Using Redemption Values Other Than Par

Why the Redemption Amount Matters

In a coupon bond, the last payment usually contains two components: the final coupon and the redemption payment. It is tempting to treat redemption as equal to par automatically. That shortcut is wrong whenever the bond’s contract specifies a different amount, such as a redemption premium or a repayment below face value. Since the terminal payment is discounted using the factor being bootstrapped, using the wrong redemption amount changes the inferred factor.

The method is the same sequential logic used in coupon-bond bootstrapping: subtract the present value of cash flows at already bootstrapped dates, then divide the remaining price by the cash flow at the new terminal date. The key is to use the cash flow the bond actually promises. The equations below assume deterministic contractual cash flows, no default, no embedded options, and prices measured at time 0 after any settlement and accrued-interest adjustments have been handled. No mortality or probability distribution is involved.

Definition: Let $D_k$ denote the time-0 value of one dollar paid at coupon date $k$. A bond with face amount $N$, coupon amount $C_j$ at date $j$, and redemption amount $R$ at maturity date $n$ has price $$ P=\sum_{j=1}^{n-1} C_jD_j+(C_n+R)D_n. $$ The redemption amount $R$ is the contractual amount paid at maturity, separate from the final coupon $C_n$. It need not equal the face amount $N$.

For a level-coupon bond with annual coupon rate $r$ stated as a proportion of face amount, the annual coupon is $C=rN$. Unless the contract says otherwise, changing the redemption amount does not change this coupon calculation. For example, a bond with face amount $100, a 5% annual coupon rate, and redemption of $110 pays a $5 coupon, not a $5.50 coupon.

Isolating the Terminal Discount Factor

Suppose all discount factors before maturity have already been bootstrapped. Their associated coupon payments have known present values. The observed bond price then determines the remaining value assigned to the terminal cash flow.

D_n=\frac{P-\displaystyle\sum_{j=1}^{n-1}C_jD_j}{C_n+R}.

The denominator is the entire maturity-date payment: final coupon plus redemption. It is not the coupon plus face amount unless the contractual redemption amount actually equals face amount. A positive terminal cash flow makes the algebraic solution unique. In an arbitrage-free valuation with positive prices and ordinary positive cash flows, the inferred discount factor should also be positive.

List the contractual payments.
Record each coupon and the stated redemption amount separately. Confirm whether the quoted coupon rate is applied to face amount.

Value earlier payments.
Multiply each earlier coupon by its already bootstrapped discount factor and add those present values.

Find the residual price.
Subtract the earlier-coupon present value from the observed bond price.

Divide by the actual terminal payment.
The residual is $(C_n+R)D_n$, so divide by $C_n+R$, not by a payment based on an assumed redemption value.

Why the Formula Holds

The pricing equation follows from valuing each promised payment separately at time 0. A payment of $C_j$ at date $j$ contributes $C_jD_j$ to the price. At maturity, both the last coupon and redemption are paid on the same date, so their combined contribution is $(C_n+R)D_n$. Adding the contributions gives the bond price. Once all earlier-date contributions are known, rearranging the equation isolates the terminal factor:

P-\sum_{j=1}^{n-1}C_jD_j=(C_n+R)D_n \quad\Longrightarrow\quad D_n=\frac{P-\sum_{j=1}^{n-1}C_jD_j}{C_n+R}.

This rearrangement depends on the stated cash flows and timing being correct. It does not apply unchanged to a bond with uncertain payments, an embedded option that changes the cash flows, or a price that has not been adjusted to the valuation date. Those cases require the relevant cash-flow or price treatment before bootstrapping.

Worked Example: A Bond Redeemed Above Face Amount

A default-free bond has face amount $100, pays annual coupons at 5% of face amount, and is redeemed for $110 at the end of year 2. Its time-0 price is $108.25. The year-1 discount factor has already been bootstrapped as $D_1=0.95$. Find $D_2$, then quantify the error from assuming redemption is $100. Assume annual payments and no accrued-interest adjustment is needed.

Method. Use the known year-1 factor to value the first coupon, then divide the price remaining for year 2 by the actual year-2 payment. This avoids substituting face amount for the stated redemption amount.

The annual coupon is $0.05(100)=\$5$. The year-1 coupon’s present value is

5(0.95)=\$4.75.

The year-2 payment is the $5 final coupon plus $110 redemption, or $115. Therefore,

D_2=\frac{108.25-4.75}{5+110} =\frac{103.50}{115} =0.9000.

Check: $4.75+115(0.9000)=4.75+103.50=\$108.25$, the quoted price.

If redemption were incorrectly treated as $100, the denominator would be $5+100=105$, giving $103.50/105=0.985714\ldots$, or 0.9857 to four decimal places. Repricing the bond’s actual cash flows with that mistaken factor gives $4.75+115(0.985714\ldots)=\$118.11$ to the nearest cent, not $108.25. The large discrepancy exposes the incorrect redemption assumption.

Using the Contractual Amount, Not a Par Convention

The redemption amount may be stated in dollars or as a percentage of face amount. Convert it to the same monetary units as the price and coupons before forming the equation. For instance, if a $1,000 bond is redeemable at 104% of face, then $R=1.04(1000)=\$1,040$. The maturity cash flow is that amount plus the final coupon.

