Why a Bond Can Sell Below Face Value
A bond promises fixed cash payments: periodic coupons and repayment of face value at maturity. But a buyer does not value those payments at face value automatically. The buyer discounts them at the return required for the bond’s risk and payment timing. If the coupon is small relative to that required return, the present value of the promised payments is less than the amount repaid at maturity. The bond therefore sells for less than face value.
The central comparison is between the coupon paid each period and the interest that would be required on the face value at the market yield. This tutorial develops that comparison and uses it to price bonds with annual and semiannual coupons. The reasoning is aligned with the SOA VEE Accounting and Finance syllabus.
Under these assumptions, the price is the present value of every remaining cash flow. Define \(v_j=1/(1+j)\), the one-period discount factor at yield \(j\). The coupon payments form an \(n\)-payment annuity, and face value is discounted from maturity:
For a bond with annual coupons, \(j\) is the annual required yield. For a bond with semiannual coupons, \(j\) is the required yield per half-year, and \(n\) counts half-year periods. The coupon amount and yield must both be expressed on the same period basis before comparing them.
The Coupon-versus-Required-Return Test
If an investor held a bond with face value \(F\) and required yield \(j\), the interest return associated with that face value for one period would be \(jF\). Thus the relevant comparison is \(C\) against \(jF\), not the bond’s annual coupon rate against an unmatched yield quote. When \(C<jF\), the bond’s coupon falls short of the required period return on face value.
The conclusion follows by comparing the bond’s cash flows with the cash flows of an investment worth \(F\) today at yield \(j\). Such an investment can be represented by payments of \(jF\) at the end of each period and \(F\) at maturity. The bond pays the same maturity amount but pays less than \(jF\) in every coupon period when \(C<jF\). Discounting those lower coupons and the same final face value at the same yield gives a lower present value.
Why the Price Is Below Face Value
The comparison can also be proved directly from the pricing formula. The geometric-series identity gives \(a_{\overline{n}|j}=(1-v_j^n)/j\). Therefore, \(j a_{\overline{n}|j}+v_j^n=1\), and multiplying by \(F\) gives the present value of a stream paying \(jF\) each period plus face value at maturity:
Subtract this expression from the bond-pricing formula. The face-value terms cancel because both streams repay \(F\) at maturity:
Since \(j>0\), \(n>0\), and \(a_{\overline{n}|j}>0\), the sign of \(P-F\) is the sign of \(C-jF\). In particular, \(C-jF<0\) implies \(P-F<0\), proving that the price is below face value. This identity also verifies the par-price condition: when \(C=jF\), the price equals \(F\). If \(j=0\), the stated annuity formula with division by \(j\) does not apply; the price is then the undiscounted sum \(nC+F\), so the positive-yield comparison above is not needed.
Worked Example: Annual Coupon Below the Required Yield
A hypothetical five-year bond has face value \(F=\$1{,}000\), pays annual coupons at a 4% coupon rate, and is valued at a required annual yield of 6%. Find its price and check whether it is below face value. Assume annual payments occur at each year-end and the bond repays face value with its fifth coupon.
Method and period setup. There is one payment period per year, so \(n=5\), \(C=0.04(\$1{,}000)=\$40\), and \(j=0.06\). The required one-year return on face value is \(jF=0.06(\$1{,}000)=\$60\). Since \(\$40<\$60\), the coupon is below the required return.
Calculate the present value. The discount factor to year five is \(v_j^5=(1.06)^{-5}=0.747258\) (shown to six decimals). The annuity factor is \((1-0.747258)/0.06=4.212367\), using the displayed rounded discount factor. Then the coupon present value is \(\$40(4.212367)=\$168.49\), and the face-value present value is \(\$1{,}000(0.747258)=\$747.26\).
Check. The discount from face value is \(\$1{,}000-\$915.75=\$84.25\). The coupon shortfall per year is \(\$60-\$40=\$20\), whose present value is \(\$20(4.212367)=\$84.25\). Thus the present-value comparison and the bond price agree. Money is rounded to the nearest cent at the final step.
Common mistake avoided. Comparing the 4% coupon rate with the \$60 payment would mix a rate with a dollar amount. Convert the coupon rate into the coupon payment first, then compare \(C\) with \(jF\).
Worked Example: Converting a Semiannual Yield Correctly
Worked Example: Converting a Semiannual Yield Correctly
A hypothetical six-year bond has face value \(F=\$4{,}000\), pays coupons twice each year, and has an annual coupon rate of 1.875%. Its required yield is quoted as 8% nominal per year, compounded semiannually. Find the price. Assume the bond makes 12 equal coupons and repays face value with the final coupon.
Method and period setup. The number of periods is \(n=6(2)=12\). Each semiannual coupon is \(C=\$4{,}000(0.01875)/2=\$37.50\). The nominal yield is divided by two to obtain the yield per half-year: \(j=0.08/2=0.04\). The required half-year return on face value is \(jF=0.04(\$4{,}000)=\$160\), so \(C=\$37.50<\$160\).
Calculate the present value. The 12-period discount factor is \(v_j^{12}=(1.04)^{-12}=0.624597\) (shown to six decimals). The annuity factor is \((1-0.624597)/0.04=9.385075\), using the displayed rounded discount factor. The coupons have present value \(\$37.50(9.385075)=\$351.94\), while the face value has present value \(\$4{,}000(0.624597)=\$2{,}498.39\).
