Time value of money

From Free net encyclopedia

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 one will prefer to receive a certain amount of money today than the same amount in the future, all else equal. As a result, when one deposits money in a bank account, one demands (and earns) interest. 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 money 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 annuity formula is used to discount a series of periodic payments of equal amounts at equal time intervals. Examples include monthly mortgage payments or semi-annual coupon bonds.

Contents

[edit]

Time value of money: conversion factors

[edit]

Present value, future value

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

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

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

[edit]

Future value, annuity amount

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

<math> \left( F / c \right) \ = \ { (1+r)^n - 1 \over r } </math>
<math> \left( A / F \right) \ = \ { r \over (1+r)^n - 1 } </math>


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

[edit]

Present value, annuity amount

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

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

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

[edit]

Examples

[edit]

Example #1: Future value

Template:Main One hundred dollars invested today at an interest rate of 5% per year will be worth:

<math> F \ = \ P \times (F/P) \ = \ P \times (1 + r )^n \ = \ \$100 \times {(1+0.05)^1}=\ \$105</math>

after one year. So, the future value of $100 in one year at 5% per year is $105.

[edit]

Example #2: Present value

Template:Main 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:

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

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

[edit]

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

<math> n = 30 {\rm \ years} \times 12 {\rm \ months \ per \ year} = 360 {\rm \ months}</math>

and the monthly interest rate is

<math> r = { 6 {\rm \% \ per \ year} \over 12 {\rm \ months \ per \ year} } = 0.5 {\rm \% \ per \ month} </math>


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

<math> 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 } </math>

<math> = \ \$200,000 \times 0.006 \ = \ \$1,200 {\rm \ per \ month} </math>
[edit]

See also

[edit]

External links

fr:Valeur temps de l'argent ru:Временная ценность денег zh:资金的时间价值

Retrieved from "http://www.netipedia.com/index.php/Time_value_of_money"