Why Maturity Matters
A payment due in one year and a payment due in two years do not necessarily face the same annual spot rate. A spot curve assigns a rate to each maturity, so each dated cash flow has its own discount factor. Applying one rate to every payment can therefore change the present value, even if that rate looks like a reasonable average of the rates in the curve.
This distinction matters whenever a schedule contains payments at multiple dates. A single rate may happen to give the same total value for one particular schedule, but that does not make it a general substitute for the spot curve. The key question is whether the rate used for each payment matches the rate for that payment’s maturity.
The formula assumes all rates are quoted as effective annual rates, the cash flows are measured from the same valuation date, and each spot rate is appropriate for the payment’s maturity and terms. The examples use certain payments and ignore differences in credit risk, liquidity, taxes, and transaction costs. If rates use another compounding convention, the matching discount factors must be used instead; rates from unlike conventions cannot be inserted directly into this formula.
Why One Flat Rate Usually Changes the Value
Suppose a flat effective annual rate \(j\) is used for every payment. The resulting value is $$ P_j=\sum_{k=1}^{n}\frac{C_{t_k}}{(1+j)^{t_k}}. $$ For this to reproduce the spot-curve value for every possible cash-flow schedule, the discount factors must agree at every maturity: $$ (1+j)^{-t}=(1+i_t)^{-t}. $$ For positive maturity \(t\) and rates greater than \(-1\), taking the same positive \(t\)-th root on both sides gives \(j=i_t\). Thus one flat rate can match the curve at every maturity only if the spot rates are the same at those maturities. If the spot rates differ, using one rate changes at least one maturity-specific discount factor.
There is an important qualification: a flat rate can match the total value of a particular schedule by coincidence. That rate is then specific to the cash flows being valued. A different schedule, with different amounts or payment dates, may not have the same value under that rate. Matching one total does not establish that the flat rate reproduces the spot curve.
Measure each date from the valuation date and identify its maturity.
Use the rate and compounding convention applicable to that maturity.
Calculate each present value before summing the amounts.
Compare the discount factors at each maturity; an average rate does not generally preserve them.
Worked Valuations
Worked Example: Comparing a Spot Curve with a Flat Rate
A hypothetical liability pays $1,050 in one year and $1,188.10 in two years. The effective annual spot rates are 5% for one year and 9% for two years. Calculate the spot-curve present value, then compare it with the value obtained by using a flat 7% effective annual rate for both payments. Report values to the nearest cent.
Method. Discount each payment at its own maturity-specific spot rate, then calculate the alternative flat-rate value using the same cash flows.
Spot-curve value. For the one-year payment, the discount factor is \(1/1.05\), so its present value is $$ \frac{1050}{1.05}=1000. $$ For the two-year payment, the discount factor is \(1/(1.09)^2=1/1.1881\), so its present value is $$ \frac{1188.10}{1.1881}=1000. $$ The total is \(1000+1000=\$2{,}000.00\).
Flat 7% value. Use the same 7% rate for both maturities: $$ \frac{1050}{1.07}=981.308411\ldots,\qquad \frac{1188.10}{(1.07)^2} =\frac{1188.10}{1.1449} =1037.732553\ldots. $$ Adding the unrounded amounts gives \(2019.040964\ldots\), or \(\$2{,}019.04\) to the nearest cent.
The flat-rate value exceeds the spot-curve value by \(2019.040964\ldots-2000=19.040964\ldots\), or \(\$19.04\) to the nearest cent. The check is that the 7% discount factors differ from the 5% one-year and 9% two-year discount factors; the cash flows therefore cannot retain both of their original present values.
Common mistake avoided: The arithmetic average of 5% and 9% is 7%, but that average does not make the two maturity-specific discount factors equal. In particular, the flat-rate value is \(\$2{,}019.04\), not \(\$2{,}004.77\).
Worked Example: Same Total Payments, Different Timing
Two hypothetical liability schedules each pay a total of $2,000. Schedule A pays $1,000 in one year and $1,000 in two years. Schedule B pays $1,500 in one year and $500 in two years. The effective annual spot rates are 4% at one year and 10% at two years. Find each schedule’s spot-curve present value, then compare the values under a flat 7% rate. Report values to the nearest cent.
Method. Apply each spot rate to its matching maturity. Then apply 7% to both dates for the comparison. The two schedules have the same nominal total, but the amounts assigned to each maturity differ.
