Why Spot-Rate Valuations Go Wrong
A spot-rate valuation discounts each future payment using the rate applicable to that payment’s maturity. The arithmetic is usually straightforward; errors tend to enter earlier, when a rate is matched to the wrong date, a time is measured in the wrong units, a quoted compounding convention is ignored, or a discount factor is applied twice. Each error can produce a plausible-looking value, so a reliable solution begins by making the timing and quotation conventions explicit.
This tutorial focuses on diagnosing and correcting those errors. It is aligned with the SOA Exam FM (Financial Mathematics) syllabus. The valuation date is time 0, and each cash flow is treated as certain and paid at its stated time.
A spot rate is not automatically one rate for every payment date. If a schedule has payments at several maturities, value each payment using its own rate or corresponding discount factor, then add the present values. A discount factor is already the multiplier that converts a future amount into its present value.
A Short Derivation and a Reliable Procedure
The effective annual spot-rate formula follows directly from the definition of accumulation. If \(P\) is invested at time 0 at the annual-effective spot rate for maturity \(t\), it accumulates to \(P(1+s_t)^t\) at time \(t\). To produce a payment \(C_t\), the present amount must satisfy \(P(1+s_t)^t=C_t\). Solving for \(P\) gives the discounting formula.
This argument also explains why the maturity matters: the accumulation factor used in the equation is the one specified for the payment’s date, not a convenient rate selected from elsewhere on the curve. Before calculating, use the following checks.
Write each amount beside its time from the valuation date. Convert months or days to the time unit required by the rate quotation.
Determine whether the quoted rate is annual effective, nominal convertible \(m\) times per year, or a force of interest. Do not substitute one convention’s formula for another.
Use the spot rate or discount factor for that payment’s date. Do not apply a single maturity’s rate to the whole schedule unless the problem specifies a flat curve.
Multiply each future payment by its applicable discount factor exactly once. Add the resulting present values and check that the units and directions make sense.
For a nominal annual rate \(i^{(m)}\) convertible \(m\) times per year, the accumulation factor over \(t\) years is \((1+i^{(m)}/m)^{mt}\), when the quoted convention applies over that interval. Thus its discount factor is the reciprocal. For a force of interest \(\delta\) that is constant over the interval, accumulation for \(t\) years is \(e^{\delta t}\), so the discount factor is \(e^{-\delta t}\). These formulas are not interchangeable without converting to a consistent convention.
Worked Valuations and the Errors They Expose
Worked Example: Using the Rate for Each Payment’s Maturity
A fund must pay $1,000 in one year, $2,000 in two years, and $3,000 in three years. The annual-effective spot rates are 4% for one year, 5% for two years, and 6% for three years. Find the present value. Identify the error in using the three-year rate for every payment.
Method. Apply the effective annual spot-rate formula separately at each maturity. The cash-flow times are already in years, and each rate is an annual-effective spot rate.
For the one-year payment, the discount factor is \(1/(1.04)^1=0.961538\), so its present value is \(1{,}000(0.961538)=\$961.54\).
For the two-year payment, the discount factor is \(1/(1.05)^2=1/1.1025=0.907029\), so its present value is \(2{,}000(0.907029)=\$1{,}814.06\).
For the three-year payment, the discount factor is \(1/(1.06)^3=1/1.191016=0.839619\), so its present value is \(3{,}000(0.839619)=\$2{,}518.86\).
Check. Each individual present value is less than its future payment because all three spot rates are positive. The total present value is therefore less than the total future payments of $6,000.
Common mistake. Applying the three-year rate to all three cash flows gives \(1{,}000/1.06+2{,}000/(1.06)^2+3{,}000/(1.06)^3=\$5{,}242.25\). The last term is correct, but the first two use a rate for the wrong maturities. The correction is to match each payment to its own spot rate, giving $5,294.46. The mistake is not repaired by the fact that all rates are annual-effective; maturity matching still matters.
Worked Example: Keeping the Time Unit Consistent
A payment schedule has $500 due in six months and $1,200 due in 18 months. The annual-effective spot rate is 3% for the six-month maturity and 4% for the 18-month maturity. Find the present value. Show why entering the times as 6 and 18 is incorrect.
Method. The rates are annual-effective, so express the payment dates in years: six months is \(0.5\) years and 18 months is \(1.5\) years.
The first discount factor is \(1/(1.03)^{0.5}=0.985329\). Its discounted payment is \(500(0.985329)=\$492.66\). The second discount factor is \(1/(1.04)^{1.5}=0.942866\). Its discounted payment is \(1{,}200(0.942866)=\$1{,}131.44\).
Check. The 18-month payment is discounted for longer and at a higher annual-effective rate; its discount factor, \(0.942866\), is lower than the six-month factor, \(0.985329\).
