How do you put a price on a stream of money over time?

A future dollar is not a present dollar

If today’s dollar can be invested at rate r and grow to $1 + r$ in one year (or to $e^r$ with continuous compounding from the logarithm module), then $1 next year is equivalent to $1/(1 + r)$ today. That’s the simplest possible present value calculation. Two payments make the same logic compound: $1 two years from now is $1/(1 + r)²$ today; $1 t years from now is $1/(1 + r)^t$ today.

The factor $1/(1 + r)^t$ — or its continuous cousin $e^(−rt)$ — is the discount factor . It shrinks with t (more time, more discount) and shrinks faster as r grows (higher interest, more aggressive discount). The whole module is one product and one sum: future cash × discount factor, summed across all the future moments money arrives.

A discrete stream — coupons, salaries, rent

Real cash flows arrive in discrete pieces. A bond pays $100 every year for N years. A salary pays every two weeks. Rent every month. For each piece, compute its present value separately, then add them all up:

$PV = Σ_{i=1}^N C_i / (1 + r)^(t_i)$

For a constant C paid annually for N years at rate r, the geometric sum has a closed form: $PV = C · (1 − (1+r)^(−N)) / r$ — but the concept is just “discount each one, add them all up.” A spreadsheet does this naturally; the formula is for paper.

The widget shows a $1/year stream. At r = 5%, the first ten years contribute about $7.72 of PV; the next ten add about $4.74 more (much less); past 30 years, additional payments contribute almost nothing. Distant money discounts away — a fact that matters when valuing very long streams (mortgages, pensions, perpetuities, eternity).

import math

# Discrete: pay $C every year for N years, discount at rate r per year.
# PV = sum_{i=1..N} C / (1 + r)^i
def pv_discrete(C, r, N):
    return sum(C / (1 + r)**i for i in range(1, N + 1))

# At r = 5%, $1/year for 10 years is worth less than $10 today.
[pv_discrete(1, 0.05, N) for N in (1, 5, 10, 20)]
# → [0.95, 4.33, 7.72, 12.46]
# Past 20 years, additional payments contribute almost nothing — distant
# money discounts away, no matter the cash flow's nominal size.

Continuous compounding — the exponential takes over

When payments arrive faster — monthly, daily, every second — the discrete sum becomes a continuous integral, and the discount factor $1/(1+r)^t$ smooths into $e^(−rt)$. Why an exponential? Take the log of $(1 + r/n)^(n·t)$ as compounding frequency n grows: $n·t · \log(1 + r/n) → r·t$, so $(1 + r/n)^(n·t) → e^(r·t)$. The logarithm module’s identity surfaces here directly — continuous growth and continuous discount are inverses of one another, and both live in the exponential.

The continuous discount factor $e^(−rt)$ matches the discrete factor at large n, but it’s easier to integrate than to sum, because of the closed form for exponential antiderivatives (from the integration module). For pricing real instruments — bonds, dividends, mortgages — both versions are used; the choice depends on whether your payments arrive in lumps or smoothly.

# Continuous version: discount factor is e^(-r*t), which is what
# 'continuous compounding' gives in the limit of (1 + r/n)^(n*t) as n → ∞.
# Why exponential? log of (1 + r/n)^(n*t) = n*t * log(1 + r/n) → r*t,
# so (1 + r/n)^(n*t) → e^(r*t). The log module's identity surfaces here
# directly: continuous discount = inverse of continuous growth.

def discount_continuous(t, r):
    return math.exp(-r * t)

[(t, round(discount_continuous(t, 0.05), 4)) for t in (0, 1, 5, 10, 30)]
# → [(0, 1.0), (1, 0.9512), (5, 0.7788), (10, 0.6065), (30, 0.2231)]
# 30 years out at 5% — a future dollar is worth ~22 cents today.

A continuous cash flow — the integral takes over

A cash flow rate $c(t)$ says: in a tiny time slice $dt$ at time $t$, you receive $c(t) · dt$ dollars. The undiscounted total over [0, T] is the integral $∫_0^T c(t) dt$. To get present value, discount each tiny slice as it arrives, then add (integrate) — the future money has to be made smaller before it can be summed:

$PV = ∫_0^T c(t) · e^(−rt) dt$

For the simplest case — constant cash flow rate c over [0, T], constant rate r — the integral is closed-form: $PV = (c/r) · (1 − e^(−rT))$. The widget shades this area on its plot; drag $r$ and $T$ and watch the answer move. Two limits matter. $r → 0$: no discount, $PV → c · T$ (just the undiscounted total). $T → ∞$: a perpetuity, $PV → c / r$ — the famous formula every finance textbook starts with.

