Why an Unknown Spot Rate Requires the Whole Schedule
A present value for several dated payments depends on the discount factor at each payment date. If all but one of those factors are known, a known present value can determine the remaining factor—and therefore the spot rate for that maturity. The key is to keep each cash flow paired with the discount factor for its own date.
This method is aligned with the SOA Exam FM (Financial Mathematics) syllabus. It uses present-value equations for dated cash flows and the relationship between a maturity-specific discount factor and its annual effective spot rate.
For a schedule with cash flows \(C_t\) at times \(t\), the time-0 present value \(P\) is the sum of each cash flow multiplied by its own discount factor. Cash-flow signs matter: receipts are positive and payments are negative, unless the problem specifies another convention. If several cash flows occur at the same maturity, their signed amounts can be combined before applying that date’s discount factor.
Isolating One Maturity’s Discount Factor
Suppose the spot rate is unknown at one positive maturity \(T\), while the discount factors at all other payment dates are known. Write \(C_T\) for the net cash flow at time \(T\), and collect the present-value contributions from all other dates into \(K\). Then \(P=K+C_TD_T\). Rearranging isolates the unknown discount factor.
The last expression is usable only if its discount-factor input is positive. A finite annual effective spot rate must satisfy \(i_T>-1\), which gives \(D_T>0\). Also, \(C_T\) must be nonzero: if the net cash flow at time \(T\) is zero, the present value contains no information about \(D_T\) or \(i_T\).
Why the Method Works
Each known-date contribution can be calculated independently. Once those amounts are added to obtain \(K\), the difference \(P-K\) is exactly the present value contributed by the unknown date. That contribution is the net cash flow \(C_T\) multiplied by its discount factor \(D_T\). Dividing by \(C_T\) recovers the factor; converting the factor to a spot rate then follows from the annual effective compounding definition.
For every date other than \(T\), multiply its cash flow by its stated discount factor. Add those signed values to obtain \(K\).
Subtract the known contributions from the stated total present value: \(P-K\).
Divide the residual by the net cash flow at time \(T\). The resulting \(D_T\) must be positive for a finite real annual effective spot rate.
Use \(i_T=D_T^{-1/T}-1\), then reprice the original schedule to confirm it gives \(P\).
The uniqueness condition follows directly from the model. For \(T>0\), as \(i_T\) ranges over rates greater than \(-1\), the factor \((1+i_T)^{-T}\) decreases strictly from arbitrarily large positive values toward zero. Thus every positive \(D_T\) corresponds to exactly one finite \(i_T>-1\). A zero or negative factor does not correspond to a finite real annual effective spot rate.
Worked Examples
Worked Example: Solving for a Three-Year Spot Rate
A cash-flow schedule has a present value of $1,207.40 at time 0. It pays $500 at the end of year 1, $400 at the end of year 2, and $600 at the end of year 3. The known discount factors are \(D_1=0.90\) and \(D_2=0.80\). Find the annual effective three-year spot rate. Assume annual effective compounding and no payments other than those listed.
Method. Calculate the known-date contributions, isolate the year-3 discount factor, convert it to a spot rate, and reprice the schedule.
The present value of the year-1 cash flow is \(500(0.90)=450\). The present value of the year-2 cash flow is \(400(0.80)=320\). Therefore, the total known contribution is \(K=450+320=770\).
The year-3 payment must contribute the remaining \(1{,}207.40-770=437.40\). Its discount factor is \(D_3=437.40/600=0.729\).
Since \(0.729=0.9^3\), the spot rate satisfies \((1+i_3)^{-3}=0.9^3\). Hence \(1+i_3=1/0.9=1.111111\ldots\), so \(i_3=0.111111\ldots\). The annual effective three-year spot rate is 11.11%, rounded to two decimal places.
Check. The year-3 contribution is \(600(0.729)=437.40\). The full present value is \(450+320+437.40=1{,}207.40\), as required. The common mistake to avoid is using a one-year discounting power for the three-year payment: the relevant factor is \((1+i_3)^{-3}\), not \((1+i_3)^{-1}\).
Worked Example: Combining Cash Flows at the Unknown Maturity
A schedule has a time-0 present value of $1,030. At year 1 it receives $200, and the discount factor is \(D_1=0.90\). At year 2 it receives $1,200 and pays $200. At year 3 it receives $300, and the discount factor is \(D_3=0.70\). Find the annual effective two-year spot rate. Assume annual effective compounding.
Method. First combine the signed cash flows at the same maturity, then solve for that maturity’s factor.
The net year-2 cash flow is \(1{,}200-200=1{,}000\). The known year-1 contribution is \(200(0.90)=180\), and the known year-3 contribution is \(300(0.70)=210\). Thus \(K=180+210=390\).
