Deriving Spot Rates from Bootstrapped Discount Factors

From Discount Factors to Spot Rates

Bootstrapping produces a discount factor for each maturity. A discount factor is useful for valuing a cash flow directly, but a spot rate expresses the same information as an annualized interest rate. Converting between them makes it possible to compare maturities on a consistent basis, provided the maturity and compounding conventions are handled correctly.

A spot rate belongs to a particular maturity. It is not necessarily the same rate at every point on the term structure. In this tutorial, maturities are measured in years, the discount factors are positive, and the requested spot rates are annual effective rates. The examples use discount factors below one.

Definition: Let $v_t$ be the discount factor for one unit payable $t$ years from the valuation date, where $t>0$. The annual effective spot rate for maturity $t$, denoted $i_t$, is the annual effective rate that satisfies $$ v_t=(1+i_t)^{-t}. $$ Thus $i_t$ is the constant annual effective rate that would produce the same present value over that particular maturity.

The subscript identifies the maturity, not a compounding frequency. For example, $i_2$ is the annual effective spot rate for a two-year cash flow. In general, it need not equal $i_1$. When $t=1$, the formula reduces to the familiar one-year relationship $v_1=1/(1+i_1)$.

Deriving the Conversion Formula

Under an annual effective rate $i_t$, one unit invested for one year accumulates to $1+i_t$. Under the annual effective convention, the accumulation over $t$ years is $(1+i_t)^t$; the corresponding present-value factor is its reciprocal. Therefore the discount-factor equation is $v_t=(1+i_t)^{-t}$. Solving this equation isolates the rate.

v_t=(1+i_t)^{-t} \quad\Longrightarrow\quad \frac{1}{v_t}=(1+i_t)^t \quad\Longrightarrow\quad i_t=v_t^{-1/t}-1.

This inversion is valid for $t>0$ and $v_t>0$. For the maturities used here, $t$ must be in years because the rate being reported is annual effective. If a maturity is given in months or quarters, convert it to years before using the exponent. Keep the full-precision discount factor through the calculation and round the rate only at the end.

Key result: For a positive bootstrapped discount factor $v_t$ at maturity $t$ years, the annual effective spot rate is $$ i_t=\left(\frac{1}{v_t}\right)^{1/t}-1. $$ To check a result, substitute the unrounded rate into $v_t=(1+i_t)^{-t}$ and confirm that it reproduces the original discount factor.

Worked Examples

Worked Example: Converting Discount Factors at Several Whole-Year Maturities

A bootstrap gives discount factors $v_1=0.940$, $v_2=0.880$, and $v_3=0.820$, for cash flows due in one, two, and three years. Find the annual effective spot rate for each maturity. Report rates as percentages to four decimal places.

Method. Apply $i_t=v_t^{-1/t}-1$ separately at each maturity. The maturity-specific exponent is essential; do not apply the one-year formula to every factor.

For the one-year maturity:

i_1=0.940^{-1/1}-1 =\frac{1}{0.940}-1 =1.063829787\ldots-1 =0.063829787\ldots.

Thus $i_1=6.3830\%$ to four decimal places.

For the two-year maturity:

i_2=0.880^{-1/2}-1 =\sqrt{\frac{1}{0.880}}-1 =\sqrt{1.136363636\ldots}-1 =0.0660036\ldots.

Thus $i_2=6.6004\%$ to four decimal places.

For the three-year maturity:

i_3=0.820^{-1/3}-1 =\left(\frac{1}{0.820}\right)^{1/3}-1 =(1.219512195\ldots)^{1/3}-1 =0.0683873\ldots.

Thus $i_3=6.8387\%$ to four decimal places. To check, use the unrounded rates: $(1+i_1)^{-1}=0.940$, $(1+i_2)^{-2}=0.880$, and $(1+i_3)^{-3}=0.820$, up to displayed precision. The increasing rates are consistent with these particular inputs; no rule requires spot rates to increase with maturity.

Common mistake avoided: using $v_t^{-1}-1$ for every maturity. That is the correct formula only when $t=1$; for $t=2$ or $t=3$, the reciprocal must be raised to the power $1/t$.

Worked Example: Fractional-Year Maturities

Two bootstrapped discount factors are $v_{0.25}=0.985$ at three months and $v_{0.75}=0.955$ at nine months. Find the annual effective spot rate for each maturity, as a percentage to four decimal places.

Method. Express the maturities in years: three months is $0.25$ years and nine months is $0.75$ years. The requested annual effective rates use those year values as the exponents.

At three months:

i_{0.25}=0.985^{-1/0.25}-1 =0.985^{-4}-1 =\frac{1}{0.985^4}-1.

The denominator is $0.985^4=0.941336550625$. Therefore

i_{0.25} =\frac{1}{0.941336550625}-1 \approx 0.0623193 =6.2319\%.

At nine months:

i_{0.75}=0.955^{-1/0.75}-1 =0.955^{-4/3}-1 =\left(\frac{1}{0.955}\right)^{4/3}-1 \approx 0.0633156 =6.3316\%.