The spot discount factors are $$ D(1)=\frac{1}{1.04}=0.9615384615\ldots,\qquad D(2)=\frac{1}{(1.10)^2}=\frac{1}{1.21}=0.8264462810\ldots. $$ For Schedule A, the one-year payment has present value \(1000/1.04=961.538461\ldots\), and the two-year payment has present value \(1000/1.21=826.446281\ldots\). Thus $$ P_A=961.538461\ldots+826.446281\ldots =1787.984742\ldots =\$1{,}787.98. $$
For Schedule B, the corresponding present values are \(1500/1.04=1442.307692\ldots\) and \(500/1.21=413.223140\ldots\). Therefore $$ P_B=1442.307692\ldots+413.223140\ldots =1855.530833\ldots =\$1{,}855.53. $$
Under a flat 7% rate, the discount factors are \(1/1.07=0.9345794393\ldots\) and \(1/(1.07)^2=1/1.1449=0.873438\ldots\). For Schedule A, $$ P_{A,7\%}=\frac{1000}{1.07}+\frac{1000}{1.1449} =934.579439\ldots+873.438\ldots =\$1{,}808.02. $$ For Schedule B, $$ P_{B,7\%}=\frac{1500}{1.07}+\frac{500}{1.1449} =1401.869159\ldots+436.719\ldots =\$1{,}838.59. $$
The flat-rate calculation does not preserve either spot-curve value: it overstates Schedule A by \(\$20.03\) and understates Schedule B by \(\$16.94\), to the nearest cent. The check is that both schedules have the same total undiscounted payments but different maturity allocations; their values respond differently to the maturity-specific discount factors.
Common mistake avoided: Equal nominal totals do not imply equal present values, and a 7% arithmetic average of the two spot rates does not correctly value either schedule. The amount paid at each maturity matters.
A Flat Rate May Fit One Payment Date but Not Another
Worked Example: Calibrating to One Maturity Misvalues the Other
A hypothetical liability pays $10,400 in one year and $11,881 in two years. The effective annual spot rates are 4% and 9%, respectively. A reviewer proposes using 4% as a single rate because it correctly values the one-year payment. Calculate the spot-curve value and the value under the proposed flat rate. Report values to the nearest cent.
Method. First value each payment with its own spot rate. Then reuse 4% at both maturities and compare. This tests whether matching one payment date is enough to replace the curve.
At one year, the spot-curve present value is $$ \frac{10400}{1.04}=10000. $$ At two years, \(1.09^2=1.1881\), so the present value is $$ \frac{11881}{1.1881}=10000. $$ The spot-curve total is therefore \(\$20{,}000.00\).
Using 4% for both payments gives $$ \frac{10400}{1.04}=10000,\qquad \frac{11881}{(1.04)^2} =\frac{11881}{1.0816} =10984.652367\ldots. $$ The flat-rate total is \(20984.652367\ldots\), or \(\$20{,}984.65\) to the nearest cent. It overstates the spot-curve value by \(\$984.65\).
The one-year payment is unchanged because 4% is its spot rate. The two-year payment is discounted at 4% instead of its 9% spot rate, producing the difference. As a further check, using 9% as a flat rate would value the two-year payment at $10,000 but value the one-year payment at \(10400/1.09=9541.284403\ldots\), giving a total of \(\$19{,}541.28\). Neither maturity’s spot rate works as a universal replacement here.
Common mistake avoided: A rate that reproduces one component of a schedule need not reproduce the full schedule. Check every distinct payment maturity rather than validating a proposed rate against only one payment.
What a Single-Rate Approximation Can and Cannot Do
A flat rate can still be useful as an approximation if its purpose and limitations are clear. For a particular schedule, it might give a value close to the spot-curve value. It might even happen to give exactly the same total, because overstatements at some maturities offset understatements at others. That coincidence is specific to the chosen cash flows: changing the amounts or dates can remove the offset.
The spot curve, by contrast, supplies a maturity-specific discount factor for each payment. This lets the valuation reflect the rates applicable to the full schedule instead of compressing the schedule into a single number. When the spot rates differ materially across the payment dates, replacing them with one rate can create a material valuation error. For exam calculations, the safest approach is to write one discounted term per payment before adding the terms.
Check Your Understanding
For each problem, use the effective annual spot rates stated and round dollar values to the nearest cent. Each problem is designed to test maturity-by-maturity discounting rather than selection of an average rate.
- A schedule pays $8,000 in one year and $9,450 in two years. The one-year and two-year spot rates are 3% and 5%. Find the spot-curve present value. (Answer: $16,338.42)
- For the schedule in question 1, calculate the value using a flat 4% effective annual rate for both payments. (Answer: $16,429.36)
- A schedule pays $5,300 in one year and $1,040.40 in two years. The one-year spot rate is 6%, and the two-year spot rate is 2%. Find the spot-curve value and the value under a flat 4% rate. (Answer: Spot-curve value $6,000.00; flat-rate value $6,058.06)
- A liability pays $5,000 at the end of each of the next two years. The one-year and two-year spot rates are 5% and 8%. Compare the spot-curve value with the value under a flat 6% rate. (Answer: Spot-curve value $9,047.91; flat-rate value $9,166.96)