Worked Example: Redemption Below Face Amount and a Distorted Factor

A three-year bond has face amount $100, pays an annual coupon of $7, and is contractually redeemed for $90. Its price is $91.92. The earlier discount factors are $D_1=0.98$ and $D_2=0.95$. Find $D_3$. Then calculate the factor obtained if redemption is mistakenly assumed to equal $100. Assume the bond has certain annual payments and its quoted price is the time-0 value.

Method. Value the first two coupons using the known factors and use the actual $90 redemption in the terminal payment. The $7 coupon is specified as an amount, so no coupon-rate conversion is needed.

The present value of the first two coupons is

7(0.98)+7(0.95)=6.86+6.65=\$13.51.

The terminal payment is the final $7 coupon plus $90 redemption, or $97. The residual price is $91.92-13.51=\$78.41$. Thus,

D_3=\frac{78.41}{97}=0.8083505\ldots=0.8084

to four decimal places. The check using the unrounded factor is $13.51+97(78.41/97)=13.51+78.41=\$91.92$.

If redemption is wrongly assumed to be $100, the assumed terminal payment becomes $7+100=\$107$. The resulting factor is

\frac{78.41}{107}=0.7328037\ldots=0.7328

to four decimal places. Using that mistaken factor with the actual $97 terminal payment gives a price of $13.51+97(0.7328037\ldots)=\$84.59$ to the nearest cent, not $91.92. The wrong redemption assumption has materially distorted the inferred terminal factor.

Keep Coupon Basis and Redemption Basis Separate

A quoted coupon rate is normally applied to the bond’s face amount. Redemption is a different contractual term. The two amounts coincide only in particular bonds; neither determines the other automatically. When a problem gives both a face amount and a redemption percentage, first convert the redemption percentage to dollars, but calculate coupons using the stated coupon basis.

Worked Example: A Redemption Premium on a Larger Face Amount

A bond with face amount $1,000 pays annual coupons at 2.5% of face amount and is redeemed for $1,040 at the end of year 2. Its time-0 price is $982.75. Given $D_1=0.97$, find $D_2$. Assume the bond has certain annual payments, and the quoted price is measured at time 0.

Method. Convert the coupon rate and redemption amount separately. This prevents the common error of calculating the coupon on the redemption amount.

The annual coupon is $0.025(1000)=\$25$, and the maturity payment is $\$25+\$1,040=\$1,065$. The present value of the year-1 coupon is

25(0.97)=\$24.25.

The residual price is $982.75-24.25=\$958.50$, so

D_2=\frac{958.50}{1,065}=0.9000.

Check: $24.25+1,065(0.9000)=24.25+958.50=\$982.75$.

A mistaken coupon calculation of $0.025(1,040)=\$26$ would produce a year-1 present value of $26(0.97)=\$25.22$ and a maturity payment of $26+1,040=\$1,066$. That incorrect convention would imply $(982.75-25.22)/1,066=957.53/1,066=0.8982$ to four decimal places. It does not reprice the bond’s actual $25 coupons and $1,040 redemption.

Common Exam Traps and Checks

Replacing redemption with face amount. Face amount is used to calculate a coupon when the stated rate is based on face. It is not a substitute for the contractual maturity repayment.
Omitting the final coupon. The denominator for $D_n$ is $C_n+R$, not $R$ alone. The final coupon is paid at maturity and must be discounted with the same factor.
Adding the coupon twice. Earlier coupons belong in the subtracted present-value sum; only the final coupon belongs in the terminal payment denominator.
Mixing units. If price and coupon are stated per $100 face but redemption is specified for a $1,000 bond, convert all amounts to a common basis before solving.
Rounding too early. Keep the residual and factor unrounded through the calculation, then round the reported factor. Repricing is most informative when performed with the unrounded value.

Key takeaway: Bootstrap the terminal factor from the bond’s actual final cash flow. After earlier coupons have been valued, divide the residual price by the final coupon plus the stated redemption amount. A par-redemption assumption is valid only when the contract specifies redemption at par.

These cash-flow equations are aligned with the SOA Exam FM (Financial Mathematics) syllabus. In a multi-bond bootstrap, apply the same rule at each new maturity: preserve the stated coupon and redemption terms, use only factors already determined at earlier dates, and check the resulting factor by repricing the bond.

Check Your Understanding

For each problem, assume certain contractual payments and a time-0 bond price. Use the stated discount factors and keep all cash amounts in the same units.

A two-year bond has face amount $100, annual coupons of $4, redemption of $108, price $108, and $D_1=0.95$. Find $D_2$. (Answer: $D_2=0.9304$ to four decimal places.)
A three-year bond pays an annual coupon of $5 and is redeemed for $97. Its price is $96, $D_1=0.96$, and $D_2=0.90$. Find $D_3$, then find the factor obtained by mistakenly assuming redemption at $100. (Answer: Correct $D_3=0.8500$; mistaken factor $=0.8257$, both to four decimal places.)
A $1,000-face bond pays annual coupons at 2.5% of face and is redeemed for $1,040 at year 2. Its price is $982.75 and $D_1=0.97$. Find $D_2$. (Answer: $D_2=0.9000$.)
A two-year bond pays $6 annually, is redeemed for $90, and has price $97.08. Given $D_1=0.98$, find $D_2$. (Answer: $D_2=0.9500$.)
A three-year bond pays $5 annually and has price $96.00. Given $D_1=0.96$, $D_2=0.91$, and $D_3=0.84$, infer its contractual redemption amount $R$. (Answer: $R=\$98.15$ to the nearest cent.)

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