Recovering A Spot Rate From Forward Rates

Why Recover a Later Spot Rate?

A spot rate summarizes the accumulation from time 0 to one maturity. A forward rate applies only over a later interval. When an earlier spot rate and the forward rate for the remaining interval are known, their accumulation factors can be combined to recover the spot rate for the full, later maturity. The key is to use each rate for the number of years it actually applies.

This calculation is useful when a term structure is described in stages rather than by directly listing every spot rate. It also provides a consistency check: the accumulation from time 0 to the later maturity must match whether it is calculated directly using the later spot rate or in two successive intervals.

Definition: Let \(s_m\) be the effective annual spot rate from time 0 to year \(m\), and let \(f_{m,n}\) be the effective annual forward rate applying from year \(m\) to year \(n\), where \(0<m<n\). The spot rate \(s_n\) is the effective annual rate that produces the same accumulation from time 0 to year \(n\) as first accumulating to year \(m\) at \(s_m\), then from year \(m\) to year \(n\) at \(f_{m,n}\).

The Recovery Formula

Assume annual compounding, integer-year maturities, and a single initial investment with no intermediate cash flows. The spot and forward rates in the formula are effective annual rates. The forward rate applies for exactly \(n-m\) years, not for \(n\) years.

(1+s_n)^n=(1+s_m)^m(1+f_{m,n})^{n-m}

The left side is the accumulation factor from time 0 to year \(n\) using the \(n\)-year spot rate. On the right, \((1+s_m)^m\) accumulates the investment to year \(m\), and \((1+f_{m,n})^{n-m}\) accumulates it over the remaining \(n-m\) years. Solving for the later spot rate gives

s_n=\left[(1+s_m)^m(1+f_{m,n})^{n-m}\right]^{1/n}-1.

The exponent \(1/n\) is necessary because a spot rate is an annual effective rate for the full \(n\)-year period. The formula does not apply unchanged if the rates are quoted under different compounding conventions, or if the stated forward rate covers an interval other than years \(m\) through \(n\). In those cases, first express the rates and interval accumulations consistently.

Why the Accumulations Must Match

The equation follows by tracking the same investment along two descriptions of the same time period. This is an accumulation identity: compounding first to year \(m\) and then to year \(n\) must give the same total accumulation as compounding directly to year \(n\) at its spot rate.

Accumulate to year \(m\).
An initial amount of 1 becomes \((1+s_m)^m\) at year \(m\).

Continue to year \(n\).
The remaining interval lasts \(n-m\) years, so the amount becomes \((1+s_m)^m(1+f_{m,n})^{n-m}\).

Equate with the full-period accumulation.
The same initial amount invested for \(n\) years at the \(n\)-year spot rate becomes \((1+s_n)^n\). Equating the two amounts and taking the \(n\)th root gives the recovery formula.

This reasoning also explains why the annual rates should not be averaged. Rates apply to periods of different lengths, and accumulation is multiplicative. The correct combination is the product of the interval accumulation factors, followed by an \(n\)th root.

Worked Example: A Two-Year Spot Rate and a Three-Year Forward Interval

A two-year spot rate is 4.0000% effective annually. The effective annual forward rate from year 2 to year 5 is 6.0000%. Find the five-year spot rate. Give the answer as a percentage to four decimal places.

Method. The first rate applies for 2 years, and the forward rate applies for \(5-2=3\) years. The total accumulation factor is therefore

$$ (1.04)^2(1.06)^3 =1.0816(1.191016) =1.2882029056. $$

Let \(s_5\) denote the five-year spot rate. Equating the five-year accumulation to this total gives

(1+s_5)^5=1.2882029056, \qquad s_5=(1.2882029056)^{1/5}-1 \approx 0.0519543.

Thus the five-year spot rate is 5.1954% effective annually to four decimal places in percentage form. Using the unrounded root, raising \(1+s_5\) to the fifth power recovers approximately \(1.2882029056\), as required.

Common mistake avoided: Averaging 4% and 6%, or treating the forward rate as if it applied for only one year, ignores the interval lengths. The correct exponents are 2 and 3.

Unequal Intervals and the Root to Use

The maturity \(n\) determines the root in the recovery calculation; the forward interval length \(n-m\) determines the exponent on the forward accumulation factor. These are different roles. If the forward rate covers two years, its factor is squared, even when the total maturity is five years. The final result still uses a fifth root when recovering a five-year spot rate.

