Comparing Curve Valuation with Single-Yield Pricing

Why Compare Two Bond Valuations?

A bond’s cash flows arrive on different dates. A bootstrapped term structure assigns a discount factor to each date, so each payment is valued using the factor for its own maturity. A single-yield calculation instead discounts every payment using one annual yield. These methods can produce different values because a non-flat yield curve does not discount every maturity at the same rate.

The comparison is useful for understanding what a yield-to-maturity does and does not summarize. For a specified bond price, its yield-to-maturity is the single rate that reproduces that price when applied to every cash flow. It is a convenient summary for that bond, but it does not replace the maturity-specific discount factors used to value other cash-flow patterns.

Definition: Let $D(t)$ be the bootstrapped discount factor from the valuation date to time $t$. It is the current value of one unit of currency payable at time $t$. For known cash flows $C_t$, the curve value is $\sum_t C_tD(t)$. A single-yield value at effective annual yield $y$ is $\sum_t C_t/(1+y)^t$, when time is measured in years and annual effective discounting applies.

The calculations below assume the bond has certain contractual cash flows, annual coupon dates, and a valuation date on a coupon date. Discount factors and the yield are measured from that same date. The curve and the single yield are both applied to the same full cash flows; accrued interest, default risk, taxes, and liquidity adjustments are not included. For payments at non-integer times, use discount factors and a yield-conversion convention appropriate to those exact times rather than treating the times as whole years.

Curve Value and Single-Yield Value

Suppose a bond pays annual coupons $c$ at times $1,\ldots,n$, and pays redemption amount $R$ with its final coupon at time $n$. Its curve value is

P_{\text{curve}} = c\sum_{k=1}^{n-1}D(k) + (c+R)D(n).

At annual effective yield $y$, the single-yield value is

P_y = c\sum_{k=1}^{n-1}\frac{1}{(1+y)^k} + \frac{c+R}{(1+y)^n}.

These are two ways of valuing the same dated payments under different discounting patterns. A price difference is not, on its own, evidence that either calculation is wrong. It indicates that the chosen maturity-specific factors do not match the single-yield factors for all cash-flow dates.

Worked Example: Pricing a Coupon Bond by Both Methods

A hypothetical three-year bond has face value and redemption amount of $100, with annual coupons of $8. The bootstrapped discount factors are $D(1)=0.98$, $D(2)=0.91$, and $D(3)=0.82$. Compare its curve value with its value at a single effective annual yield of 6%.

Method. List each payment by date, then apply the factor for that date. For the single-yield value, use $1/(1.06)^t$ at each date.

Curve value. The cash flows are $8 at years 1 and 2, and $108 at year 3:

P_{\text{curve}} =8(0.98)+8(0.91)+108(0.82) =7.84+7.28+88.56 =\$103.68.

Single-yield value. The 6% discount factors are $1/1.06=0.9433962$, $1/1.06^2=0.8899964$, and $1/1.06^3=0.8396193$. Thus

P_{6\%} =8(0.9433962)+8(0.8899964)+108(0.8396193) =7.5472+7.1200+90.6789 =\$105.35.

The curve value is $103.68, which is $1.67 less than the 6% single-yield value. The curve factors are larger than the 6% factors at years 1 and 2, but the year-3 curve factor is smaller. The relatively large final payment makes the later-date difference important.

Check. The three curve-valued payments add to $7.84+7.28+88.56=103.68$. The three single-yield-valued payments add to approximately $105.3460$, which rounds to $105.35.

Why the Difference Depends on Cash-Flow Timing

The comparison can be made directly, date by date. Define $v_y=1/(1+y)$. Subtracting the two valuations gives

P_{\text{curve}}-P_y = \sum_t C_t\left[D(t)-v_y^t\right].

Each term is the cash flow at a date multiplied by the difference between the curve factor and the single-yield factor at that date. A payment on a date where $D(t)>v_y^t$ contributes positively to the difference; a payment where $D(t)<v_y^t$ contributes negatively. The amount of each cash flow therefore matters as well as the sign of the factor difference.

Justification: By definition, $D(t)$ is the current value of one unit payable at time $t$. A known payment of $C_t$ units at that date can therefore be valued as $C_tD(t)$. Adding the present values of all the dated payments gives $\sum_t C_tD(t)$. Under a single effective annual yield $y$, one unit at time $t$ has value $(1+y)^{-t}$, so the same cash flows have value $\sum_t C_t(1+y)^{-t}$. Subtracting the sums term by term gives the stated difference formula.

For a level-coupon bond with annual coupon $c$ and redemption $R$, the difference can also be written as

P_{\text{curve}}-P_y = c\sum_{k=1}^{n}\left[D(k)-v_y^k\right] + R\left[D(n)-v_y^n\right].

This form separates the coupon contribution from the redemption contribution. If the sum of the coupon-date factor differences is positive, increasing the coupon increases the curve-minus-yield difference, all else held fixed. If that sum is negative, increasing the coupon decreases it. There is no general rule that a higher coupon must make the curve value larger or smaller than the single-yield value: the answer depends on the factor differences at the coupon dates and at redemption.

Worked Example: Comparing Two Coupon Patterns

Use $D(1)=0.98$, $D(2)=0.91$, and $D(3)=0.82$, and compare each bond’s curve value with its value at a 6% annual effective yield. Both bonds have $100 redemption at the end of year 3. Bond A pays an annual coupon of $2; Bond B pays an annual coupon of $10.

Method. Calculate the value at each date under both methods. This shows which payments create the difference rather than relying only on the total prices.

The 6% discount factors are $0.9433962$, $0.8899964$, and $0.8396193$, as in the previous calculation.

