Time value of money

Future value of a present sum

The future value (FV) formula is similar and uses the same variables.

${\displaystyle FV\ =\ PV\cdot (1+i)^{n}}$

Present value of a future sum

The present value formula is the core formula for the time value of money; each of the other formulas is derived from this formula. For example, the annuity formula is the sum of a series of present value calculations.

The present value (PV) formula has four variables, each of which can be solved for by numerical methods:

${\displaystyle PV\ =\ {\frac {FV}{(1+i)^{n}}}}$

The cumulative present value of future cash flows can be calculated by summing the contributions of FV_t, the value of cash flow at time t:

${\displaystyle PV\ =\ \sum _{t=1}^{n}{\frac {FV_{t}}{(1+i)^{t}}}}$

Note that this series can be summed for a given value of n, or when n is ∞.^[18] This is a very general formula, which leads to several important special cases given below.

Present value of an annuity for n payment periods

In this case the cash flow values remain the same throughout the n periods. The present value of an annuity (PVA) formula has four variables, each of which can be solved for by numerical methods:

${\displaystyle PV(A)\,=\,{\frac {A}{i}}\cdot \left[{1-{\frac {1}{\left(1+i\right)^{n}}}}\right]}$

To get the PV of an annuity due, multiply the above equation by (1 + i).

Present value of a growing annuity

In this case, each cash flow grows by a factor of (1 + g). Similar to the formula for an annuity, the present value of a growing annuity (PVGA) uses the same variables with the addition of g as the rate of growth of the annuity (A is the annuity payment in the first period). This is a calculation that is rarely provided for on financial calculators.

Where i ≠ g :

${\displaystyle PV(A)\,=\,{A \over (i-g)}\left[1-\left({1+g \over 1+i}\right)^{n}\right]}$

Where i = g :

${\displaystyle PV(A)\,=\,{A\times n \over 1+i}}$

To get the PV of a growing annuity due, multiply the above equation by (1 + i).

Present value of a growing annuity and a fixed-benefit pension

How to calculate the present value of the Cost Of Living Adjustment that you will add to your fixed-benefit pension payment. The growth factor (1+g) applies to both cash flows and the fixed pension F. A is the first payment.

${\displaystyle PV(A)\,=\,{A+F \over i-g}\left[1-\left({1+g \over 1+i}\right)^{n}\right]-{F \over i}\left[1-{1 \over (1+i)^{n}}\right]}$

To find the Present Value (PV) necessary to provide a Cost Of Living Adjustment for a pension, set A = 0 so:

• Nothing is paid in the first year
• The second year pays an amount equal to ${\displaystyle F\times g}$
• The third year pays ${\displaystyle F(1+g)^{2}{\text{ minus }}F}$
• The fourth year pays ${\displaystyle F(1+g)^{3}{\text{ minus }}F}$, and so on

If PV is more than what’s needed, the extra amount gets spread across future payments, increasing each year by a factor of (1+g) which is a growing annuity.
If PV is too small, the payment becomes negative, meaning the pension must be reduced by that amount in order to add the Cost Of Living Adjustment.

Present value of a perpetuity

A perpetuity is payments of a set amount of money that occur on a routine basis and continue forever. When n → ∞, the PV of a perpetuity (a perpetual annuity) formula becomes a simple division.

${\displaystyle PV(P)\ =\ {A \over i}}$

Present value of a growing perpetuity

When the perpetual annuity payment grows at a fixed rate (g, with g < i) the value is determined according to the following formula, obtained by setting n to infinity in the earlier formula for a growing perpetuity:

${\displaystyle PV(A)\,=\,{A \over i-g}}$

In practice, there are few securities with precise characteristics, and the application of this valuation approach is subject to various qualifications and modifications. Most importantly, it is rare to find a growing perpetual annuity with fixed rates of growth and true perpetual cash flow generation. Despite these qualifications, the general approach may be used in valuations of real estate, equities, and other assets.

This is the well known Gordon growth model used for stock valuation.

Future value of an annuity

The future value (after n periods) of an annuity (FVA) formula has four variables, each of which can be solved for by numerical methods:

${\displaystyle FV(A)\,=\,A\cdot {\frac {\left(1+i\right)^{n}-1}{i}}}$

To get the FV of an annuity due, multiply the above equation by (1 + i).

Future value of a growing annuity

The future value (after n periods) of a growing annuity (FVA) formula has five variables, each of which can be solved for by numerical methods:

Where i ≠ g :

${\displaystyle FV(A)\,=\,A\cdot {\frac {\left(1+i\right)^{n}-\left(1+g\right)^{n}}{i-g}}}$

Where i = g :

${\displaystyle FV(A)\,=\,A\cdot n(1+i)^{n-1}}$

Formula table

The following table summarizes the different formulas commonly used in calculating the time value of money.^[19] These values are often displayed in tables where the interest rate and time are specified.