Worked Example: A Three-Year Spot Rate and a Two-Year Forward Interval

A three-year spot rate is 5.0000% effective annually. The effective annual forward rate from year 3 to year 5 is 4.0000%. Find the five-year spot rate, to four decimal places in percentage form.

The known spot rate applies for 3 years; the forward rate applies for \(5-3=2\) years. First calculate the two-stage accumulation:

$$ (1.05)^3(1.04)^2 =1.157625(1.0816) =1.2520872. $$

The five-year spot rate must accumulate 1 to the same value:

(1+s_5)^5=1.2520872, \qquad s_5=(1.2520872)^{1/5}-1 \approx 0.0459885.

The five-year spot rate is therefore 4.5989% effective annually to four decimal places in percentage form. With the unrounded root, the fifth power of the accumulation factor agrees with 1.2520872.

Common mistake avoided: The forward factor is raised to 2, not 5, because the forward rate applies only from year 3 to year 5. The fifth root is used because the target spot rate covers the full five years.

Worked Example: Checking a Later Spot Rate by a Square Root

A two-year spot rate is 2.0000% effective annually. The effective annual forward rate from year 2 to year 4 is 6.0000%. Find the four-year spot rate, to four decimal places in percentage form, and check the accumulation.

The first interval lasts 2 years and the forward interval also lasts \(4-2=2\) years. Their accumulation factors are

(1.02)^2=1.0404, \qquad (1.06)^2=1.1236,

so the combined accumulation is

$$ 1.0404(1.1236)=1.16899344. $$

For a four-year spot rate \(s_4\),

(1+s_4)^4=1.16899344, \qquad s_4=(1.16899344)^{1/4}-1 \approx 0.0398077.

The four-year spot rate is 3.9808% effective annually to four decimal places in percentage form. As a convenient check, \(1.16899344=(1.0812)^2\), so its fourth root is \(\sqrt{1.0812}\), approximately 1.0398077. Raising the unrounded root to the fourth power returns 1.16899344.

Common mistake avoided: Taking a square root would be appropriate only for the two-year intervals considered individually. The target spot rate spans four years, so the combined accumulation must be raised to the power \(1/4\).

Common Errors to Check

Before accepting a calculated spot rate, verify the timeline and units. Most errors arise from assigning a correct rate to the wrong number of years or taking a root based on the interval rather than the total maturity.

Using \(n\) years for the forward rate. The forward interval from year \(m\) to year \(n\) lasts \(n-m\) years.
Taking the wrong root. To recover \(s_n\), take the \(n\)th root of the total accumulation factor, even if the forward interval has a different length.
Averaging rates arithmetically. Successive accumulation factors multiply; the annual spot rate is recovered from their product.
Mixing quotation conventions. The displayed formula assumes effective annual rates. Rates under another convention must be converted or used through their appropriate accumulation factors.

A useful final check is to compute both sides of \((1+s_n)^n=(1+s_m)^m(1+f_{m,n})^{n-m}\) using the unrounded calculated rate. They should agree apart from calculator precision. If they do not, recheck the interval lengths, the rate quotation, and the root.

Key takeaway: Multiply the earlier spot accumulation factor by the forward accumulation factor, raising each to the number of years it covers. Then take the \(n\)th root of the product and subtract 1 to recover the spot rate at maturity \(n\).

Check Your Understanding

In each problem, rates are effective annual rates under annual compounding. Use the full maturity for the root and the actual forward interval length for its exponent.

A two-year spot rate is 3%. The forward rate from year 2 to year 4 is also 3%. Find the four-year spot rate. (Answer: 3.0000%)
A one-year spot rate is 2%, and the forward rate from year 1 to year 2 is 4%. Find the two-year spot rate. (Answer: 2.9951%)
A two-year spot rate is 4%, and the forward rate from year 2 to year 5 is 4%. Find the five-year spot rate. (Answer: 4.0000%)
A three-year spot rate is 5%, and the forward rate from year 3 to year 5 is 2%. Find the five-year spot rate. (Answer: 3.7896%)
A two-year spot rate is 2%, and the forward rate from year 2 to year 6 is 5%. Find the six-year spot rate. (Answer: 3.9903%)

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