# Continuous cash flow at rate c (dollars/year) over [0, T]:
# PV = ∫_0^T c · e^(-r*t) dt = (c/r) * (1 - e^(-r*T))
# For r → 0, the limit is c*T (no discount). For T → ∞, the limit is
# c/r — the perpetuity formula. Both are readable from the closed form.
def pv_continuous(c, r, T):
    if abs(r) < 1e-9:
        return c * T
    return (c / r) * (1 - math.exp(-r * T))

# Compare a $1/year stream's PV under different (r, T):
[(r, T, round(pv_continuous(1, r, T), 2))
 for r in (0.02, 0.05, 0.10) for T in (10, 30, 100)]
# → [(0.02, 10, 9.06),  (0.02, 30, 22.55),  (0.02, 100, 43.23),
#    (0.05, 10, 7.87),  (0.05, 30, 15.54),  (0.05, 100, 19.87),
#    (0.10, 10, 6.32),  (0.10, 30, 9.50),   (0.10, 100, 9.99)]
# At 10% the perpetuity limit is c/r = 10. At 100 years we're already
# at $9.99 — basically the limit. *5/r is roughly when extending
# matters less than the next decimal place.*

The 5/r rule — distant money barely matters

The closed form $(c/r)(1 − e^(−rT))$ hides one of finance’s most useful rules of thumb. The factor $1 − e^(−rT)$ reaches 99% of its perpetuity limit at $r·T = \ln(100) ≈ 4.6$, or roughly $T ≈ 5/r$. So at r = 10%, the perpetuity limit is essentially reached after 50 years; at r = 5%, it takes 100 years; at r = 2%, it takes 250 years. Once you are past 5/r, extending the horizon barely adds anything.

This is why infinite-life valuations (perpetuities, real estate, sovereign debt) and very-long-life valuations (insurance reserves, climate damage, multi-generation projects) are not that different at typical discount rates. The future just isn’t worth that much in present-value terms. The whole intergenerational-economics debate is about whether r itself should differ across generations — because the answer to “what does a benefit 200 years from now cost us today?” is almost entirely a choice of r, not a choice of c.

Sensitivity — how PV moves when the rate moves

Up to here, $r$ has been a number you set. In practice it is a number that moves — central banks change it, markets re-price it, your view on it shifts. The natural next question is: if $r$ shifts by a small amount $\Delta r$, how much does PV move? That is a question about a derivative — specifically $\partial PV / \partial r$, the slope of PV viewed as a function of $r$.

For the continuous case $PV = \int_0^T c(t) · e^{-rt} dt$, differentiating under the integral sign gives

$\partial PV / \partial r \; = \; -\int_0^T t · c(t) · e^{-rt} dt$

— the same integral as PV, except each cash-flow slice is multiplied by an extra $t$. Far-future money contributes more to the sensitivity. Define the modified duration $D = -(\partial PV / \partial r) / PV$. Then a small rate move predicts a price move by linearization:

$\Delta PV \;\approx\; -D · PV · \Delta r$

A 10-year zero-coupon bond has $D \approx 10$; a 1% rise in rates predicts about a 10% drop in price. A 2-year bond has $D \approx 2$; the same rate move predicts only about 2%. Duration is a derivative of price with respect to rate, normalized — and it is what the bond market uses to price interest-rate risk every day.

The same shape runs the Greeks in option pricing — $\Delta$ is $\partial \text{Price} / \partial S$, $\Gamma$ is the second derivative, $\Theta$ is $\partial \text{Price} / \partial t$, $\mathcal{V}$ is $\partial \text{Price} / \partial \sigma$. Every financial instrument has a family of sensitivities — partial derivatives of the price with respect to each input. The word derivative in “financial derivative” and the word derivative in calculus are the same word for a deep reason: the instrument’s price is the derivative of something else, and its risk is captured by its own derivatives.