\$1{,}000\), the bond trades at a discount. For a positive-coupon bond redeemable at face value, a discount means \(c<y\), and the current yield lies between them. Therefore \(6.0000\%<6.2500\%<y\): its YTM is greater than \(6.2500\%\). The exact YTM is not obtained from this ordering alone; it would require solving the bond-price equation. Common mistake avoided: a quoted market price below face value identifies the discount case, but it does not by itself supply a numerical YTM.
Common Interpretation Traps
- Using face value in the current-yield denominator. \(C/F\) is the coupon rate; \(C/P\) is current yield. They are equal only when the bond is at par.
- Reversing the premium ordering. For a premium bond, the correct order is \(y<CY<c\), not \(c<CY<y\). Current yield lies between YTM and coupon rate.
- Treating current yield as total return. Current yield does not include any gain or loss from the bond’s price changing toward face value, so it is not a substitute for YTM.
- Mixing rate periods. The derivation uses annual coupons and an annual effective yield. With more frequent coupons, use the yield per coupon period and the matching coupon amount in the price equation; compare quoted annual rates only after expressing them on a consistent basis.
- Assuming the ordering applies unchanged to every bond. The result relies on fixed positive coupons and redemption at face value. A zero-coupon bond has no positive coupon rate or current yield, and features such as nonstandard redemption terms require a different analysis.
The practical value of the ordering is that it lets an investor or analyst interpret a price and rate quote without confusing cash income with total yield. Given the price relative to face value, one can check whether the quoted coupon rate, current yield, and YTM are logically ordered. Given the coupon rate and YTM, one can predict whether the bond should trade at a premium or discount under the stated assumptions. These relationships are aligned with the SOA VEE Accounting and Finance syllabus.
Check Your Understanding
For each problem, keep coupon rate, current yield, and YTM distinct, and use a consistent annual basis.
- A bond with face value \(\$5{,}000\) pays an annual coupon of \(\$240\) and sells for \(\$4{,}800\). Calculate its coupon rate and current yield, classify its price, and give the ordering relative to YTM. (Answer: \(c=4.8000\%\), \(CY=5.0000\%\), discount, and \(4.8000\%<5.0000\%<y\).)
- A two-year bond has face value \(\$1{,}000\), annual coupon \(\$75\), and annual effective YTM \(6\%\). Find the price and current yield, then order all three rates. Round the price to cents and rates to four decimals. (Answer: \(P=\$1{,}027.50\), \(CY=7.2993\%\), and \(6.0000\%<7.2993\%<7.5000\%\).)
- A bond with face value \(\$2{,}000\) pays an annual coupon of \(\$90\) and is quoted at \(\$1{,}985\). Find the coupon rate and current yield, then state the inequality its YTM must satisfy. (Answer: \(c=4.5000\%\), \(CY=4.5340\%\), the bond is at a discount, and \(4.5000\%<4.5340\%<y\).)
- An analyst says, “This bond sells above face value, so its current yield must exceed its coupon rate.” Is the statement correct? Give the correct ordering for a standard positive-coupon bond redeemable at face value. (Answer: No. A premium bond has \(y<CY<c\), so current yield is below the coupon rate.)