Time value of money

"Time is money."

It is well clear for everyone that $100 received today is worth more than the same amount received in one year. The main reason is that, at the very least, an amount received today can be placed in a savings account and earn interest. Furthermore, anyone who has to wait to receive an amount of money in the future should be compensated for that delayed gratification. Two other reasons can explain the Time Value of Money (see Capinsky and Zastawniak (2003)): firstly, due to inflation, the longer we wait, the less a given amount of money will perhaps be worth (due to loss of purchasing power). Secondly, there is always a possibility (risk) that we might never receive the promised amount of money.

Definitions

The time value of money (TVM) is a necessary concept of finance that allows us to equate

1. (PV) the value of one monetary unit (MU) received now or the present value of one MU at time t discounted back to today.
2. (FV): the future value of an investment of one MU plus all interest earned since the deposit.
3. (PVPA) the present value of a perpetual annuity: the present value of a stream of fixed periodic payments continuing until infinity (see perpetuity).
4. (PVA) the present value of a finite stream of periodic payments of one MU: i.e. how much mortgage you can afford.
5. (FVA) the future value of a finite stream of payments (annuity) in between : the value of your portfolio after a lifetime of saving.

The premise is that you prefer to receive money today, rather than the same amount in the future, all else being equal. As a result, you demand interest, paid either along the way or at the end. The interest compensates you for several things, including the time in which the money could be put to some other productive use (opportunity cost), the risk of not actually getting the money in the future (default), and the risk of loss of purchasing power (inflation) .

Equations

There are four basic equations that represent the four equalities listed above. The variables can be input into a financial calculator (and solved for any unknown). Or the variables can be input at any online calculator.

• The present value (PV) formula has four variables, each of which can be solved for:
  1. PV the value of one MU a time = 0
  2. FV the value of one MU at time = n in the future
  3. r equals the discount rate that would be compounded for each period of time
  4. n is number of time periods over which cash flow are compounded.

     $\left(PV\right)\ =\ {FV \over (1+r)^{n}}$

• The future value (FV) formula is the same and uses the same variable.

$\left(FV\right)\ =\ PV(1+r)^{n}$

• The PV of a perpetual annuity formula is simple division.

$\left(PVPA\right)\ =\ {A \over r}$

• The present value of an annuity (PVA) formula has four variables, each of which can be solved for:
  1. PVA the value of the annuity at time=0
  2. A the value of the the individual payments in each compounding period
  3. r equals the interest rate that would be compounded for each period of time
  4. n is the number of payment periods.

     $PVA\,=\,A\cdot {\frac {1-{\frac {1}{\left(1+r\right)^{n}}}}{r}}$

• The future value of an annuity (FVA) formula has four variables, each of which can be solved for:
  1. FVA the value of the annuity at time=n
  2. A the value of the the individual payments in each compounding period
  3. r equals the interest rate that would be compounded for each period of time
  4. n is the number of payment periods.

     $FVA\,=\,A\cdot {\frac {\left(1+r\right)^{n}-1}{r}}$

For both calculations of annuities, you must decide whether the payments are made at the beginning of each time period, or (as in the formulas above) at the end. The calculator you use will allow the input somehow. You must also be clear that the time period used between payments is the same time period for the interest rate. e.g. The calculation for a mortgage paid monthly uses the monthly interest rate.

Examples

#1: Present value

One hundred MU 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 MU one year from now at 5% is 95.23 MU.

#2: Present value of an Annuity - solving for the payment 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:

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

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

References

Capinski M. and Zastawniak T. (2003), "Mathematics for Finance-An Introduction to Financial Engineering", Springer-Verlag

See also

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

External links

• Time Value of Money from studyfinance.com at the University of Arizona
• Damodaran's note on Time Value of Money