Building A Spot Curve From A Forward Strip

From a Forward Strip to a Spot Curve

A sequence of annual forward rates describes successive one-year periods, not separate rates that can be read as spot rates. To build a spot curve, start at the valuation date and chain the gross accumulation factor for each year. The resulting factor to year $n$ gives the annual effective spot rate and discount factor for that maturity.

This reconstruction is useful whenever rates are supplied as a period-by-period strip but a calculation requires a rate or discount factor for a particular maturity. The essential discipline is to keep each forward rate attached to its own interval and to multiply accumulation factors in chronological order.

Definition: Assume annual periods and a single valuation date. Let $f_k$ be the effective annual forward rate for the one-year period from time $k-1$ to time $k$, for $k=1,2,\ldots$. Let $A_n$ be the accumulation factor from time 0 to time $n$ implied by the spot curve, with $A_0=1$. Then $$ A_n=\prod_{k=1}^{n}(1+f_k). $$ The annual effective spot rate $i_n$ for maturity $n$ and its discount factor $v_n$ are $$ 1+i_n=A_n^{1/n}, \qquad v_n=\frac{1}{A_n}=(1+i_n)^{-n}. $$

The formula assumes that all rates use the same valuation date, annual time intervals, and effective annual compounding. Each $f_k$ applies only to its stated interval. If quoted rates use different compounding conventions or nonannual intervals, they must first be put on a consistent basis. The formulas above do not directly apply to a strip whose periods or rate conventions have not been reconciled.

Why the Forward Factors Multiply

The forward rate for year $k$ takes an amount at time $k-1$ and accumulates it to time $k$. The factor $1+f_k$ therefore relates the two spot accumulation factors, as shown below.

\frac{A_k}{A_{k-1}}=1+f_k

Rearrange this relation to obtain $A_k=A_{k-1}(1+f_k)$. Starting from $A_0=1$, repeated substitution gives

\begin{aligned} A_1&=1(1+f_1),\\ A_2&=(1+f_1)(1+f_2),\\ A_3&=(1+f_1)(1+f_2)(1+f_3),\\ A_n&=\prod_{k=1}^{n}(1+f_k). \end{aligned}

This is a product of gross accumulation factors, not a sum or product of the rate percentages themselves. Once $A_n$ is known, the spot rate is the single annual effective rate that would produce that same total accumulation over $n$ years. Hence $(1+i_n)^n=A_n$, which gives $i_n=A_n^{1/n}-1$.

Building the Curve Step by Step

A practical calculation keeps a running product. At each maturity, multiply the previous accumulation factor by the next year’s gross forward factor. Then take the reciprocal for the discount factor, or take the $n$th root to get the annual effective spot rate.

Set the starting factor.
At the valuation date, use $A_0=1$. This represents one unit already held at time 0.

Chain the forward factors.
For each maturity $n$, calculate $A_n=A_{n-1}(1+f_n)$, keeping the forward rate matched to its one-year interval.

Convert to spot quantities.
Calculate $i_n=A_n^{1/n}-1$ and $v_n=1/A_n$ when the spot rate or discount factor is needed.

Check the chain.
Confirm that $A_n/A_{n-1}=1+f_n$ for every year. This catches a misplaced rate or an omitted factor.

Worked Examples

Worked Example: Constructing Spot Rates at Four Maturities

A valuation model supplies annual forward rates of 4% for year 1, 6% for year 2, 8% for year 3, and 5% for year 4. Build the spot accumulation factors and annual effective spot rates through year 4. Report spot rates to four decimal places in percentage terms.

Method. Chain the gross annual forward factors, then take the appropriate root at each maturity. This method avoids treating an individual one-year forward rate as though it were the spot rate for a longer term.

At year 1, the factor is $1+0.04=1.04$, so $A_1=1.04$ and $i_1=1.04-1=0.04$, or 4.0000%.

At year 2, $A_2=A_1(1+f_2)=1.04(1.06)=1.1024$. Therefore $i_2=(1.1024)^{1/2}-1=0.049952\ldots$, or 4.9952%.

At year 3, $A_3=A_2(1+f_3)=1.1024(1.08)=1.190592$. Therefore $i_3=(1.190592)^{1/3}-1=0.059874\ldots$, or 5.9874%.

At year 4, $A_4=A_3(1+f_4)=1.190592(1.05)=1.2501216$. Therefore $i_4=(1.2501216)^{1/4}-1=0.057397\ldots$, or 5.7397%.

The spot curve is thus 4.0000%, 4.9952%, 5.9874%, and 5.7397% at years 1 through 4, respectively. As a check, the final factor satisfies $A_4/A_3=1.2501216/1.190592=1.05$, the gross forward factor for year 4. All rates have been rounded only at the final step, to four decimal places.

Worked Example: Discounting a Payment Using a Forward Strip

A hypothetical contract pays $5,000 at the end of year 3. The annual forward rates for years 1, 2, and 3 are 3%, 5%, and 4%, respectively. Find the present value at time 0 using the strip, and find the equivalent annual effective spot rate to year 3. Report money to the nearest cent and the rate to four decimal places in percentage terms.

Method. First reconstruct the year-3 accumulation factor. Its reciprocal discounts the payment to time 0; its cube root gives the annual effective spot rate.

The successive factors are $A_1=1.03$, $A_2=1.03(1.05)=1.0815$, and $A_3=1.0815(1.04)=1.12476$. Thus the year-3 discount factor is $v_3=1/1.12476=0.88907856\ldots$.

The present value is $5{,}000(0.88907856\ldots)=\$4{,}445.3928\ldots$, which is $4,445.39 to the nearest cent.

