Reading a Nominal Yield Quote
A bond yield quote is useful only when its rate period is clear. If a bond pays coupons twice a year and its quoted yield is a nominal annual rate convertible semiannually, the rate used for each half-year of discounting is half the quoted annual rate. Using the annual quote itself for every coupon period would apply the wrong rate and misprice the bond.
This tutorial focuses on a matching compounding frequency: the nominal annual yield is convertible as many times per year as the bond pays coupons. The central task is to convert the quote into the rate per coupon period, count the periods to each cash flow, and discount using that periodic rate. This rate conversion is aligned with the SOA Exam FM (Financial Mathematics) syllabus.
The word nominal matters. The quote \(i^{(m)}\) is an annual rate divided into \(m\) equal periodic rates; it is not itself the effective annual rate. The effective annual yield equivalent to that quote is \(\left(1+i^{(m)}/m\right)^m-1\). Conversely, a given effective annual yield \(i\) converts to a coupon-period rate of \((1+i)^{1/m}-1\), not \(i/m\). The earlier material on converting effective annual yields covers that distinct conversion.
From the Quote to the Discount Rate
Once the quote and coupon frequencies match, the calculation has three linked parts: identify \(m\), divide the nominal annual yield by \(m\), and discount each cash flow according to its number of coupon periods from the valuation date. If the bond pays a level coupon \(C\) at each of \(n\) remaining dates and redemption amount \(R\) at the final date, its price on a coupon date is
Here \(P\) denotes the bond price, and \(a_{\overline{n}|\,r}=\bigl(1-(1+r)^{-n}\bigr)/r\) is the present-value annuity factor at periodic rate \(r\). The formula presumes fixed coupons, a fixed redemption amount, a constant yield for all periods, and valuation on a coupon date. Between-coupon-date pricing requires additional timing treatment.
Determine whether the rate is nominal and how many times per year it is convertible. Do not infer this from the percent alone.
For a matching quote, the conversion frequency \(m\) equals the bond’s coupon frequency. Divide \(i^{(m)}\) by \(m\) to get the yield per coupon period.
Multiply years to maturity by \(m\) to obtain the number of periods, then use the periodic yield for every coupon and redemption payment.
Why the Periodic Rate Is the Right Rate
A nominal annual yield is defined through its accumulation over the specified compounding periods. In one coupon period, an amount accumulates by the factor \(1+i^{(m)}/m\). After \(k\) such periods, the accumulation factor is the \(k\)th power of that one-period factor. Therefore, the present value of a payment \(k\) periods from the valuation date is the payment multiplied by the reciprocal of that factor.
This also justifies the coupon-bond expression. Each coupon \(C\) is discounted for its own number of periods, so the coupon values sum to \(C\sum_{k=1}^{n}(1+r)^{-k}=C a_{\overline{n}|\,r}\), where \(r=i^{(m)}/m\). The redemption payment is made at period \(n\), so its present value is \(R(1+r)^{-n}\). Adding these components gives the price formula above.
Worked Applications
Worked Example: Semiannual Yield on a Two-Year Bond
A hypothetical bond has face value $1,000, pays coupons semiannually at a 6% annual coupon rate, and redeems at face value in two years. Its yield is quoted as 8% nominal per year, convertible semiannually. Find the price on a coupon date. Use the matching-period method and avoid treating the 8% quote as a half-year rate.
Step 1: Convert the quoted yield. The quote is \(i^{(2)}=0.08\), and there are two coupon periods per year. The semiannual yield is \(0.08/2=0.04\), or 4% per half-year.
Step 2: Determine the cash flows. The semiannual coupon is \(1{,}000(0.06)/2=\$30\). Two years with two payments per year gives \(n=2(2)=4\) periods. The bond pays $30 at each of periods 1 through 4 and an additional $1,000 redemption at period 4.
Step 3: Find the discount factors. At 4% per period, \((1.04)^{-4}\approx0.8548041910\), and \(a_{\overline{4}|\,0.04}=[1-(1.04)^{-4}]/0.04\approx3.6298952243\).
Step 4: Value the coupons and redemption. The coupons have present value \(30(3.6298952243)=\$108.8969\). The redemption has present value \(1{,}000(0.8548041910)=\$854.8042\). Thus \(P\approx108.8968567+854.8041910=\$963.7010\), or $963.70 to the nearest cent.
Check. The annual coupon rate is 6%, below the yield quote of 8% nominal convertible semiannually. A price below the $1,000 redemption value is consistent with that relationship.
Worked Example: Quarterly Quote and a Different Redemption Amount
A hypothetical bond has face value $2,000, pays $30 quarterly, and redeems $2,050 after six months. The yield is 8% nominal per year, convertible quarterly. Find the price on a coupon date. The method is to convert the quote first, then count and discount the two quarterly cash flows.