The displayed decimal rates are rounded to seven decimal places and the percentages to four decimal places. As a check, applying the unrounded rates for $0.25$ and $0.75$ years gives discount factors $0.985$ and $0.955$, respectively, to the shown precision.

Common mistake avoided: using $t=3$ or $t=9$ because the maturities were stated in months. That would treat the rates as annual over the wrong time units. Convert months to years first; equivalently, use $1/t=4$ for three months and $1/t=4/3$ for nine months.

Worked Example: Converting a Nominal Spot-Rate Convention

A hypothetical market quote gives a $10\%$ nominal annual spot rate convertible semiannually for a cash flow due in $1.5$ years. Find the discount factor and the equivalent annual effective spot rate for that maturity. Report the discount factor to seven decimal places and the annual effective rate as a percentage to two decimal places.

Method. First use the quoted nominal rate and its semiannual frequency to calculate the discount factor. Then express the same accumulation as an annual effective rate over $1.5$ years.

The nominal rate per half-year is $0.10/2=0.05$. There are $2(1.5)=3$ half-year periods, so the discount factor is

v_{1.5} =\left(1+\frac{0.10}{2}\right)^{-2(1.5)} =(1.05)^{-3} =\frac{1}{1.157625} \approx 0.8638376.

The annual effective rate equivalent to a nominal rate of $10\%$ convertible semiannually is

$$ (1+0.10/2)^2-1 =1.05^2-1 =1.1025-1 =0.1025. $$

Thus the annual effective spot rate is $10.25\%$. To verify it directly from the discount factor, use the maturity conversion:

i_{1.5}=v_{1.5}^{-1/1.5}-1 =\big((1.05)^{-3}\big)^{-2/3}-1 =1.05^2-1 =0.1025.

The discount-factor check is $(1.1025)^{-1.5}=(1.05)^{-3}$, as required. The two expressions agree because one annual effective period contains two half-year periods.

Common mistake avoided: reporting the quoted nominal rate $10\%$ as though it were the annual effective rate. The nominal rate is defined with a conversion frequency; its equivalent annual effective rate here is $10.25\%$.

Conventions, Checks, and Exam Traps

A bootstrapped discount factor already incorporates the cash-flow timing used in the bootstrap. Once the factor for a maturity is known, converting it to an annual effective spot rate is a separate step: identify the maturity in years, apply the inverse-maturity power, subtract one, and report the requested units. Do not alter the cash-flow dates or coupon timing during this conversion.

Identify the factor and its date.
Use the discount factor corresponding to the cash flow's exact maturity, not a factor for a nearby coupon date.

Express maturity in years.
For example, convert months by dividing by 12. Use the resulting year value as $t$.

Apply the annual effective conversion.
Calculate $i_t=v_t^{-1/t}-1$, retaining sufficient precision during the calculation.

Check the result and round.
Substitute the unrounded rate into $(1+i_t)^{-t}$, compare with the input factor, and round only the final answer.

If the question gives a nominal rate convertible $m$ times per year rather than a discount factor, its annual effective equivalent is $(1+i^{(m)}/m)^m-1$. For a maturity of $t$ years, the associated discount factor under that convention is $(1+i^{(m)}/m)^{-mt}$. These formulas agree with the annual effective spot-rate method when both describe the same accumulation. Always distinguish a quoted nominal rate from the annual effective rate requested.

An especially tempting error is to use $v_t^{-t}-1$ instead of $v_t^{-1/t}-1$. For example, with $v_2=0.880$, the incorrect calculation gives

0.880^{-2}-1 =\left(\frac{1}{0.880}\right)^2-1 \approx 0.2913.

That is not the two-year annual effective spot rate: it uses the maturity as the exponent rather than its reciprocal. The correct calculation is

0.880^{-1/2}-1 \approx 0.0660036,

or $6.6004\%$. Repricing confirms the correct result: $(1.0660036\ldots)^{-2}\approx0.880$. A final repricing check is a reliable way to catch both exponent errors and mistakes in converting maturity units.

Key takeaway: A discount factor is tied to a maturity, and so is the annual effective spot rate derived from it. Convert the maturity to years, use the reciprocal maturity as the exponent, preserve the stated compounding convention, and verify the result by recovering the original factor.

Check Your Understanding

For each problem, use the stated maturity and convention carefully. Rates should be reported as percentages to four decimal places unless the question specifies otherwise.

A bootstrapped factor is $v_1=0.920$. Find the one-year annual effective spot rate. (Answer: $8.6957\%$)
A two-year cash flow has discount factor $v_2=0.900$. Find its annual effective spot rate. (Answer: $5.4093\%$)
A factor of $0.970$ applies to a cash flow due in six months. Find the annual effective spot rate. (Answer: $6.2812\%$)
A nine-month cash flow has discount factor $0.955$. Find the annual effective spot rate, using annual effective compounding. (Answer: $6.3316\%$)
A hypothetical quote is $6\%$ nominal annual interest convertible semiannually for a $2.5$-year maturity. Find the equivalent annual effective spot rate. (Answer: $6.0900\%$)

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