How Maturity-Specific Rates Affect Present Value

Why a Maturity-Specific Rate Matters

A spot curve assigns a discount factor to each maturity. If the spot rate for one maturity changes while the other discount factors are held fixed, only the cash flows discounted at that maturity are directly revalued. This makes it possible to isolate the effect of a particular rate change rather than comparing two entire spot curves.

The key distinction is between an exact change, found by discounting at the old and new rates, and a local sensitivity estimate, found from a derivative. The derivative is useful for understanding how present value responds to a small change; for a material change, calculate the new present value directly.

Definition: Let $C_t$ be the net cash flow due at time $t$, measured in years, and let $i_t$ be the effective annual spot rate for maturity $t$. Under annual effective compounding, the maturity-$t$ discount factor is $v_t=(1+i_t)^{-t}$, and that cash flow's present value is $C_t v_t$. A change in $i_t$, with all other discount factors fixed, changes the present value of the cash flow at $t$ but not the present values of cash flows at other maturities.

This model assumes the spot rate for maturity $t$ is quoted as an effective annual rate and that its discount factor is calculated using the exponent $t$. It requires $i_t>-1$, so the base $1+i_t$ is positive. Cash flows may be positive receipts or negative payments; their signs must be retained in the calculation. For several payments at the same maturity, first combine them into a single net cash flow.

Exact Change and Local Sensitivity

Suppose the rate for maturity $t$ changes from $i_t$ to $i_t+\Delta i_t$. The exact change in present value for the cash flow due at $t$ is the new value minus the old value. A derivative gives the local rate of change at the original rate.

\Delta \operatorname{PV}_t = C_t\left[(1+i_t+\Delta i_t)^{-t}-(1+i_t)^{-t}\right]

\frac{\partial \operatorname{PV}_t}{\partial i_t} = \frac{d}{di_t}\left[C_t(1+i_t)^{-t}\right] = -tC_t(1+i_t)^{-t-1}

For a small rate change, the derivative-based estimate is $\Delta \operatorname{PV}_t\approx -tC_t(1+i_t)^{-t-1}\Delta i_t$. Here $\Delta i_t$ is written as a decimal: a rise of one percentage point means $\Delta i_t=0.01$, not $1$.

Why the Sensitivity Formula Holds

The present value at maturity $t$ is $C_t$ multiplied by its discount factor. The cash flow is fixed, so differentiating with respect to the spot rate means differentiating only the discount factor. The chain rule gives the derivative shown above.

Write the affected value.
For a fixed cash flow $C_t$, its present value as a function of its maturity-specific rate is $C_t(1+i_t)^{-t}$.

Differentiate the rate-dependent factor.
The power rule gives $\frac{d}{di_t}(1+i_t)^{-t}=-t(1+i_t)^{-t-1}$.

Multiply by the fixed cash flow.
The derivative of the present value is therefore $-tC_t(1+i_t)^{-t-1}$.

For a positive receipt, the derivative is negative: increasing its spot rate decreases its present value. For a negative payment, the derivative is positive: increasing the rate makes the present value of that negative amount less negative. The derivative describes a local change, not necessarily the exact result for a large rate movement.

Key result: If a schedule has net cash flows $C_s$ at maturities $s$, and only $i_t$ changes, then the change in the schedule's total present value is exactly the change in $C_t(1+i_t)^{-t}$. All terms at maturities $s\ne t$ remain unchanged when their discount factors are held fixed.

Worked Examples

Worked Example: Repricing Only the Two-Year Cash Flow

A portfolio pays \$1,000 in one year, \$2,000 in two years, and \$3,000 in three years. The effective annual spot rates are 3% for one year, 4% for two years, and 5% for three years. The two-year spot rate rises to 5%; both other discount factors are held fixed. Find the exact change in the portfolio's present value.

Method. Isolate the cash flow due in two years. The other two present values do not change under the stated assumption.

At the original two-year rate, the affected present value is $\$2{,}000/(1.04)^2$. Since $1.04^2=1.0816$, this is approximately $\$1{,}849.11$.

At the new two-year rate, it is $\$2{,}000/(1.05)^2$. Since $1.05^2=1.1025$, this is approximately $\$1{,}814.06$.

Subtract the old value from the new value: $\$1{,}814.06-\$1{,}849.11=-\$35.05$. Using the unrounded discount factors gives the same change to the nearest cent.

Check. The changed rate applies to a positive receipt, and the rate rose, so the present value must fall. The negative change is consistent. A common mistake is to re-discount all three payments at 5%; that would change discount factors the problem says to hold fixed.

