Comparing Values Under Different Spot Curves

A dated cash-flow stream does not have a single present value independent of its valuation assumptions. Each payment is discounted using the factor for its own maturity, so changing the spot curve can change the stream’s value. A useful comparison does more than report two totals: it shows which maturities create the difference and whether their effects reinforce or offset one another.

The comparison is meaningful only when both valuations use the same valuation date, the same dated cash flows, and the same sign convention. In particular, a liability payment made by the entity being valued is a negative cash flow from that entity’s perspective. Keeping that convention throughout prevents the difference from being assigned the wrong sign.

Definition: Let $C_t$ be the signed cash flow at time $t$, measured from a common valuation date $t=0$. Let $D_A(t)$ and $D_B(t)$ be the discount factors at time $t$ under spot curves A and B. The present value under curve $j$ is $PV_j=\sum_t C_tD_j(t)$, where $j$ is A or B. The contribution at maturity $t$ to $PV_A-PV_B$ is $C_t[D_A(t)-D_B(t)]$.

A discount factor gives the value at time 0 of one unit payable at the stated maturity. Under annual effective spot rates, for example, a payment at integer time $t$ is discounted by the factor corresponding to the $t$-year spot rate, not by a rate selected from another maturity. The same comparison formulas apply when the factors have been supplied directly or obtained under another stated quotation convention.

For this tutorial, cash flows are deterministic, dated amounts and both curves provide a discount factor for every cash-flow date. The valuation date is common, and each factor is applied to the cash flow at its own date. The comparison does not require one curve to have higher rates or higher discount factors at every maturity. If the factors move in different directions across maturities, contributions may have different signs.

Why the Maturity Contributions Add Up

The contribution formula follows by subtracting the two present-value sums term by term. Because the cash-flow dates are identical, each term in the first sum has a matching term in the second:

\begin{aligned} PV_A-PV_B &=\sum_t C_tD_A(t)-\sum_t C_tD_B(t)\\ &=\sum_t \left[C_tD_A(t)-C_tD_B(t)\right]\\ &=\sum_t C_t\left[D_A(t)-D_B(t)\right]. \end{aligned}

Thus, each maturity’s contribution is determined by two items: the signed cash flow $C_t$ and the difference between the two discount factors at that maturity. A maturity with equal factors has zero contribution, regardless of the size of its cash flow. The contributions can be added to check the overall difference, and their signs show whether each date raises or lowers $PV_A-PV_B$.

The sign depends on whether the cash flow is an inflow or an outflow. For a positive receipt, a larger discount factor gives a larger present value. For a negative liability payment, a larger discount factor makes the present value more negative. This is why the cash-flow sign must remain part of the calculation rather than being discarded when comparing factor differences.

Worked Example: A Short Liability Schedule

A hypothetical benefit schedule requires the valued entity to pay $12,000 at the end of year 1 and $4,000 at the end of year 3. Under curve A, the discount factors at those dates are 0.96 and 0.88. Under curve B, they are 0.95 and 0.90. Calculate both present values, the difference $PV_A-PV_B$, and each maturity’s contribution.

Method. Record each required payment as a negative cash flow. Discount it under each curve, then use the contribution formula to show how each maturity affects the difference.

Year 1. The cash flow is $-\$12{,}000$. Its present values are $ -\$12{,}000(0.96)=-\$11{,}520$ under A and $ -\$12{,}000(0.95)=-\$11{,}400$ under B. Its contribution to $PV_A-PV_B$ is $ -\$12{,}000(0.96-0.95)=-\$120$.

Year 3. The cash flow is $-\$4{,}000$. Its present values are $ -\$4{,}000(0.88)=-\$3{,}520$ under A and $ -\$4{,}000(0.90)=-\$3{,}600$ under B. Its contribution is $ -\$4{,}000(0.88-0.90)=+\$80$.

The total values are $PV_A=-\$11{,}520-\$3{,}520=-\$15{,}040$ and $PV_B=-\$11{,}400-\$3{,}600=-\$15{,}000$. Therefore, $PV_A-PV_B=-\$15{,}040-(-\$15{,}000)=-\$40$. The contributions reconcile: $-\$120+\$80=-\$40$.

