###### Introduction to Statistics With Releva ...

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Time value of money is a concept that refers to the greater benefit of receiving a given amount of money at present rather than in the future due to its earning potential. For example, money could be invested in a bank account and earn interest even overnight. Interest earned will depend on the rate of return offered by government bonds (risk-free assets), inflation, liquidity risk, default risk, time to maturity, and other factors.

In a nutshell, time value calculations allow people to establish the future value of a given amount of money at present. The present value (PV) is the money you have today. The future value (FV) is the accumulated amount of money you get after investing the original sum at a certain interest rate and for a given time period, say 2 years. The concept has a wide range of applications that incorporate financial matters-bonds, shares, loan facilities, among others.

**Fundamental Formulas in Time Value of Money Calculations**

Let,

FV = future value

PV = present value

r = interest rate

n = number of investment periods per year

t = number of years

Note: besides annual interest payments, interest could be compounded monthly, quarterly or semi-annually. If for instance, interest is payable quarterly, then we have 12/3 i.e 4 investment periods per year.

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

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

QuestionSuppose an individual invests $10,000 in a bank account that pays interest at a rate of 10% compounded annually. What will be the future value after 2 years?

A. $12,000

B. $12,100

C. $22,000

SolutionThe correct answer is B.

PV=10,000, r=0.1, n=1, t=2

$$ \begin{align*}

FV & =10,000{ \left( 1+\frac { 0.1 }{ 1 } \right) }^{ 1*2 } \\

& =10,000(1.1)^2 \\

& =12,100 \end{align*} $$To confirm our answer, we could work out the PV of a future value of 12,100 invested under similar terms, starting with the FV of $12100.

$$ \begin{align*}

PV & =12,100 \left(1+0.1/1 \right)^{-(1*2)} \\

& =12,100(1.1)^-2 \\

& =$10,000 \\

\end{align*} $$

**Points to Note**

First, establish all the components of the relevant formula before commencing actual calculation. Secondly, only the term within the brackets is subject to the power function.

The concept of the time value of money serves as the foundation for more concrete financial calculations such as simple interest, compound interest, and the value of stocks or bonds at any given point in time.

*Reading 6 LOS 6a:*

*Interpret interest rates as required rates of return, discount rates, or opportunity costs*