Continuous Perpetuities Under Constant Force

What Makes a Continuous Perpetuity Different?

A continuous annuity pays over a specified time interval; a continuous perpetuity continues without a final payment time. The payment is not a sequence of installments at discrete dates. Instead, payments flow continuously at a stated rate, called the payment intensity. To value the stream, discount the payments at each time and integrate over the full payment period.

This tutorial assumes a level payment intensity and a constant force of interest. The force must be positive for the present value of an indefinitely continuing level stream to be finite. The formula also gives a useful comparison: the value of a finite stream approaches the perpetuity value as its term increases.

Definition: A level continuous perpetuity with payment intensity $c$ pays at a constant rate of $c$ currency units per unit of time, continuously from time $0$ onward. Under a constant force of interest $\delta>0$, its present value at time $0$ is the integral of the discounted payment intensity over all future time.

The time unit must agree across the payment intensity, force, and term. For example, if $c$ is dollars per year, then $\delta$ is an annual force and a term $n$ is measured in years. A payment intensity is a rate of payment, not a total payment made at a single date.

Deriving the Perpetuity Formula

Under a constant force $\delta$, a payment made at time $t$ has present-value discount factor $e^{-\delta t}$. During a short interval of length $dt$, a level stream with intensity $c$ pays approximately $c\,dt$. Its present value is therefore approximately $c e^{-\delta t}dt$. Integrating these discounted amounts gives the exact value.

V(\infty)=\int_0^\infty c e^{-\delta t}\,dt =c\left[-\frac{1}{\delta}e^{-\delta t}\right]_0^\infty =\frac{c}{\delta}, \qquad \delta>0.

The last equality uses $\lim_{t\to\infty}e^{-\delta t}=0$ when $\delta>0$. The condition matters: if $\delta=0$, the integral is $\int_0^\infty c\,dt$, which diverges for $c>0$. If $\delta<0$, the discount factor grows over time and the integral also diverges. Thus $c/\delta$ is not a finite perpetuity value for a zero or negative force.

Theorem: For a level continuous payment intensity $c$ beginning at time $0$, under a constant positive force $\delta$, the present value is $c/\delta$. If payments stop after $n$ time units, their present value is $c(1-e^{-\delta n})/\delta$. The difference between the perpetuity value and the finite-term value is $ce^{-\delta n}/\delta$.

For a finite term $n$, the same integral has a finite upper limit. Subtracting the finite value from the perpetuity value shows exactly how much value lies beyond time $n$:

V(n)=\int_0^n c e^{-\delta t}\,dt =\frac{c}{\delta}(1-e^{-\delta n}), \qquad V(\infty)-V(n)=\frac{c}{\delta}e^{-\delta n}.

The proportion of the perpetuity value captured by the first $n$ time units is $1-e^{-\delta n}$; the omitted proportion is $e^{-\delta n}$. As $n$ increases, the finite-term value increases toward $c/\delta$, while the omitted tail decreases toward zero. This describes convergence under a constant force, not a rule to use when the force varies over time.

Worked Examples

Worked Example: Comparing a 12-Year Stream with Its Perpetuity

A hypothetical research fund pays continuously at an intensity of $1,800 per year. The fund earns interest at a constant force of 0.06 per year. Find the present value of payments over 12 years, the perpetuity value, and the value omitted by stopping after 12 years.

Method. Use the finite-term integral and the infinite-term formula with the same intensity and force. The force is already stated, so no rate conversion is needed.

The perpetuity value is $V(\infty)=c/\delta=1{,}800/0.06=\$30{,}000$.

For the 12-year stream, $\delta n=0.06(12)=0.72$, and $e^{-0.72}\approx0.4867522560$. Thus $1-e^{-0.72}\approx0.5132477440$, giving $V(12)=30{,}000(0.5132477440)=\$15{,}397.4323$.

The omitted tail is $30{,}000e^{-0.72}=30{,}000(0.4867522560)=\$14{,}602.5677$. Rounded to cents, the finite value is $15,397.43, the perpetuity value is $30,000.00, and the omitted tail is $14,602.57. As a check, the finite value plus the omitted tail equals $30,000.00 before rounding.

Common mistake avoided. The stream pays for 12 years, not forever. Using $30,000 as the 12-year value would include payments after year 12; the tail calculation isolates that extra value.

Worked Example: Converting an Effective Annual Rate Before Valuation

A hypothetical endowment pays continuously at an intensity of $2,400 per year. Its effective annual interest rate is 5%. Find the perpetuity value and the value of the first 20 years of payments. Round final dollar amounts to cents.

Method. Convert the effective annual rate $i$ to the equivalent constant force $\delta$, then use the continuous-stream formulas. The relationship follows because one year of accumulation under a constant force is $e^\delta$, which must equal $1+i$. Therefore $\delta=\ln(1+i)$.

