Time value of money

A separate article treats the option time value.

The time value of money (TVM) or the discounted present value is one of the basic concepts of finance, developed by Leonardo Fibonacci in 1202.

The time value of money is based on the premise that person prefers to receive a certain amount of money today, rather than the same amount in the future, all else equal. As a result, he demands interest when depositing money in a bank account or making any similar investment. Money received today is more valuable than money received in the future by the amount of interest the money can earn. If $90 today will accumulate to $100 a year from now, then the present value of $100 to be received one year from now is $90.

TVM also takes into account risk aversion - both default risk and inflation risk. 100 monetary units today is a sure thing and can be enjoyed now. In 5 years that money could be worthless or not returned to the investor. There is a residual time value of money, beyond compensation for default and inflation risk, that represents simply the preference for consumption now versus later. Inflation-indexed bonds notably carry no inflation risk. In the United States for instance, Treasury Inflation-Protected Securities carry neither inflation nor default risk, but pay interest.

Three formulas are used to adjust for this time value:

1. The present value formula is used to discount future money streams: that is, to convert future amounts to their equivalent present day amounts.
2. The future value formula is used to compound today's money into the equivalent amount at some time in the future (i.e., to compound money...either a lump sum or streams of payments).
3. The present value of an annuity formula is used to discount a series of periodic payments of equal amounts to the present day. Variations of this formula can find the future value of the annuity, or solve for the annuity given the present value (for example, finding monthly mortgage payments) or find the annuity given the future value (for example finding a monthly payment needed to reach a retirement savings goal).

Time value of money: conversion factors

Present value, future value

The following factors can convert between present value P and future value F:

$\left(F/P\right)\ =\ (1+r)^{n}$

$\left(P/F\right)\ =\ {1 \over (1+r)^{n}}$

where r is the required rate of return per time period and n is the number of time periods.

Future value, annuity amount

The following factors can convert between future value F and annuity amount A:

$\left(F/A\right)\ =\ {(1+r)^{n}-1 \over r}$

$\left(A/F\right)\ =\ {r \over (1+r)^{n}-1}$

where r is the required rate of return per time period and n is the number of time periods.

Present value, annuity amount

The following factors can convert between present value P and annuity amount A:

$\left(P/A\right)\ =\ {(1+r)^{n}-1 \over r(1+r)^{n}}$

$\left(A/P\right)\ =\ {r(1+r)^{n} \over (1+r)^{n}-1}$

where r is the required rate of return per time period and n is the number of time periods.

Examples

Example #1: Future value

Main article: Future value

Example #2: Present value

Main article: Present value

One hundred euros to be paid 1 year from now, where the expected rate of return is 5% per year, is worth in today's money:

$P\ =\ F\times (P/F)\ =F\times \ {1 \over (1+r)^{n}}\ =\ {\frac {\ 100}{1.05}}\ =\ 95.23$

So the present value of €100 one year from now at 5% is €95.23.

Example #3: Annuity amount

Consider a 30 year mortgage where the principal amount P is $200,000 and the annual interest rate is 6%.

The number of monthly payments is

$n=30{\rm {\ years}}\times 12{\rm {\ months\ per\ year}}=360{\rm {\ months}}$

and the monthly interest rate is

$r={6{\rm {\%\ per\ year}} \over 12{\rm {\ months\ per\ year}}}=0.5{\rm {\%\ per\ month}}$

The annuity formula for (A/P) calculates the monthly payment:

$A\ =\ P\times \left(A/P\right)\ =\ P\times {r(1+r)^{n} \over (1+r)^{n}-1}\ =\ \$200,000\times {0.005(1.005)^{360} \over (1.005)^{360}-1}$

$=\ \$200,000\times 0.006\ =\ \$1,200{\rm {\ per\ month}}$

See also

• Discounting
• Discounted cash flow
• Exponential growth
• Internal rate of return
• Perpetuity
• Real versus nominal value
• Time preference theory of interest

External links

• TeachMeFinance
• Time Value of Money from studyfinance.com at the University of Arizona
• Time Value of Money I from DiskLectures.com.
• Time Value of Money Concepts from David R. Frick & Co.
• Time Value of Money, with Examples — Explains concisely the present value and future value of money, which is used to compare investments; includes formulas and examples.
• Annuities: Ordinary? Due? What Do I Do? -- Annuity tutorial (with quiz) from Prof. John Wachowicz at the University of Tennessee.
• Pamphlet explaining concepts with examples - by Charles Welty