Common mistake. Treating the month counts as years would produce \(500/(1.03)^6+1{,}200/(1.04)^{18}\), approximately $1,011.09. That calculation treats six months as six years and 18 months as 18 years, so it over-discounts both payments. The correction is to use \(t=0.5\) and \(t=1.5\) with annual-effective rates. The exponent’s unit is part of the formula, not a free choice.
Worked Example: Respecting a Nominal Compounding Convention
A hypothetical settlement pays $10,000 exactly two years from now. The applicable spot rate is a nominal annual rate of 12% convertible monthly. Find the present value and show the error from treating 12% as an annual-effective rate.
Method. A nominal annual rate of 12% convertible monthly has a monthly rate of \(0.12/12=0.01\). Two years contain \(12(2)=24\) monthly periods, so the two-year accumulation factor is \((1.01)^{24}=1.269735\).
The present value is the payment divided by this accumulation factor:
The equivalent annual-effective rate is \((1.01)^{12}-1=0.126825\), or about 12.6825%. Using that effective rate for two years gives the same result: \(10{,}000/(1.126825)^2=\$7{,}875.66\), apart from rounding.
Check. The nominal-rate accumulation factor exceeds 1, so the present value is below $10,000. The equivalent-rate calculation independently confirms the convention-specific calculation.
Common mistake. Treating the stated 12% nominal rate as an annual-effective rate gives \(10{,}000/(1.12)^2=\$7{,}971.94\). That calculation uses a different accumulation convention. The correction is either to use 24 monthly periods at 1% per month or to convert to the equivalent annual-effective rate before discounting.
Worked Example: Avoiding a Second Discount
At time 0, the discount factors are \(D(0,1)=0.96\) and \(D(0,2)=0.90\). A liability pays $500 in one year and $1,000 in two years. Find its present value and explain why multiplying by the discount factors twice is wrong.
Method. A discount factor is the present value at time 0 of one dollar payable at its stated maturity. Therefore, multiply each future payment by its factor once.
The one-year payment has present value \(500(0.96)=\$480\). The two-year payment has present value \(1{,}000(0.90)=\$900\). Adding the two amounts gives:
Check. Both discount factors are less than 1, and the total present value of $1,380 is less than the total future payments of $1,500.
Common mistake. Discounting the already-discounted amounts again gives \(500(0.96)^2+1{,}000(0.90)^2=\$1{,}270.80\). The first multiplication by each \(D(0,t)\) has already moved the corresponding payment back to time 0. A second multiplication changes the calculation to an unintended extra discount; it is not part of the spot-rate valuation.
Practical Checks Before Accepting a Value
A correct calculation should be traceable from each payment to its own present value. A small table makes that trace explicit and helps reveal a rate or timing mismatch before it contaminates the total.
| Check | Question to ask | Typical warning sign |
|---|---|---|
| Maturity | Does this rate or factor correspond to this payment’s date? | The same rate is used for every payment without a stated flat-curve assumption. |
| Time unit | Is the exponent in years, months, or compounding periods as required? | A rate quoted per year is raised to a month count. |
| Compounding convention | Is the rate effective, nominal with a stated frequency, or continuous? | A nominal quote is inserted as though it were annual effective. |
| Discounting count | Has the future amount already been multiplied by its discount factor? | A present value is multiplied by the same discount factor again. |
The magnitude of a result is a useful diagnostic, but it cannot by itself prove the valuation correct. For example, an incorrectly chosen rate may still give a present value below the total payments. The stronger check is structural: verify every date, quote convention, and factor in the line that values that cash flow.
- Keep an unrounded factor or intermediate value for each payment; round money only in the final reported total.
- Label a rate with its maturity and convention in working, rather than writing only a percentage.
- When a discount factor is supplied directly, use it as the present-value multiplier once.
- If rates are quoted in a different compounding convention, convert consistently or use that convention’s accumulation formula throughout.
Check Your Understanding
For each problem, identify the relevant dates and quotation convention before calculating. Report money to the nearest cent.
- A liability pays $800 in two years and $1,500 in three years. The annual-effective spot rates are 5% for two years and 6% for three years. Find the liability’s present value. (Answer: $1,985.05)
- A single payment of $900 is due in nine months. The annual-effective spot rate for that maturity is 4%. Express the time in years and find the present value. (Answer: $873.91)
- A hypothetical asset pays $5,000 in three years. Its applicable spot rate is nominal annual 8% convertible quarterly. Find the present value using the stated convention, and do not treat 8% as annual effective. (Answer: $3,942.47)
- The discount factors are \(D(0,1)=0.97\) and \(D(0,2)=0.91\). A contract pays $600 in one year and $1,400 in two years. Find its present value and state how many times each factor is applied. (Answer: $1,856.00; each factor is applied once)
- A fund pays $700 in six months and $1,300 in 18 months. The annual-effective spot rates are 3% for six months and 4% for 18 months. Convert both times to years and find the present value. (Answer: $1,915.46)