Here, $\delta=\ln(1.05)\approx0.04879016417$ per year. The perpetuity value is $2{,}400/0.04879016417\approx\$49{,}190.24235$, or $49,190.24.

For 20 years, $\delta n=20\ln(1.05)$, so $e^{-\delta n}=1.05^{-20}\approx0.376889482$. The finite-term value is $(2{,}400/\ln(1.05))(1-1.05^{-20})$. The finite proportion is $1-0.376889482=0.623110518$, so the value is approximately $49{,}190.24235(0.623110518)=\$30{,}650.95734$, or $30,650.96.

The omitted tail is approximately $49{,}190.24235(0.376889482)=\$18{,}539.28501$, or $18,539.29. The check is $30{,}650.96+18{,}539.29=\$49{,}190.25$, a one-cent difference caused by rounding the two components separately; before rounding, they sum to the perpetuity value.

Common mistake avoided. Substituting the effective rate $0.05$ for the force would give $2{,}400/0.05=\$48{,}000$. That is not the correct continuous-perpetuity calculation: the force is $\ln(1.05)$, so the correct perpetuity value is $49,190.24.

Worked Example: A Deferred Stream and Its Finite-Term Limit

A hypothetical grant begins paying continuously at $1,200 per year at time 3 and continues forever. The constant force is 0.04 per year. Find its present value at time 0. Then find the time-0 value if instead payments stop 10 years after they begin, at time 13. Round final dollar amounts to cents.

Method. First value each stream at its start time, then discount that value back to time 0. At time 3, the perpetuity value is $1{,}200/0.04=\$30{,}000$. The discount factor from time 3 to time 0 is $e^{-0.04(3)}=e^{-0.12}\approx0.8869204367$. Thus the time-0 perpetuity value is $30{,}000(0.8869204367)=\$26{,}607.61310$, or $26,607.61.

For the 10-year stream, its value at time 3 is $30{,}000(1-e^{-0.04(10)})=30{,}000(1-e^{-0.4})$. Since $e^{-0.4}\approx0.6703200460$, this is $30{,}000(0.3296799540)=\$9{,}890.39862$. Discounting back three years gives $9{,}890.39862(0.8869204367)=\$8{,}771.99666$, or $8,772.00.

As a check, the omitted payments after time 13 have time-0 value $30{,}000e^{-0.04(13)}=30{,}000e^{-0.52}$. Using $e^{-0.52}\approx0.5945205480$, this is $17,835.61644. Adding it to the finite value gives $26,607.61310 before rounding, equal to the deferred perpetuity value.

Common mistake avoided. The 10-year term starts at time 3, not at time 0. The finite stream therefore ends at time 13, and its time-0 value includes a separate three-year discount.

Common Traps and Practical Interpretation

A continuous perpetuity formula is compact, but applying it correctly requires attention to the payment horizon, rate basis, and time units.

Using an effective rate as a force. When the rate is effective annually, first use $\delta=\ln(1+i)$ for a constant-force model. Do not substitute $i$ directly into $c/\delta$.
Confusing intensity with a lump sum. An intensity of $1,800 per year describes payments spread continuously through time. It is not a single $1,800 payment at the beginning or end of each year.
Ignoring the finite horizon. The formula $c/\delta$ includes payments for all future time. For a stream ending after $n$ years, use $c(1-e^{-\delta n})/\delta$.
Mixing time units. A yearly intensity and yearly force require $n$ in years. If the time unit changes, express the intensity, force, and term consistently before integrating.
Applying the formula when the force is not positive. The integral over an infinite horizon is finite only when $\delta>0$ for a positive level intensity.

Key takeaway: Under a positive constant force, a level continuous perpetuity has value $c/\delta$. A finite stream of length $n$ has value $c(1-e^{-\delta n})/\delta$, and the remaining tail is $ce^{-\delta n}/\delta$. These formulas also show precisely how the finite value approaches its infinite-term limit.

Check Your Understanding

For each problem, use the stated constant force and keep the payment intensity and term in consistent time units.

A hypothetical stream pays continuously at $750 per year forever under a constant force of 0.05 per year. Find its present value. (Answer: $15,000.00)
A stream pays at an intensity of $1,800 per year under a constant force of 0.06 per year, but stops after 8 years. Find its finite value and the omitted perpetuity tail, both at time 0. (Answer: finite value $11,436.50; omitted tail $18,563.50)
A hypothetical grant pays continuously at $1,200 per year starting at time 2, under a constant force of 0.04 per year. Find the time-0 value of the perpetuity and of the stream if it stops 8 years after starting. (Answer: perpetuity value $27,693.49; finite-stream value $7,583.89)
Under a constant force of 0.05 per year, how long must a level continuous stream run to accumulate 90% of its perpetuity value? (Answer: $\ln(10)/0.05\approx46.052$ years)
A stream pays continuously at $1,000 per year under a constant force of 0.10 per year for 10 years. Find its value and the percentage of the perpetuity value captured. (Answer: $6,321.21; approximately 63.21%)

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