Three Rates, Three Different Questions
A bond can have a stated coupon rate, a current yield, and a yield to maturity, and these rates need not be equal. The coupon rate describes the contractual coupon relative to face value. Current yield compares the coupon income with the bond’s market price. Yield to maturity (YTM) uses the price and all promised cash flows, including repayment of face value, to express a single annual discount rate. Distinguishing their bases prevents a common error: treating a bond’s coupon as the return an investor earns at the market price.
Unless stated otherwise, the examples use a conventional fixed-rate bond with annual coupon payments, a single repayment of face value at maturity, no default, and a purchase immediately after a coupon date. Prices are therefore shown without accrued interest. The annual YTM is an effective annual rate because the cash flows occur annually. For a bond with a different payment frequency, the yield period and cash-flow timing must be matched.
What Each Measure Includes
The coupon rate is set by the bond’s terms. It does not change when the bond’s market price changes. For example, a bond with face value $1,000 and annual coupons of $60 has a coupon rate of \(60/1{,}000=6\%\), whether it trades for $950, $1,000, or $1,050.
Current yield uses the market price in its denominator, so it changes when the price changes. It measures annual coupon income as a fraction of the price paid. It does not include any gain or loss between the purchase price and the face value received at maturity. It also does not account for when the coupons arrive.
YTM uses the full schedule of promised payments and the price. It accounts for both coupon payments and the difference, if any, between the purchase price and the face value repaid at maturity. It is a yield calculated from specified cash flows and price, not a guarantee of the return an investor will actually realize. For example, default or selling before maturity can change the investor’s outcome.
| Measure | Calculation basis | What it leaves out or includes |
|---|---|---|
| Coupon rate | Annual coupon divided by face value | Does not change with market price; does not measure price gain or loss |
| Current yield | Annual coupon divided by market price | Includes coupon income relative to price; excludes the gain or loss at redemption |
| Yield to maturity | Rate that discounts promised cash flows to market price | Includes coupons and repayment of face value in the pricing calculation |
Worked Examples
Worked Example: A Premium Bond
A hypothetical bond has face value $1,000, an annual coupon of $80, and two years remaining. Its annual YTM is 5.00%. Find its price, coupon rate, and current yield, then compare the three rates. Assume annual payments and purchase immediately after a coupon date.
Method. First discount each promised payment at the given YTM. Then calculate the coupon rate using face value and current yield using the calculated market price.
The bond pays $80 at the end of year 1 and $1,080 at the end of year 2. Its price is
The price is $1,055.78 to the nearest cent. The coupon rate is
The current yield uses the unrounded price:
Thus the coupon rate is 8.00%, the current yield is about 7.577%, and the YTM is 5.00%. Since the price exceeds face value, this is a premium bond. The ordering is \(5.00\% < 7.577\% < 8.00\%\). The common mistake avoided here is to use face value, rather than market price, in the current-yield denominator.
Worked Example: A Discount Bond
A hypothetical bond has face value $5,000, an annual coupon of $200, two years remaining, and an annual YTM of 7.00%. Calculate its price, coupon rate, and current yield. Classify the bond and compare the rates.
Method. Discount the year-1 coupon and the year-2 coupon-plus-face-value payment. Use the resulting price, not face value, to find current yield.
The promised payments are $200 after one year and $5,200 after two years. Therefore,
To the nearest cent, the price is $4,728.80. The coupon rate is
The current yield, using the unrounded price, is
The price is below face value, so the bond is at a discount. The rate ordering is \(4.00\% < 4.229\% < 7.00\%\). A common mistake is to round an intermediate price too early or incorrectly: the displayed discounting gives $4,728.797276, which rounds to $4,728.80, not $4,728.79. The coupon rate remains based on $5,000 even though the bond trades below face value.
Worked Example: Comparing Premium, Par, and Discount Prices
Consider three hypothetical one-year bonds, each with face value $1,000 and an annual coupon of $60. Their respective annual YTMs are 4.00%, 6.00%, and 8.00%. For each bond, calculate price and current yield, and identify the price category. Assume the next and final payment, one year from now, is the $60 coupon plus $1,000 face value.
Method. For a one-year bond, discount the single $1,060 payment at the stated YTM. The coupon rate is \(60/1{,}000=6.00\%\) for all three bonds.
At a 4.00% YTM,
This bond trades at a premium. Its rate ordering is \(4.00\% < 5.8868\% < 6.00\%\).
At a 6.00% YTM,
This bond trades at par, and all three measures equal 6.00%.