The year-1 contribution makes A’s value lower than B’s, while the year-3 contribution offsets $80 of that difference. This result is consistent with the factors: at year 1, A’s factor is larger, making the negative liability more negative; at year 3, A’s factor is smaller, making it less negative. The reconciliation is the check. The common mistake this method avoids is treating the liability amounts as positive receipts, which would reverse the signs of the values and contributions.

Reading the Difference Across Maturities

A total present-value difference can conceal offsetting effects. For example, one curve may have a higher discount factor at an early date and a lower one later. In that case, early and late cash flows can pull the total difference in opposite directions. A maturity-by-maturity table makes the source of the result visible.

Worked Example: Identifying Offsetting Maturities

A hypothetical entity has liability payments of $600 at the end of year 1, $1,000 at the end of year 2, and $2,000 at the end of year 4. The discount factors under curve A at those maturities are 0.97, 0.93, and 0.84. The corresponding factors under curve B are 0.98, 0.91, and 0.86. Find both present values and the contribution from each maturity.

Method. Use signed cash flows of $-\$600$, $-\$1{,}000$, and $-\$2{,}000$. Compute each value and contribution at its matching maturity.

At year 1, the present values are $-\$600(0.97)=-\$582$ under A and $-\$600(0.98)=-\$588$ under B. The contribution is $-\$600(0.97-0.98)=+\$6$.

At year 2, the present values are $-\$1{,}000(0.93)=-\$930$ under A and $-\$1{,}000(0.91)=-\$910$ under B. The contribution is $-\$1{,}000(0.93-0.91)=-\$20$.

At year 4, the present values are $-\$2{,}000(0.84)=-\$1{,}680$ under A and $-\$2{,}000(0.86)=-\$1{,}720$ under B. The contribution is $-\$2{,}000(0.84-0.86)=+\$40$.

Adding the discounted cash flows gives $PV_A=-\$582-\$930-\$1{,}680=-\$3{,}192$ and $PV_B=-\$588-\$910-\$1{,}720=-\$3{,}218$. Thus $PV_A-PV_B=-\$3{,}192-(-\$3{,}218)=+\$26$. The independent contribution check gives $+\$6-\$20+\$40=+\$26$. The year-4 contribution is the largest in absolute value; the year-2 effect offsets part of the positive total.

The method avoids comparing the curves as if one were uniformly more favorable. At year 2, curve A has the larger factor, so the negative payment makes A’s present value lower at that maturity. At years 1 and 4, curve A has the smaller factor, making those liability values less negative than under B.

Comparing Curves When Cash Flows Have Different Signs

A stream can include both receipts and payments. The same formulas apply, provided each cash flow is signed from one consistent perspective. A receipt is positive and a payment is negative for the entity whose net present value is being calculated. The effect of a factor difference therefore depends on both its direction and the sign of the cash flow.

Worked Example: A Mixed-Sign Cash-Flow Stream

A hypothetical project has a receipt of $5,000 at the end of year 1, a payment of $8,000 at the end of year 2, and a receipt of $2,000 at the end of year 4. Curve A has discount factors 0.97, 0.91, and 0.80 at those dates; curve B has factors 0.96, 0.93, and 0.78. Calculate both net present values and explain the contributions to $PV_A-PV_B$.

Method. Use the signed stream $+\$5{,}000$, $-\$8{,}000$, $+\$2{,}000$, and discount each amount at its own maturity.

At year 1, the values are $\$5{,}000(0.97)=\$4{,}850$ under A and $\$5{,}000(0.96)=\$4{,}800$ under B. The contribution is $\$5{,}000(0.97-0.96)=+\$50$.

At year 2, the values are $-\$8{,}000(0.91)=-\$7{,}280$ under A and $-\$8{,}000(0.93)=-\$7{,}440$ under B. The contribution is $-\$8{,}000(0.91-0.93)=+\$160$.

At year 4, the values are $\$2{,}000(0.80)=\$1{,}600$ under A and $\$2{,}000(0.78)=\$1{,}560$ under B. The contribution is $\$2{,}000(0.80-0.78)=+\$40$.

