How Discount Rates and Timing Affect Present Value

Why the Discount Rate and Payment Date Both Matter

Present value depends not only on how much money is paid, but also on when it is paid and on the rate used to discount it. Raising the rate reduces the present value of a positive future payment. The size of that reduction, however, depends on the payment date. Understanding this dependence helps with comparing values and checking rate-sensitivity claims.

This tutorial assumes a single effective annual rate applies at every date. Time $t$ is measured in years from the valuation date, and a payment at time $t$ is discounted using annual compounding, including fractional-year exponents when needed. If cash flows have different spot rates by maturity, each cash flow must instead be discounted at its applicable spot rate.

Definition: For a cash flow $C$ due at time $t$, its present value at effective annual rate $i$, where $i>-1$, is $PV(i)=C(1+i)^{-t}$. For dated cash flows $C_k$ at times $t_k$, the stream’s present value is $PV(i)=\sum_k C_k(1+i)^{-t_k}$. A positive $C_k$ is a receipt and a negative $C_k$ is a payment.

Rate Sensitivity of One Payment

For a fixed positive payment $C$, increasing $i$ reduces $PV$. Differentiating with respect to the annual rate gives the change in present value per unit change in rate:

\frac{dPV}{di}=-tC(1+i)^{-t-1}

The derivative is negative when $C>0$ and $t>0$. Its magnitude, $tC/(1+i)^{t+1}$, is the absolute dollar sensitivity at the current rate. It is a local measure: for a sufficiently small change $\Delta i$, the first-order estimate is

\Delta PV \approx -\frac{tC}{(1+i)^{t+1}}\Delta i

A rate increase of one percentage point means $\Delta i=0.01$, not $\Delta i=1$. The derivative is stated per unit change in the decimal rate, so multiply it by $0.01$ to estimate the dollar effect of a one-percentage-point increase.

Why Later Payments Have Greater Proportional Sensitivity

Divide the derivative by the present value $C(1+i)^{-t}$. For a positive payment, this gives the rate of change in present value relative to its current level:

\frac{1}{PV}\frac{dPV}{di}=-\frac{t}{1+i}

For a small absolute rate change $\Delta i$, this implies

\frac{\Delta PV}{PV}\approx-\frac{t}{1+i}\Delta i

This is the approximate percentage change in present value for a given percentage-point change in the rate. If instead the rate changes by a small proportion of its current level, its elasticity is the relative sensitivity multiplied by $i$:

\frac{i}{PV}\frac{dPV}{di}=-\frac{it}{1+i}

When $i>0$, the magnitude of this elasticity increases with $t$. Thus, for the same small relative change in the rate, a later positive payment has a greater proportional response. This does not imply that it always has a greater dollar response: the payment’s present value is also smaller when it is further away.

The distinction can be seen in the absolute sensitivity $tC/(1+i)^{t+1}$. As $t$ grows, the factor $t$ increases but the discounting factor in the denominator also grows. Their combined effect need not increase. At a positive rate, a sufficiently distant payment can have less dollar sensitivity than a nearer one of the same nominal amount.

Worked Example: Comparing Two Rates for a Dated Payment

A hypothetical investment will pay $C=\$10{,}000$ at the end of year 2. Compare its present value at effective annual rates of 5% and 8%, then find the dollar change. Use the exact present values rather than treating the local derivative estimate as an exact result.

Method. Apply $PV(i)=C(1+i)^{-t}$ at each rate. The payment is positive, so a rate increase should produce a decrease in present value.

At 5%, $t=2$, and $(1.05)^2=1.1025$:

PV(0.05)=\frac{\$10{,}000}{1.1025}=\$9{,}070.2948\ldots=\$9{,}070.29

At 8%, $(1.08)^2=1.1664$:

PV(0.08)=\frac{\$10{,}000}{1.1664}=\$8{,}573.3889\ldots=\$8{,}573.39

The change, calculated as the new value minus the original value, is $\$8{,}573.39-\$9{,}070.29=-\$496.90$ using the displayed rounded values. Using unrounded values first gives $-\$496.9059\ldots$, or −$496.91 to the nearest cent.

Check. The rate rose from 5% to 8%, and the present value fell, as expected for a positive future payment. The common mistake is to use $t=2$ as a multiplier outside the discount factor; the payment is discounted by $(1+i)^2$, not by $2(1+i)$.

Applying Sensitivity to a Cash-Flow Stream

For a stream, differentiate each discounted cash flow and add the results. Since the number of cash flows is finite, this is equivalent to differentiating the sum term by term:

\frac{dPV}{di} =\frac{d}{di}\sum_k C_k(1+i)^{-t_k} =-\sum_k \frac{t_kC_k}{(1+i)^{t_k+1}}

If every cash flow is positive and occurs after the valuation date, each term in the derivative is negative, so the stream’s present value decreases as the rate rises. With both receipts and payments, the terms can have different signs. The overall effect then depends on their amounts and dates; it should not be inferred from one cash flow alone.

The derivative provides a local estimate. For a finite rate change, calculate the present value at both rates when accuracy matters. A derivative multiplied by a rate change is an approximation, not generally the exact difference.

Worked Example: Estimating and Checking a Stream’s Rate Effect

A hypothetical contract provides receipts of $5,000 at the end of year 1 and $10,000 at the end of year 3. Find its present value at an effective annual rate of 5%, calculate its local rate sensitivity, and check the effect of increasing the rate to 6%.

Method. Discount the two receipts separately and add them. Then differentiate the stream formula. Because both receipts are positive, the derivative should be negative.