At an 8.00% YTM,
This bond trades at a discount. Its rate ordering is \(6.00\% < 6.1132\% < 8.00\%\). Across the three cases, the coupon rate stays fixed at 6.00%; changes in the required yield change the price, which in turn changes current yield.
Why the Rates Have These Orderings
For a standard annual coupon bond with positive face value, annual coupon \(C\), maturity \(n\), and positive YTM \(y\), write \(v=1/(1+y)\). Its price can also be written
This follows by summing the present values of the coupons as a geometric series and adding the present value of the face value. Multiplying by \(y\) and subtracting \(C\) gives
Because \(v^n\) is positive, the sign of \(yP-C\) is the sign of \(yF-C\). If \(y>C/F\), then \(yP>C\), so \(C/P<y\). If \(y<C/F\), then \(yP<C\), so \(C/P>y\). Also, \(P>F\) for a premium bond and \(P<F\) for a discount bond, so comparing \(C/P\) with \(C/F\) gives the other side of each ordering. The resulting standard pattern is:
| Bond price | Rate ordering for a conventional positive-coupon bond |
|---|---|
| Premium: \(P>F\) | YTM < current yield < coupon rate |
| Par: \(P=F\) | YTM = current yield = coupon rate |
| Discount: \(P<F\) | Coupon rate < current yield < YTM |
The ordering reflects what current yield omits. For a premium bond, the investor pays more than the face value that will be repaid at maturity. That price difference offsets some coupon income in the YTM calculation, so YTM is below current yield. For a discount bond, repayment of face value is greater than the purchase price, adding a gain to the coupons in the YTM calculation; YTM is therefore above current yield. At par, there is no price difference at redemption, and the three measures coincide for the conventional bond described here.
These orderings rely on the stated standard bond setup. They should not be applied without checking the cash-flow structure—for example, if coupons or principal repayment are not fixed as assumed. In all cases, current yield remains the simple ratio of annual coupon to current price; it is not a substitute for calculating YTM from the full cash-flow schedule.
Common Mistakes and Practical Use
- Using the wrong denominator. Coupon rate uses face value; current yield uses market price. Changing the denominator changes the measure.
- Calling current yield the bond’s total yield. Current yield excludes the price gain or loss that occurs when a bond is redeemed for face value.
- Assuming the coupon rate changes with the market. The coupon rate is fixed by the bond’s terms, while market price and current yield can change.
- Comparing mismatched yield periods. A yield and its cash flows must use compatible periods and compounding conventions. The calculations here use annual coupons and annual effective YTM.
- Rounding before the final calculation. Carry sufficient precision through the price calculation, then round money to cents and rates to an appropriate number of significant figures.
For an actuarial or corporate-finance analysis, the three measures answer different questions. Coupon rate describes contractual terms; current yield gives a quick coupon-income comparison at today’s price; YTM incorporates the full promised payment schedule and price under its stated assumptions. Labeling each measure and its denominator makes comparisons more reliable and is aligned with the SOA VEE Accounting and Finance syllabus.
Check Your Understanding
For each problem, keep coupon rate, current yield, and YTM tied to their distinct definitions.
- A hypothetical bond has face value $2,000 and annual coupons of $110. Its market price is $960. Find its coupon rate and current yield, and classify it as premium, par, or discount. (Answer: coupon rate 5.50%; current yield 11.458%; discount.)
- A hypothetical bond has face value $1,000, annual coupons of $70, and a market price of $1,050. Find its coupon rate and current yield. Is it at a premium or discount, and which of these two rates is larger? (Answer: coupon rate 7.00%; current yield 6.667%; premium; coupon rate is larger.)
- A hypothetical bond with face value $1,000 pays $90 annually, sells for $1,080, and has a stated annual YTM of 7.50%. Calculate the coupon rate and current yield, then order all three rates. (Answer: coupon rate 9.00%; current yield 8.333%; YTM 7.50% < current yield 8.333% < coupon rate 9.00%; premium.)
- A two-year hypothetical bond has face value $2,000, annual coupons of $120, and an annual YTM of 5.00%. Calculate its price and current yield, and identify the price category. (Answer: price $2,037.19; current yield approximately 5.890%; premium.)
- A two-year hypothetical bond has face value $1,000, annual coupons of $50, and an annual YTM of 6.00%. Discount both promised payments to find its price, then calculate current yield and compare the three rates. (Answer: price approximately $981.67; current yield approximately 5.093%; coupon rate 5.00% < current yield < YTM 6.00%; discount.)