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Find	Given	Formula
Future value (F)	Present value (P)	${\displaystyle F=P\cdot (1+i)^{n}}$
Present value (P)	Future value (F)	${\displaystyle P=F\cdot (1+i)^{-n}}$
Repeating payment (A)	Future value (F)	${\displaystyle A=F\cdot {\frac {i}{(1+i)^{n}-1}}}$
Repeating payment (A)	Present value (P)	${\displaystyle A=P\cdot {\frac {i(1+i)^{n}}{(1+i)^{n}-1}}}$
Future value (F)	Repeating payment (A)	${\displaystyle F=A\cdot {\frac {(1+i)^{n}-1}{i}}}$
Present value (P)	Repeating payment (A)	${\displaystyle P=A\cdot {\frac {(1+i)^{n}-1}{i(1+i)^{n}}}}$
Future value (F)	Initial gradient payment (G)	${\displaystyle F=G\cdot {\frac {(1+i)^{n}-in-1}{i^{2}}}}$
Present value (P)	Initial gradient payment (G)	${\displaystyle P=G\cdot {\frac {(1+i)^{n}-in-1}{i^{2}(1+i)^{n}}}}$
Fixed payment (A)	Initial gradient payment (G)	${\displaystyle A=G\cdot \left[{\frac {1}{i}}-{\frac {n}{(1+i)^{n}-1}}\right]}$
Future value (F)	Initial exponentially increasing payment (D) Increasing percentage (g)	${\displaystyle F=D\cdot {\frac {(1+g)^{n}-(1+i)^{n}}{g-i}}}$   (for i ≠ g) ${\displaystyle F=D\cdot {\frac {n(1+i)^{n}}{1+g}}}$   (for i = g)
Present value (P)	Initial exponentially increasing payment (D) Increasing percentage (g)	${\displaystyle P=D\cdot {\frac {\left({1+g \over 1+i}\right)^{n}-1}{g-i}}}$   (for i ≠ g) ${\displaystyle P=D\cdot {\frac {n}{1+g}}}$   (for i = g)
Present value (P)	Initial exponentially increasing payment(D) with a fixed-benefit pension(F) Increasing percentage (g)	${\displaystyle P=(D+F)\cdot {\frac {\left({1+g \over 1+i}\right)^{n}-1}{g-i}}-{\frac {F}{i}}\left[1-{1 \over (1+i)^{n}}\right]}$

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Notes:

• A is a fixed payment amount, every period
• G is the initial payment amount of an increasing payment amount, that starts at G and increases by G for each subsequent period.
• D is the initial payment amount of an exponentially (geometrically) increasing payment amount, that starts at D and increases by a factor of (1 + g) each subsequent period.

Annuity derivation

The formula for the present value of a regular stream of future payments (an annuity) is derived from a sum of the formula for future value of a single future payment, as below, where C is the payment amount and n the period.

A single payment C at future time m has the following future value at future time n:

${\displaystyle FV\ =C(1+i)^{n-m}}$

Summing over all payments from time 1 to time n, then reversing the order of terms and substituting k = n − m:

${\displaystyle FVA\ =\sum _{m=1}^{n}C(1+i)^{n-m}\ =\sum _{k=0}^{n-1}C(1+i)^{k}}$

Note that this is a geometric series, with the initial value being a = C, the multiplicative factor being 1 + i, with n terms. Applying the formula for geometric series, we get:

${\displaystyle FVA\ ={\frac {C(1-(1+i)^{n})}{1-(1+i)}}\ ={\frac {C(1-(1+i)^{n})}{-i}}}$

The present value of the annuity (PVA) is obtained by simply dividing by ${\displaystyle (1+i)^{n}}$:

${\displaystyle PVA\ ={\frac {FVA}{(1+i)^{n}}}={\frac {C}{i}}\left(1-{\frac {1}{(1+i)^{n}}}\right)}$

Another simple and intuitive way to derive the future value of an annuity is to consider an endowment, whose interest is paid as the annuity, and whose principal remains constant. The principal of this hypothetical endowment can be computed as that whose interest equals the annuity payment amount:

${\displaystyle {\text{Principal}}\times i=C}$

${\displaystyle {\text{Principal}}={\frac {C}{i}}+{\text{goal}}}$

Note that no money enters or leaves the combined system of endowment principal + accumulated annuity payments, and thus the future value of this system can be computed simply via the future value formula:

${\displaystyle FV=PV(1+i)^{n}}$

Initially, before any payments, the present value of the system is just the endowment principal, ${\displaystyle PV={\frac {C}{i}}}$. At the end, the future value is the endowment principal (which is the same) plus the future value of the total annuity payments (${\displaystyle FV={\frac {C}{i}}+FVA}$). Plugging this back into the equation:

${\displaystyle {\frac {C}{i}}+FVA={\frac {C}{i}}(1+i)^{n}}$

${\displaystyle FVA={\frac {C}{i}}\left[\left(1+i\right)^{n}-1\right]}$

Growing Annuity and a fixed-benefit pension derivation

• C = First payment increasing by g each year
• F = Fixed pension

View this as a lender/borrower scenario with the lender on the left side of the equal sign and the borrower on the right.