At 5%, the discount factors use $1.05$, $1.05^3=1.157625$:

PV(0.05)=\frac{\$5{,}000}{1.05}+\frac{\$10{,}000}{1.157625} =\$4{,}761.9048\ldots+\$8{,}638.3760\ldots =\$13{,}400.28

The derivative at 5% is

\left.\frac{dPV}{di}\right|_{i=0.05} =-\frac{1(\$5{,}000)}{1.05^2} -\frac{3(\$10{,}000)}{1.05^4} =-\$4{,}535.15-\$24{,}681.07 =-\$29{,}216.22

This derivative is in dollars per unit change in the decimal rate. For a one-percentage-point increase, the local estimate is $-\$29{,}216.22(0.01)=-\$292.16$.

Now calculate the exact value at 6%. Since $1.06^3=1.191016$:

PV(0.06)=\frac{\$5{,}000}{1.06}+\frac{\$10{,}000}{1.191016} =\$4{,}716.9811\ldots+\$8{,}396.1928\ldots =\$13{,}113.17

The exact change, using unrounded values, is $\$13{,}113.1739\ldots-\$13{,}400.2807\ldots=-\$287.11$. It is close to, but not equal to, the local estimate of $-\$292.16$.

Check. Both cash flows are positive, so the exact value falls when the rate rises. The common mistake is to report the derivative-based estimate as the exact change; the calculation at 6% confirms that the finite change differs.

Absolute Sensitivity Can Fall for a Distant Payment

The following comparison isolates the difference between dollar and proportional sensitivity. Consider the same hypothetical $1,000 payment at year 1 and year 50, valued at an effective annual rate of 10%. The derivative magnitude for a payment at year $t$ is $tC/(1+i)^{t+1}$.

Worked Example: Dollar Sensitivity Versus Proportional Sensitivity

Calculate the present values and local dollar sensitivities for both payment dates. Then compare their proportional sensitivities.

Method. Use the single-payment present value and derivative formulas at $i=0.10$. The payment amount and rate are identical; only the date changes.

For the year-1 payment:

PV_1=\frac{\$1{,}000}{1.1}=\$909.09,\qquad \left|\frac{dPV_1}{di}\right|=\frac{1(\$1{,}000)}{1.1^2}=\$826.45

For the year-50 payment, $1.1^{51}\approx129.1299$:

PV_{50}=\frac{\$1{,}000}{1.1^{50}}\approx\$8.52,\qquad \left|\frac{dPV_{50}}{di}\right|=\frac{50(\$1{,}000)}{1.1^{51}}\approx\$387.21

The year-50 payment has lower absolute dollar sensitivity at this rate, despite being more distant. Its proportional sensitivity is greater: the magnitude of $(1/PV)(dPV/di)$ is $t/(1+i)$, which is $1/1.1\approx0.9091$ for year 1 and $50/1.1\approx45.4545$ for year 50, each per unit rate change.

Check. The far-future payment has a much smaller present value, so its dollar change can be smaller even though its proportional response is larger. The common mistake is to claim that a later date always means a larger dollar derivative. It does not; compare $t/(1+i)^{t+1}$ for the actual dates and rate.

Practical Interpretation and Common Traps

Rate sensitivity is useful when assessing how a stated valuation changes under a different discount rate. The derivative answers a local question: how quickly is present value changing near the current rate? An exact recalculation answers a finite-change question. Use the measure that matches the question.

Do not confuse the two kinds of sensitivity. Dollar sensitivity is measured by $|dPV/di|$; relative sensitivity is measured by $|(1/PV)(dPV/di)|$. One does not guarantee the other will be larger.
Use decimal rate changes. A move from 5% to 6% is $\Delta i=0.01$. It is not a change of 1 in the derivative formula.
Respect cash-flow signs. A positive future receipt has a negative derivative. A negative future payment has a positive derivative, because a higher rate reduces the present-value magnitude of that payment.
Match the rate model to the cash flows. A single uniform rate is appropriate only when the problem specifies that convention. If maturity-specific spot rates apply, discount each date with its own rate.
Check dates and units. Use $t$ in years when $i$ is an effective annual rate. A payment due in six months has $t=0.5$ under the stated annual compounding convention, not $t=6$.

For a single positive payment and a positive rate, a later date always increases proportional sensitivity to a relative rate change. But its absolute dollar sensitivity depends on the balance between the time multiplier and discounting. That distinction is the key to interpreting claims about how strongly distant payments respond to rates.

Check Your Understanding

Use effective annual rates and round money to the nearest cent. Treat derivative-based changes as local estimates unless the question requests exact present values.

A positive payment of $10,000 is due in year 2. At 10%, calculate its present value and the approximate dollar change for a one-percentage-point increase in the rate. (Answer: $8,264.46; approximately −$150.26.)
A hypothetical benefit pays $4,000 at year 1 and $6,000 at year 3. Find its present value at 8%. (Answer: $8,466.70.)
Equal positive payments are each $5,000, one due in year 2 and one in year 5. At 6%, find the ratio of the year-5 present value to the year-2 present value. (Answer: $1/(1.06)^3\approx0.8396$.)
A receipt of $3,000 is due in year 1 and a receipt of $4,000 in year 4. Calculate the stream’s present value at 5%, then state whether it rises or falls if the rate increases to 6%. (Answer: approximately $6,147.95 at 5%; it falls at 6% because both receipts are positive.)
At a positive rate, compare two positive payments with equal nominal amounts but different dates. Which has greater proportional sensitivity to a small relative rate change, and what condition would be needed to determine which has greater dollar sensitivity? (Answer: the later payment has greater proportional sensitivity; compare $t/(1+i)^{t+1}$ at the stated rate and dates to determine dollar sensitivity.)

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