Therefore, $PV_A=\$4{,}850-\$7{,}280+\$1{,}600=-\$830$, while $PV_B=\$4{,}800-\$7{,}440+\$1{,}560=-\$1{,}080$. Their difference is $-\$830-(-\$1{,}080)=+\$250$, which agrees with the contribution total $+\$50+\$160+\$40=+\$250$.

The year-2 factor difference applies to a negative cash flow, so it contributes positively to $PV_A-PV_B$. The year-1 and year-4 factor differences apply to positive receipts. This example shows why a rule stated only in terms of “higher factors” is incomplete: the cash-flow sign also matters.

Common Sign Trap and a Practical Check

A frequent error is to calculate liability present values as positive amounts and then report the difference as though the values were signed cash flows. For the year-1 and year-3 liabilities in the first example, that incorrect approach would give $+\$15{,}040$ and $+\$15{,}000$, with a reported difference of $+\$40$. But those are not the present values from the liable entity’s perspective. The signed values are $-\$15{,}040$ and $-\$15{,}000$, so $PV_A-PV_B=-\$40$. The incorrect calculation has reversed the comparison’s sign.

If a question instead explicitly asks for the positive present value of liabilities as an amount owed, that amount is the negative of the signed present value. State which quantity is being reported and do not mix the two conventions. For the signed cash-flow comparison in this tutorial, use the following sequence:

Set the perspective.
Record receipts as positive and payments as negative for the entity being valued.

Match each date to its factors.
Use the discount factor at that cash flow’s maturity under each curve.

Calculate the two totals and the contributions.
Use $C_tD_A(t)$, $C_tD_B(t)$, and $C_t[D_A(t)-D_B(t)]$ at each date.

Reconcile.
Check that the sum of the maturity contributions equals $PV_A-PV_B$.

This comparison is useful when the valuation question is specifically about the effect of using one spot curve rather than another. It separates the total result into maturity-level effects, while preserving the full cash-flow signs. A zero contribution means that the two curves assign the same discount factor at that date; it does not mean that the cash flow itself has zero present value.

Key takeaway: For identical dated cash flows, subtract the two present-value sums term by term. Each signed contribution is the cash flow multiplied by the difference in the two curves’ discount factors at that maturity, and all contributions must sum to the total present-value difference.

Check Your Understanding

Use signed cash flows from the perspective of the entity being valued. For each question, calculate $PV_A-PV_B$; where requested, also identify the maturity contributions.

A liability schedule pays $2,000 at year 1 and $3,000 at year 3. Curve A’s factors are 0.96 and 0.86; curve B’s are 0.95 and 0.88. Find both present values and the contributions at each date. (Answer: $PV_A=-\$4{,}500$, $PV_B=-\$4{,}540$; contributions $-\$20$ and $+\$60$; $PV_A-PV_B=+\$40$.)
A liability schedule pays $4,000 at year 2 and $1,000 at year 4. Curve A’s factors are 0.91 and 0.79; curve B’s are 0.92 and 0.77. Find the two present values and their difference. (Answer: $PV_A=-\$4{,}430$, $PV_B=-\$4{,}450$, and $PV_A-PV_B=+\$20$.)
A liability schedule pays $1,000 at year 1, $2,500 at year 2, and $4,000 at year 3. Curve A’s factors are 0.99, 0.94, and 0.87; curve B’s are 0.98, 0.95, and 0.85. Calculate each maturity contribution and the total difference. (Answer: contributions $-\$10$, $+\$25$, and $-\$80$; $PV_A-PV_B=-\$65$.)
A stream has a receipt of $5,000 at year 1, a payment of $8,000 at year 2, and a receipt of $2,000 at year 4. Curve A’s factors are 0.97, 0.91, and 0.80; curve B’s are 0.96, 0.93, and 0.78. Find the net present value under each curve and the difference. (Answer: $PV_A=-\$830$, $PV_B=-\$1{,}080$, and $PV_A-PV_B=+\$250$.)

Comparing Values Under Different Spot Curves

What You'll Learn