At the end of a 1 year loan, a saver would have his money grow by 1+i or PV(1+i). If he instead loaned this money out to a borrower, the borrower would need to make one payment C back to the lender. Setting these equal would give

${\displaystyle n=1:PV(1+i)=(C+F)-F}$

At the end of a 2 year loan, a saver would have his money grow by another 1+i for a total amount of ${\displaystyle PV(1+i)^{2}}$. If he instead loaned this money out, the borrower would need to make two payments. The first payment (C+F)-F would start earning interest since it has now been paid back. His second payment (C+F) would grow by 1+g. The pension F would remain the same

${\displaystyle {\begin{aligned}n=2:PV(1+i)^{2}=&(C+F)(1+i)-F(1+i)\\&+(C+F)(1+g)-F\end{aligned}}}$

At the end of a 3 year loan, a saver would have his money grow by another 1+i for a total amount of ${\displaystyle PV(1+i)^{3}}$. If he instead loaned this money out, the borrower would need to make three payments. The first two payments would earn an additional (1+i) interest. His third payment would grow by an additional 1+g and the pension would still remain the same.

${\displaystyle {\begin{aligned}n=3:PV(1+i)^{3}=&(C+F)(1+i)^{2}-F(1+i)^{2}\\&+(C+F)(1+g)(1+i)-F(1+i)\\&+(C+F)(1+g)^{2}-F\end{aligned}}}$

Simplify the 3 year loan and generalize it at the end for n years. Start by combining the (C+F) and F terms

${\displaystyle {\begin{aligned}PV(1+i)^{3}=&(C+F)(1+g)^{2}\left[\left({\frac {1+i}{1+g}}\right)^{2}+{\frac {1+i}{1+g}}+1\right]\\&-F\left[(1+i)^{2}+(1+i)+1\right]\end{aligned}}}$

Write these terms in summation notation

${\displaystyle {\begin{aligned}PV(1+i)^{3}=&(C+F)(1+g)^{2}\sum _{k=0}^{2}\left({\frac {1+i}{1+g}}\right)^{k}\\&-F\sum _{k=0}^{2}(1+i)^{k}\end{aligned}}}$

Rewrite this is closed form using the geometric Series formula${\displaystyle \sum _{k=0}^{n}r^{k}={\frac {r^{n+1}-1}{r-1}}}$

${\displaystyle {\begin{aligned}PV(1+i)^{3}=&(C+F)(1+g)^{2}\left[{\frac {\left({\frac {1+i}{1+g}}\right)^{3}-1}{\left({\frac {1+i}{1+g}}\right)-1}}\right]\\&-F\left[{\frac {(1+i)^{3}-1}{(1+i)-1}}\right]\end{aligned}}}$

Multiply the top and bottom of the C+F term by (1+g) and cancel the 1’s on the F term

${\displaystyle {\begin{aligned}PV(1+i)^{3}=&(C+F)(1+g)^{3}\left[{\frac {\left({\frac {1+i}{1+g}}\right)^{3}-1}{1+i-(1+g)}}\right]\\&-F\left[{\frac {(1+i)^{3}-1}{i}}\right]\end{aligned}}}$

Multiply the ${\displaystyle (1+g)^{3}}$ on the C+F term through the square brackets.

${\displaystyle {\begin{aligned}PV(1+i)^{3}=&(C+F)\left[{\frac {(1+i)^{3}-(1+g)^{3}}{i-g}}\right]\\&-F\left[{\frac {(1+i)^{3}-1}{i}}\right]\end{aligned}}}$

Divide both sides by ${\displaystyle (1+i)^{3}}$ and generalize by replacing the exponent 3 with n

${\displaystyle {\begin{aligned}PV={\frac {C+F}{i-g}}\left[1-\left({\frac {1+g}{1+i}}\right)^{n}\right]-{\frac {F}{i}}\left[1-{\frac {1}{(1+i)^{n}}}\right]\end{aligned}}}$

Perpetuity derivation

Without showing the formal derivation here, the perpetuity formula is derived from the annuity formula. Specifically, the term:

${\displaystyle \left({1-{1 \over {(1+i)^{n}}}}\right)}$

can be seen to approach the value of 1 as n grows larger. At infinity, it is equal to 1, leaving ${\displaystyle {C \over i}}$ as the only term remaining.

Examples

Using continuous compounding yields the following formulas for various instruments:

Annuity

${\displaystyle \ PV\ =\ {A(1-e^{-rt}) \over e^{r}-1}}$

Perpetuity

${\displaystyle \ PV\ =\ {A \over e^{r}-1}}$

Growing annuity

${\displaystyle \ PV\ =\ {Ae^{-g}(1-e^{-(r-g)t}) \over e^{(r-g)}-1}}$

Growing perpetuity

${\displaystyle \ PV\ =\ {Ae^{-g} \over e^{(r-g)}-1}}$

Annuity with continuous payments

${\displaystyle \ PV\ =\ {1-e^{(-rt)} \over r}}$

These formulas assume that payment A is made in the first payment period and annuity ends at time t.^[20]