Worked Example: Comparing a Derivative Estimate with an Exact Change

A receipt of \$5,000 is due in three years. Its three-year spot rate is initially 0% and rises to 1%. Find (a) the derivative-based estimate of the present-value change and (b) the exact change.

Method. Evaluate the derivative at the original rate for the estimate, then calculate both present values for the exact change. The rate change is $0.01$, because one percentage point is $0.01$ in decimal form.

At $i_3=0$, the derivative is $-3(\$5{,}000)(1+0)^{-4}=-\$15{,}000$ per unit change in the rate. The estimated change is therefore $-\$15{,}000(0.01)=-\$150.00$.

For the exact calculation, the original present value is $\$5{,}000/(1.00)^3=\$5{,}000.00$. The new present value is $\$5{,}000/(1.01)^3$. Since $1.01^3=1.030301$, the new value is approximately $\$4{,}852.95$. Thus the exact change is $\$4{,}852.95-\$5{,}000.00=-\$147.05$.

Check. The derivative estimate is close to, but not identical to, the exact change. The positive receipt loses value when its rate rises. A common mistake is to report the derivative estimate as the exact change; here the derivative estimate is $-\$150.00$, while the exact change is $-\$147.05$.

Worked Example: Combining Same-Maturity Cash Flows Before Repricing

A schedule contains a \$6,000 receipt and a \$3,500 payment, both due in four years. The four-year effective annual spot rate falls from 3% to 2%. Other cash flows in the schedule are at different maturities, and their discount factors are held fixed. Find the exact change in the schedule's total present value.

Method. Combine the two cash flows due at the same maturity: $C_4=\$6{,}000-\$3{,}500=\$2{,}500$. Only this net amount is revalued.

At 3%, its present value is $\$2{,}500/(1.03)^4$. Since $1.03^4=1.12550881$, this is approximately $\$2{,}221.22$.

At 2%, its present value is $\$2{,}500/(1.02)^4$. Since $1.02^4=1.08243216$, this is approximately $\$2{,}309.61$.

The exact change is approximately $\$2{,}309.61-\$2{,}221.22=+\$88.40$, rounding the unrounded difference to cents.

Check. The net cash flow is a positive receipt, and its rate fell, so its present value rises. A common mistake is to revalue the \$6,000 receipt but leave the \$3,500 payment out of the sensitivity calculation; both amounts are due at the affected maturity, so the correct amount to revalue is their net \$2,500.

Interpreting and Applying the Result

The factor $t$ in the derivative shows why, at a given rate and cash-flow amount, a later-dated payment can be more sensitive to a small rate change. The amount of sensitivity also depends on the discounting term $(1+i_t)^{-t-1}$ and on the size and sign of the cash flow. The derivative is measured in dollars per unit change in the rate; multiplying it by a rate change expressed as a decimal gives an estimated dollar change.

The effect is maturity-specific. If the rate for year two changes, a payment due in year three is not affected by that change in this calculation. This conclusion depends on holding the other discount factors fixed. If a problem instead gives a new curve in which several maturity-specific rates change, each affected cash flow must be revalued using its own new discount factor.

An exact calculation is particularly important when the rate change is not small, when the maturity is long, or when the result is needed to the nearest cent. A derivative is a tangent-line approximation at the starting rate; as the rate moves, the discount factor's slope changes. Use the derivative to estimate and explain sensitivity, and use the old and new discount factors to obtain the exact change.

Exam takeaway: For a change in one spot rate, identify its maturity, combine all cash flows due at that maturity, and change only that maturity's discount factor. Use a decimal rate change in any derivative estimate, and distinguish the estimate from the exact revaluation. This treatment is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, keep all discount factors not explicitly changed fixed. Round monetary answers to the nearest cent.

A \$1,200 receipt is due in two years. Its two-year effective annual spot rate rises from 4% to 5%. Find the exact present-value change. (Answer: $-\$21.03$)
A \$4,000 receipt is due in two years at an initial spot rate of 0%. The rate rises by 0.5 percentage points. Find the derivative-based estimate and the exact present-value change. (Answer: estimate $-\$40.00$; exact change $-\$39.70$)
A schedule has a net receipt of \$1,800 due in three years. Its three-year spot rate rises from 2% to 3%. Find the exact present-value change. (Answer: $-\$48.93$)
A \$5,000 receipt and a \$1,200 payment are both due in four years. The four-year spot rate falls from 3% to 2.5%. Find the exact change in their combined present value. (Answer: $+\$66.36$)
For a positive cash flow due at maturity $t$, with $t>0$ and $i_t>-1$, state the sign of the derivative of its present value with respect to $i_t$, and explain what that sign means. (Answer: The derivative is negative, $-tC_t(1+i_t)^{-t-1}<0$, so increasing the rate decreases the present value.)

What You'll Learn