Bond A, with $2 coupons. Its cash flows are $2, $2, and $102:

P_{\text{curve,A}} =2(0.98)+2(0.91)+102(0.82) =1.96+1.82+83.64 =\$87.42.

P_{6\%,A} =2(0.9433962)+2(0.8899964)+102(0.8396193) =1.8868+1.7800+85.6412 =\$89.31.

Thus, the curve-minus-yield difference for Bond A is $87.42-89.3079=-\$1.89$, rounded to the nearest cent.

Bond B, with $10 coupons. Its cash flows are $10, $10, and $110:

P_{\text{curve,B}} =10(0.98)+10(0.91)+110(0.82) =9.80+9.10+90.20 =\$109.10.

P_{6\%,B} =10(0.9433962)+10(0.8899964)+110(0.8396193) =9.4340+8.9000+92.3581 =\$110.69.

The curve-minus-yield difference for Bond B is $109.10-110.6920=-\$1.59$. The higher coupon makes the difference less negative in this example. To see why, the factor differences for years 1, 2, and 3 are approximately $0.0366038$, $0.0200036$, and $-0.0196193$. The first two coupon-date differences are positive, while the final-date difference is negative. Increasing the coupon adds payments at all three dates, and here the combined coupon-date contribution is positive.

Check. The total curve values include the $100 redemption discounted at year 3: $100(0.82)=\$82.00$ in each case. The remaining curve value is the discounted coupons: $5.42 for Bond A and $27.10 for Bond B.

A Bond’s Yield-to-Maturity Is Not the Whole Curve

For a given positive price and specified bond cash flows, the yield-to-maturity is the annual effective rate $y$ that solves the single-yield pricing equation. Because each cash flow is positive and its discount factor decreases as $y$ increases, the calculated price decreases as the yield increases. A numerical solution can therefore be found by testing yields and narrowing an interval around the price that matches the target.

When a yield is calculated from a curve-valued price, it reproduces that price for the particular bond used in the calculation. It does not generally reproduce the curve values of other bonds. Different bonds place different weights on different dates, so their equivalent single yields can differ even when they are valued from the same curve.

Worked Example: One Bond’s Yield Does Not Price Every Cash Flow Pattern

A hypothetical two-year bond pays a $10 coupon after one year and $110 after two years. Its bootstrapped factors are $D(1)=0.95$ and $D(2)=181/220$. Find its curve value and show that a 10% single yield reproduces that value. Then compare a two-year zero-coupon payment of $100 under the same two valuation methods.

Method. First value the coupon bond by date. Then apply the proposed single yield to those same payments. Finally, use that same yield on the zero-coupon payment to test whether it also reproduces the curve value.

Coupon bond curve value. Since $181/220=0.8227273$,

P_{\text{curve}} =10(0.95)+110\left(\frac{181}{220}\right) =9.50+90.50 =\$100.00.

Coupon bond value at 10%. The single-yield discount factors are $1/1.10=0.9090909$ and $1/1.10^2=0.8264463$:

P_{10\%} =10(0.9090909)+110(0.8264463) =9.0909+90.9091 =\$100.00.

The bond’s yield-to-maturity is therefore 10%: applying that yield to this bond’s cash flows reproduces its curve value.

Zero-coupon comparison. The curve value of $100 payable in two years is $100(181/220)=\$82.27$. At 10%, its single-yield value is $100/1.10^2=\$82.64$. The difference is approximately $-\$0.37$ when expressed as curve value minus single-yield value.

Check. The 10% yield matches the coupon bond’s $100 curve value, but it does not match the zero-coupon bond’s curve value. The single yield summarizes one cash-flow pattern, not every maturity-specific factor.

Common Mistakes and a Practical Check

Using one spot rate for every payment. A bootstrapped curve valuation uses the factor for each payment’s own maturity. Replacing all factors with one yield changes the method.
Discounting the redemption but omitting the final coupon. If the coupon and redemption are both paid at maturity, the final cash flow is $c+R$.
Comparing different payment dates or valuation dates. Both methods must use the same valuation date and the same dated cash flows.
Assuming a yield is a replacement for the curve. A yield that reproduces one bond’s price need not reproduce another bond’s price.
Assuming the direction of the difference from the coupon rate alone. Check $D(t)-v_y^t$ at each relevant date and weight each difference by its cash flow.

A reliable exam check is to write the cash-flow dates first, then keep a separate column for the curve factor and the single-yield factor. Compute the two present values independently and subtract them only after both totals are obtained. This makes a misplaced coupon or an incorrect maturity factor easier to detect.

Check Your Understanding

For each problem, use the stated valuation date and annual timing. Round money to the nearest cent.

A three-year bond pays $6 at the end of years 1 and 2, and $106 at the end of year 3. The discount factors are $0.97$, $0.92$, and $0.84$. Find its curve value. (Answer: $100.38)
A three-year bond pays annual coupons of $4 and $100 redemption. Its curve factors are $0.96$, $0.91$, and $0.83$. Compare its curve value with its value at a 5% annual effective yield, and give curve value minus yield value. (Answer: curve value $93.80; yield value $97.28; difference $-\$3.48$)
A two-year bond pays annual coupons of $7 and $100 redemption. Its curve factors are $0.95$ and $0.89$. Compare its curve value with its value at a 6% annual effective yield. (Answer: curve value $101.88; yield value $101.83; difference $+\$0.05$)
A two-year bond pays a $6 coupon after one year and $106 after two years. Its curve factors are $0.94$ and $0.86$. Compare the curve value with the value at an 8% annual effective yield. (Answer: curve value $96.80; yield value $96.43; difference $+\$0.37$)

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