Step 1: Convert the yield. Here \(m=4\), so the periodic yield is \(0.08/4=0.02\), or 2% per quarter. The six-month term contains \(4(0.5)=2\) quarters.
Step 2: List the payments. The bond pays $30 at the end of quarter 1. At the end of quarter 2, it pays the $30 coupon plus $2,050 redemption, for a total of $2,080.
Step 3: Calculate the factors. The factors are \((1.02)^{-1}\approx0.9803921569\) and \((1.02)^{-2}\approx0.9611687812\). The two-payment annuity factor is \(a_{\overline{2}|\,0.02}=[1-0.9611687812]/0.02\approx1.9415609381\).
Step 4: Value each component. The coupons have present value \(30(1.9415609381)=\$58.2468\). The redemption has present value \(2{,}050(0.9611687812)=\$1{,}970.3960\). The total is approximately \(\$58.2468281+\$1{,}970.3960015=\$2{,}028.6428\), or $2,028.64 to the nearest cent.
Check. The final payment is the coupon plus redemption, but the $2,050 redemption amount itself is discounted separately in the calculation. The price is above face value, which is possible here because the redemption amount exceeds face value; that fact alone does not determine whether the bond is above or below its price under other terms.
Worked Example: A Monthly Quote on a Short-Term Note
A hypothetical two-month note has face value $10,000, pays a $75 coupon at the end of each month, and repays $10,000 at the end of month 2. Its yield is 12% nominal per year, convertible monthly. Find its price on the issue date. This example checks that both the periodic yield and the number of periods use months.
Step 1: Convert the quote. There are \(m=12\) periods per year, so the monthly yield is \(0.12/12=0.01\), or 1%.
Step 2: Count and identify the payments. Two months gives two monthly periods. The note pays $75 at month 1. At month 2 it pays $75 plus $10,000, for a total of $10,075.
Step 3: Find the factors. The factors are \((1.01)^{-1}\approx0.9900990099\) and \((1.01)^{-2}\approx0.9802960494\). Thus \(a_{\overline{2}|\,0.01}=0.9900990099+0.9802960494\approx1.9703950593\).
Step 4: Calculate the price. The coupon value is \(75(1.9703950593)=\$147.7796\). The redemption value is \(10{,}000(0.9802960494)=\$9{,}802.9605\). Hence the price is approximately \(\$147.7796294+\$9{,}802.9604941=\$9{,}950.7401\), or $9,950.74 to the nearest cent.
Check. The note has a 1% monthly yield and two monthly periods. Using a two-year or two-quarter period count would be a time-unit error.
Common Rate-Unit Trap
Suppose the first example is misread and 8% is used as the rate for each half-year. That wrong calculation would give \(30a_{\overline{4}|\,0.08}+1{,}000(1.08)^{-4}\approx30(3.3121268)+1{,}000(0.7350299)=\$834.39\). This is not the price at the quoted yield: it treats the yield as 8% every six months, even though the quote says 8% nominal annually convertible semiannually.
The correction is to divide the nominal quote by its stated conversion frequency: \(0.08/2=0.04\) per half-year. The correct price is therefore \(30a_{\overline{4}|\,0.04}+1{,}000(1.04)^{-4}\approx\$963.70\), as calculated above. Another error is to use half the effective annual yield as the semiannual rate. If the quote is nominal, divide the nominal rate by \(m\); if it is effective annual, convert using the \(m\)th root.
Check Your Understanding
For each problem, convert the quote before setting up or evaluating the cash flows.
- A bond pays monthly coupons and has a yield of 7.2% nominal per year, convertible monthly. What is the monthly yield? (Answer: 0.6% per month.)
- A three-year bond has a yield of 5.4% nominal per year, convertible quarterly. State its quarterly yield and equivalent effective annual yield. (Answer: 1.35% per quarter; approximately 5.5103% effective annually.)
- A bond pays $15 at the end of each quarter for three quarters and redeems $980 with its third coupon. The yield is 8% nominal per year, convertible quarterly. Find the price on a coupon date. (Answer: approximately $966.73.)
- A two-period semiannual bond pays $40 at the end of each half-year and redeems $1,020 with its second coupon. Its yield is 6% nominal per year, convertible semiannually. Find its price. (Answer: approximately $1,037.99.)
- A bond’s yield is 8.16% effective annually, and coupons are semiannual. What is the equivalent yield per half-year? Explain why dividing 8.16% by two is not the correct conversion. (Answer: 4.00% per half-year, since \((1.04)^2=1.0816\); the quote is effective annual, not nominal.)