Why Solve for Factors Before Rates?
A term structure problem may give information over overlapping periods: an investment from time 0 to time 2, another from time 1 to time 3, and perhaps a spot rate from time 0 to time 1. The unknown rates are linked because the accumulation over a longer period must agree with the accumulation over its component intervals. Solving directly with rate percentages can obscure those links; interval accumulation factors make them explicit.
The method here is to translate each quote into a factor over its stated interval, write one equation for each available path, and solve for the unknown factors. Convert factors back to rates only after the equations are solved. The method assumes a no-arbitrage term structure, consistent time units, and annual effective rates unless a quote explicitly states otherwise.
The last identity is the central accounting rule. Investing from \(s\) to \(u\) directly must accumulate the same amount as investing from \(s\) to \(t\), then reinvesting from \(t\) to \(u\). It is an equality of factors, not an equality or sum of rates.
Build and Solve the Factor Equations
A useful first step is to name the unknown accumulation factors, rather than immediately writing unknown rates. For example, let \(x=A(0,1)\), \(y=A(1,2)\), and \(z=A(2,3)\). Then \(A(0,2)=xy\), \(A(1,3)=yz\), and \(A(0,3)=xyz\). Each market observation becomes an equation involving these factors. Once the factors are known, an annual effective rate over a one-year interval is the factor minus 1.
For intervals longer than one year, the rate conversion must reflect the interval length. If \(A(s,t)\) is known and the annual effective forward rate is constant over that interval, then \(f_{s,t}=A(s,t)^{1/(t-s)}-1\). The same root principle gives the spot rate from \(A(0,t)\). Do not treat a two-year accumulation factor as though it were a one-year factor.
Write each interval with its start and end time, and record whether its quoted rate is spot or forward.
Use the stated interval length; for an annual effective rate \(r\) over \(n\) years, the factor is \((1+r)^n\).
Equate a direct accumulation factor to the product of factors along each available route.
Solve for the unknown factors, convert them to rates, then verify all original equations.
Why the Product Rule Holds
Suppose one unit is invested at time \(s\). At time \(t\), it has grown to \(A(s,t)\) units. Reinvesting that amount from \(t\) to \(u\) produces \(A(s,t)A(t,u)\) units at time \(u\). A direct investment from \(s\) to \(u\) produces \(A(s,u)\) units. If these amounts differ, an investor could choose the route with the higher accumulation while borrowing or selling the lower-accumulation route, creating an arbitrage under the assumed frictionless market. Therefore the two factors must agree:
This argument requires that the investments have matching cash-flow timing, currency, and risk, and that borrowing and lending at the relevant terms are available. If those conditions fail, the equality may not describe the actual market opportunity. In the standard deterministic Exam FM term-structure model, it is the no-arbitrage relation used to link the intervals.
Worked Example: Recovering Two Adjacent Forward Rates
A hypothetical market has \(A(0,1)=1.02\), \(A(0,2)=1.0608\), and \(A(1,3)=1.1024\). Find the one-year forward rates for years 1 to 2 and years 2 to 3, and check the implied accumulation from time 0 to time 3. All rates are annual effective.
Method. Divide factors over nested intervals to isolate the missing interval factor. This avoids treating overlapping-period rates as if they could be subtracted.
First, \(A(0,2)=A(0,1)A(1,2)\), so
The one-year forward rate for years 1 to 2 is therefore \(1.04-1=0.04\), or 4%.
Next, \(A(1,3)=A(1,2)A(2,3)\). Hence
so the one-year forward rate for years 2 to 3 is \(1.06-1=0.06\), or 6%.
Check the full path: \(A(0,3)=A(0,1)A(1,3)=1.02(1.1024)=1.124448\). The adjacent-interval path gives \(1.02(1.04)(1.06)=1.124448\), the same factor. The recovered rates are consistent.
Solving a System With Several Unknown Rates
When several rates are unknown, the equations should be solved in an order that isolates one factor at a time. The data need to supply enough independent information: if there are three unknown factors but only two independent equations, there is generally no unique solution. A third equation that is merely a rearrangement of the first two does not add new information.
Worked Example: Solving Three Linked Unknown Rates
A hypothetical term structure supplies \(A(0,2)=1.0712\), \(A(1,3)=1.092\), and \(A(0,3)=1.12476\). Find the year-1 spot rate, the forward rate for years 1 to 2, and the forward rate for years 2 to 3. Assume annual effective rates.
Method. Use the product identities to form three equations in the one-year factors \(x=A(0,1)\), \(y=A(1,2)\), and \(z=A(2,3)\):
Divide the third equation by the second to isolate \(x\):
Thus the year-1 spot rate is \(1.03-1=0.03\), or 3%.