The annual effective spot rate is $i_3=(1.12476)^{1/3}-1=0.039968\ldots$, or 3.9968% to four decimal places. To check the result, accumulating the unrounded present value by the factor $1.12476$ returns $5,000, up to rounding. The key trap avoided is discounting the year-3 payment using only the year-3 forward rate: that rate applies to year 3 alone, not to the full three-year period.

Worked Example: Accumulating a Deposit and Finding Its Spot Yield

A deposit of $2,500 is invested at time 0. The annual forward rates for years 1, 2, and 3 are 2%, 4%, and 6%. Find the accumulated value at time 3 and the equivalent annual effective spot rate for the three-year investment. Report the accumulated value to the nearest cent and the rate to four decimal places in percentage terms.

Method. Apply each forward factor to the amount carried into that year, then find the annual rate that produces the same three-year accumulation.

After year 1, the amount is $\$2{,}500(1.02)=\$2{,}550$. After year 2, it is $\$2{,}550(1.04)=\$2{,}652$. After year 3, it is $\$2{,}652(1.06)=\$2{,}811.12$. Equivalently, the total accumulation factor is $A_3=1.02(1.04)(1.06)=1.124448$, and $\$2{,}500(1.124448)=\$2{,}811.12$.

The annual effective spot rate satisfies $(1+i_3)^3=1.124448$, so $i_3=(1.124448)^{1/3}-1=0.0398718\ldots$, or 3.9872% to four decimal places. As a check, the unrounded spot rate compounded for three years produces the same factor, 1.124448, and therefore the same accumulated value. The calculation avoids the mistake of adding the forward rates and applying that sum as a single three-year interest rate.

Reading and Checking the Reconstructed Curve

The accumulation factors are often the most direct representation of the reconstructed curve. They preserve the exact chained result, while spot rates express that result as equivalent annual effective rates. Discount factors are simply the reciprocals of those same factors.

Maturity	Accumulation factor	Annual effective spot rate	Discount factor
1 year	$A_1=1+f_1$	$i_1=A_1-1$	$v_1=1/A_1$
2 years	$A_2=(1+f_1)(1+f_2)$	$i_2=A_2^{1/2}-1$	$v_2=1/A_2$
$n$ years	$A_n=\prod_{k=1}^{n}(1+f_k)$	$i_n=A_n^{1/n}-1$	$v_n=1/A_n$

A spot rate is an equivalent rate over the whole term; it is not generally equal to any one forward rate in the strip. It is also not generally the arithmetic average of the forward rates. The spot rate is determined by the geometric average of their gross accumulation factors.

Common Exam Trap: Averaging the Rates

Suppose the first three annual forward rates are 4%, 6%, and 8%. A tempting but incorrect calculation is to average the percentages and claim that the three-year spot rate is 6%. That would imply a three-year accumulation factor of $(1.06)^3=1.191016$.

The correct accumulation factor is $(1.04)(1.06)(1.08)=1.190592$. The implied three-year spot rate is therefore $(1.190592)^{1/3}-1=5.9874\%$, not 6%. The small difference here does not make averaging valid; with other rates, the error can be larger. The correction is always to multiply gross factors first, and only then take the maturity root.

Key takeaway: To build a spot curve from successive annual forwards, carry forward the accumulation factor by multiplying by $1+f_k$ at each year. For maturity $n$, use $A_n=\prod_{k=1}^{n}(1+f_k)$, $i_n=A_n^{1/n}-1$, and $v_n=1/A_n$. Check each step with $A_n/A_{n-1}=1+f_n$.

Check Your Understanding

For each problem, use annual effective forward rates for the stated consecutive one-year periods.

The forward rates for years 1, 2, and 3 are 2%, 4%, and 6%. Find the year-3 accumulation factor and annual effective spot rate, giving the rate to four decimal places in percentage terms. (Answer: $A_3=1.124448$; $i_3\approx3.9872\%$.)
The forward rates for years 1 and 2 are 5% and 3%. Find $A_2$ and the two-year annual effective spot rate to four decimal places in percentage terms. (Answer: $A_2=1.0815$; $i_2\approx3.9952\%$.)
A hypothetical payment of $2,400 is due at the end of year 3. The forward rates for years 1, 2, and 3 are 1%, 4%, and 5%. Find its present value to the nearest cent. (Answer: $A_3=1.10292$; present value $\approx\$2,176.04$.)
The forward rates for years 1, 2, and 3 are 3%, 0%, and 6%. Find the year-1 and year-3 accumulation factors and the three-year spot rate to four decimal places in percentage terms. (Answer: $A_1=1.03$, $A_3=1.0918$; $i_3\approx2.9709\%$.)
A deposit of $3,000 is invested at time 0. The forward rates for years 1, 2, and 3 are 2%, 4%, and 6%. Find its accumulated value at time 3, to the nearest cent, and the equivalent three-year annual effective spot rate to four decimal places in percentage terms. (Answer: accumulated value $=\$3,373.34$; spot rate $\approx3.9872\%$.)

Maturity	Accumulation factor	Annual effective spot rate	Discount factor
1 year	\(A_1=1+f_1\)	\(i_1=A_1-1\)	\(v_1=1/A_1\)
2 years	\(A_2=(1+f_1)(1+f_2)\)	\(i_2=A_2^{1/2}-1\)	\(v_2=1/A_2\)
\(n\) years	\(A_n=\prod_{k=1}^{n}(1+f_k)\)	\(i_n=A_n^{1/n}-1\)	\(v_n=1/A_n\)

What You'll Learn