The year-2 contribution must be \(1{,}030-390=640\). Therefore, \(D_2=640/1{,}000=0.64\).
The two-year spot rate satisfies \((1+i_2)^{-2}=0.64\), so \((1+i_2)^2=1/0.64=1.5625\). Taking the positive square root gives \(1+i_2=1.25\), hence \(i_2=0.25\). The annual effective two-year spot rate is 25.00%.
Check. The year-2 contribution is \(1{,}000(0.64)=640\), and \(180+640+210=1{,}030\). The common mistake is to discount the $1,200 receipt and $200 payment separately using different factors even though they occur at the same maturity. Their combined amount is $1,000, and both use \(D_2\).
Worked Example: Recognizing a Negative Spot Rate
A schedule has a time-0 present value of $1,000. It pays $100 at year 1, with \(D_1=0.95\); receives $500 at year 2; and receives $600 at year 3, with \(D_3=0.50\). Find the annual effective two-year spot rate. Assume annual effective compounding.
Method. Subtract the known contributions, then check the resulting factor before converting it to a rate.
The year-1 contribution is \(100(0.95)=95\). The year-3 contribution is \(600(0.50)=300\). Thus \(K=95+300=395\), leaving a year-2 contribution of \(1{,}000-395=605\).
The year-2 discount factor is \(D_2=605/500=1.21\), which is positive and therefore valid. The rate satisfies \((1+i_2)^{-2}=1.21\). Thus \(1+i_2=1/\sqrt{1.21}=1/1.1=0.9090909\ldots\), and \(i_2=-0.0909090\ldots\). The annual effective two-year spot rate is -9.09%, rounded to two decimal places.
Check. The year-2 contribution is \(500(1.21)=605\), and \(95+605+300=1{,}000\). The common mistake is to reject a discount factor greater than 1 as impossible. It implies a negative spot rate, not an invalid factor; the rate is still greater than \(-1\), as the annual effective model requires.
Validity Checks and Exam Traps
Before reporting a rate, verify the structure and the result. A correct-looking power calculation cannot repair a cash-flow equation with a wrong sign, date, or net amount.
| Check | What to verify | Why it matters |
|---|---|---|
| Timing | Use the exponent \(T\) for the unknown payment’s maturity. | The spot rate is annual effective, but the discount factor spans \(T\) years. |
| Cash-flow signs | Add receipts and payments with their stated signs. | The residual present value must be divided by the net cash flow at the unknown date. |
| Same-date amounts | Combine all amounts at maturity \(T\), or apply \(D_T\) to each and then add. | All cash flows at one maturity share the same maturity-specific factor. |
| Factor validity | Confirm \(D_T>0\) and \(C_T\ne0\). | A finite real annual effective rate requires a positive factor and a nonzero net cash flow. |
| Final repricing | Substitute the solved factor into the entire schedule. | The resulting total must equal the stated present value. |
A particularly important trap is to treat a spot rate as if it were the same at every maturity. The equation for a year-3 payment is \(C_3(1+i_3)^{-3}\), not \(C_3(1+i_1)^{-3}\) or \(C_3(1+i_3)^{-1}\). Another trap is to take a root before isolating the discount factor. First form \((P-K)/C_T\); only then apply the maturity root.
Check Your Understanding
For each problem, assume annual effective compounding, use the stated discount factors for known maturities, and solve for the single unknown maturity-specific spot rate.
- A schedule has present value $336. It pays $100 at year 1, where \(D_1=0.96\), and $300 at year 2. Find the two-year spot rate. (Answer: 11.80%)
- A schedule has present value $1,009. It pays $100 at year 1 with \(D_1=0.90\), $250 at year 2 with \(D_2=0.76\), and $1,000 at year 3. Find the three-year spot rate. (Answer: 11.11%)
- A schedule has present value $700. Its known contributions are $90 at year 1 and $160 at year 2. It pays $500 at year 3, the only remaining cash flow. Find the three-year spot rate. (Answer: 3.57%)
- A schedule has present value $200. Its known-date cash flows contribute $520 in total. At year 2, the only cash flow is a payment of $400. Find the two-year spot rate and explain why a negative cash flow does not by itself make the result invalid. (Answer: 11.80%; the residual and the cash flow are both negative, giving \(D_2=0.80>0\).)
- A schedule has present value $500. All known-date contributions total $650, and the net cash flow at the unknown year-4 maturity is $300. Determine whether a finite real annual effective four-year spot rate exists. (Answer: No; the implied factor is \((500-650)/300=-0.50\), which is not positive.)