Use \(xy=1.0712\) to find \(y\):
The forward rate for years 1 to 2 is \(1.04-1=0.04\), or 4%. Finally, use \(yz=1.092\):
The forward rate for years 2 to 3 is therefore \(1.05-1=0.05\), or 5%. Check the original data: \(1.03(1.04)=1.0712\), \(1.04(1.05)=1.092\), and \(1.03(1.04)(1.05)=1.12476\). Every equation is satisfied.
Mixing Spot and Forward Quotes
A spot rate covers an interval beginning at time 0, while a forward rate can cover an interval beginning later. They can be combined in the same system, but the interval and compounding period must be kept visible. If a quoted forward rate spans two years, its factor is squared; that factor must not be mistaken for a one-year factor.
Worked Example: Combining a Spot Rate and a Two-Year Forward Rate
Suppose \(A(0,1)=1.02\), \(A(0,2)=1.0816\), and a constant annual effective forward rate \(f_{1,3}\) applies from year 1 through year 3, with \(A(1,3)=1.1025\). Find the two-year spot rate \(i_2\), \(f_{1,3}\), and the one-year forward rate for years 2 to 3. Give rates to four decimal places as percentages.
Method. Convert each multi-year factor using its actual term, then divide interval factors to find the remaining one-year accumulation.
The two-year spot rate satisfies \((1+i_2)^2=A(0,2)=1.0816\). Since \(1.04^2=1.0816\), \(i_2=0.04\), or 4.0000%.
The forward rate from year 1 to year 3 applies for two years:
Thus \(f_{1,3}=0.05\), or 5.0000%.
Now find the full factor from time 0 to time 3: \(A(0,3)=A(0,1)A(1,3)=1.02(1.1025)=1.12455\). Since \(A(0,3)=A(0,2)A(2,3)\),
The one-year forward rate for years 2 to 3 is \(1.039709689\ldots-1=0.039709689\ldots\), or 3.9710% to four decimal places. As a check, the two-year factor from year 1 to year 3 is \(A(1,2)A(2,3)\). Here \(A(1,2)=A(0,2)/A(0,1)=1.0816/1.02=1.060392157\ldots\), and multiplying by \(1.039709689\ldots\) gives \(1.1025\), subject to displayed rounding.
Common Mistakes and Checks
The factor system helps identify errors before a rate is reported. For each result, verify both the interval and the accumulation path that produced it.
- Subtracting quoted rates instead of dividing factors. If \(A(0,2)=A(0,1)A(1,2)\), then \(A(1,2)=A(0,2)/A(0,1)\). It is not the two-year spot rate minus the one-year spot rate.
- Adding annual rates across years. A two-year accumulation at annual effective rate \(r\) is \((1+r)^2\), not \(1+2r\), except as an approximation in a separate context.
- Ignoring the interval length. For a two-year forward quote, take the square root of the factor to recover its annual effective rate. Do not subtract 1 from the two-year factor and call that an annual rate.
- Using too few independent equations. Count the unknown factors and check that the supplied relationships determine them. Repeating a consequence of an existing equation does not resolve an underdetermined system.
- Failing to test consistency. Multiply the recovered factors along at least one alternate path and compare with the supplied factor. A mismatch beyond rounding indicates an algebra, timing, or data error.
These systems are useful because a candidate can often recover rates even when no single quote directly supplies the desired interval. The essential discipline is to keep the maturity dates attached to every factor, solve for factors first, and only then convert to annual effective rates. That approach is aligned with the SOA Exam FM (Financial Mathematics) syllabus.
Check Your Understanding
For each problem, assume a no-arbitrage term structure and annual effective rates unless stated otherwise.
- A hypothetical curve has \(A(0,1)=1.04\) and \(A(0,2)=1.0816\). Find the forward rate for years 1 to 2 and the two-year spot rate. (Answer: 4% and 4%)
- Given \(A(0,1)=1.01\) and \(A(0,2)=1.0403\), find the one-year forward rate for years 1 to 2. (Answer: 3%)
- Given \(A(0,1)=1.02\), \(A(0,2)=1.0608\), and \(A(1,3)=1.092\), find the forward rates for years 1 to 2 and years 2 to 3. (Answer: 4% and 5%)
- Let \(x=A(0,1)\), \(y=A(1,2)\), and \(z=A(2,3)\). Market data give \(A(0,2)=1.0605\), \(A(1,3)=1.071\), and \(A(0,3)=1.08171\). Find the corresponding three one-year rates. (Answer: year-1 spot rate 1%, forward rate for years 1 to 2 is 5%, and forward rate for years 2 to 3 is 2%)
- A one-year spot factor is \(A(0,1)=1.01\), and the adjacent one-year forward factors are \(A(1,2)=1.02\) and \(A(2,3)=1.03\). A separate quote claims \(A(0,3)=1.061\). Is the quote consistent, and what should the three-year factor be? (Answer: No; the consistent factor is \(1.01(1.02)(1.03)=1.061106\))