Check. The bond is below its \(\$4{,}000\) face value. The per-period coupon shortfall is \(\$160-\$37.50=\$122.50\). Its present value is \(\$122.50(9.385075)=\$1{,}149.67\), and \(\$4{,}000-\$1{,}149.67=\$2{,}850.33\), matching the calculated price. Money is rounded to the nearest cent at the final step.
Common mistake avoided. Using 8% as the yield for each half-year would discount the cash flows at twice the appropriate per-period rate. The required yield is 4% per half-year because the quote is nominal 8% compounded twice per year.
Worked Example: Measuring the Discount for a Different Bond
Worked Example: Measuring the Discount for a Different Bond
A hypothetical five-year bond has face value \(F=\$2{,}500\), pays an annual coupon rate of 4%, and has a required annual yield of 5.5%. Determine its price and quantify how far below face value it sells. Assume five year-end coupon payments, with face value repaid at the end of year five.
Method and comparison. The annual coupon is \(C=0.04(\$2{,}500)=\$100\). Here \(n=5\) and \(j=0.055\), so the required annual return on face value is \(jF=0.055(\$2{,}500)=\$137.50\). Since \(\$100<\$137.50\), the bond must sell below face value.
Calculate the price. The five-year discount factor is \(v_j^5=(1.055)^{-5}=0.765134\) (shown to six decimals). The annuity factor is \((1-0.765134)/0.055=4.270291\), using the displayed rounded discount factor. The coupon present value is \(\$100(4.270291)=\$427.03\), and the face-value present value is \(\$2{,}500(0.765134)=\$1{,}912.84\).
Check. The coupon shortfall is \(\$137.50-\$100=\$37.50\) per year. Its present value is \(\$37.50(4.270291)=\$160.14\), so face value less that amount is \(\$2{,}500-\$160.14=\$2{,}339.86\). A one-cent discrepancy can arise here because the displayed six-decimal factors are rounded. Using unrounded factors gives \(P\approx\$2{,}339.86\); this is the price to report to the nearest cent. The displayed component amounts also sum to \(\$2{,}339.87\) because each was rounded separately. Money is rounded only at the final step, and the checked price is \(\$2{,}339.86\).
Common mistake avoided. Do not subtract the coupon rate from the yield and apply that percentage directly to face value to obtain the price discount. The coupon shortfall is a cash amount each period, and its present value depends on the number and timing of payments.
Common Errors and Practical Checks
The below-face-value conclusion depends on comparing cash flows at a consistent yield and period. A price is not below face value merely because a coupon rate “looks low”; it is below face value when the coupon payment is less than the required return on face value per matching period, under the stated bond assumptions.
- Mixing annual and semiannual units. With semiannual coupons, use the semiannual coupon amount, semiannual yield, and total number of half-year periods. Do not compare an annual coupon amount with a half-year required return.
- Discounting only the face value. The bond price includes the present value of all coupons as well as the present value of face value.
- Misplacing the final coupon. The final coupon and face-value repayment are both paid at maturity. Both are discounted for \(n\) periods, not \(n-1\).
- Applying the result beyond its assumptions. The comparison is for fixed promised cash flows valued at one constant required yield, with no default, embedded option, or accrued interest. Bonds with different cash flows or valuation conventions require those features to be handled explicitly.
A useful exam check is to calculate \(C\), \(jF\), and their difference before entering a pricing formula. If \(C<jF\), the computed price should be below face value. If it is not, check the rate conversion, number of periods, coupon amount, and final payment timing.
Check Your Understanding
For each problem, use the required yield per coupon period and state whether the price is below, equal to, or above face value. Prices are rounded to the nearest cent.
- A hypothetical bond has face value \(\$1{,}000\), four annual periods remaining, an annual coupon rate of 3%, and a required annual yield of 5%. Calculate its price and classify it relative to face value. (Answer: \(C=\$30\), \(jF=\$50\), and \(P\approx\$929.08\), below face value.)
- A hypothetical four-year bond has face value \(\$2{,}000\), semiannual coupons at a 6% annual coupon rate, and an 8% nominal annual required yield compounded semiannually. Calculate the price using eight periods. (Answer: \(C=\$60\), \(j=4\%\), \(jF=\$80\), and \(P\approx\$1{,}865.35\), below face value.)
- A hypothetical three-year bond has face value \(\$1{,}500\), annual coupons of \(\$75\), and a required annual yield of 4%. Calculate the price and explain why it exceeds face value. (Answer: \(jF=\$60<\$75\), and \(P\approx\$1{,}541.63\), above face value.)
- A hypothetical bond has face value \(\$3{,}000\), makes six equal annual coupons, and has a required annual yield of 5%. What annual coupon payment makes its price equal to face value? (Answer: \(C=jF=0.05(\$3{,}000)=\$150\) per year; its price is \(\$3{,}000\).)
- A hypothetical five-year bond has face value \(\$1{,}000\), annual coupons of \(\$40\), and a required annual yield of 6%. Find the coupon shortfall per year and its present value, then state the bond price. (Answer: \(jF=\$60\), shortfall \(=\$20\), \(a_{\overline{5}|0.06}\approx4.212364\), shortfall present value \(\approx\$84.25\), and \(P\approx\$915.75\), below face value.)