# Equivalence between the Present and Future value of a single amount of money

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## Lecture Note 8

This lecture note was originally written by a student and partly modified and certified by the course instructor.

## Time Value of Money

Time value of money expresses the purchasing power of money at a given period. For example, suppose you can buy one ball of Ga kenkey this year for GH¢1.00. Do you think the same ball of Ga kenkey would go for GH¢1.00 next year?

In another example, imagine a neighbor borrows your block making machine for her construction business. Do you think it’s fair for her to return the machine to you next year without any

compensation because she is a friend? Three major economic factors should be considered here which is all related to ‘time value of money’.

1. Opportunity cost: You missed the chance of making money from the machine within the year. In other words, you could have earned some interest had you sold your machine and put the money in a savings account.

2. Depreciation: The machine would be older next year and probably less productive compared to this year.

3. Inflation: The price of the machine might be higher next year than it is this year.

If one considers all three factors above, then it is not fair for the neighbor to return the machine next year without any compensation.

This course will consider only factor 1 and will primarily consider cases involving an individual/entity and the bank.

Equivalence between the Present and Future value of a single amount of money

Because of time value of money, the present value of a future amount of money, and vice versa is of great importance in business. For example, if you invested Ghc 1000 in an asset that offers an interest rate of 20% per year, you will be interested to know (today) what you will receive next year. Today, the Ghc 1000 is a Present value, and what you will receive next year is a Future value. On the other hand, if you are promised a certain amount two years to come (i.e. future value), you would be wondering the worth of that money today (i.e. present value). Perhaps you would want to borrow some money today so that you could pay it off with what you’ve been promised at the end of year 2.

Let and be the present (current time period) and future (i.e. time period ) value of a single amount assuming interest rate will remain at within the period considered. Then, and are related by the formula: ( )

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present and later consider time period 3 as the future to be taken to time zero. This will become clear later.

Illustrations: Example 1

Find the present value of the cash flow shown above when = 5%.

( )

( )

For the future amount Ghc 3000 at period 2, it has to move back two time periods to the present (period zero). Thus, and therefore

( )

However, for the future amount Ghc 2000 at period 1, and therefore ( )

The total present value of the cash flow above is: ( )

( ) Example 2

Find the present value of the cash flow shown below if interest rate ( ) is 10% per year.

Year Cash

1 2000

2 5000

3 0

4 -2000

5 3000

Solution

( )

( ) ( )

( )

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## Types of Cash Flows

### Uniform Cash Flows

Amount of cash inflow or outflow is the same each time period.

Let be the present value of the uniform cash flows above. Then:

( ) ( ) ( ) ( ) ( ) ( ) [

( ) ( ) ( ) ( ) ( ) ( ) ]

When is large (say ), it will be too tedious to compute . In what follows, we seek a shortcut to find such that unlike equation (1) above which requires computations, only 1 computation would be needed to find .

To do this, first form Equation (2) by dividing Equation (1) by ( )

( ) [( ) ( ) ( ) ( ) ( ) ( ) ]

Next, subtract equation (2) from equation (1): ( ) ( ) By cancelling out like terms, we have:

( ) [( ) ( ) ]

Note that the emphasis is on finding . Therefore, we rearrange to make the subject.

( )

( ) ( )[ ( ) ]

[

( ) ] [ ( ) ]

[( )

( ) ]

Thus, the present value of a continuous uniform cash flow that begins at the end of period 1 to end of period is as shown in equation (3).

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Note also that since ( ) , we can find the future value of all uniform cash flows by simply replacing with that from equation (3). Then:

[( )

( ) ] ( )

[( ) ]

Example 1: In this example, we test the power of the shortcut. Find the net present value of the cash flow shown below with =5%.

Alternative 1: Brute-force approach

( )

( )

( )

( )

( )

( )

Alternative 2: Using the shortcut.

*( ) ( ) +

It is clear that when is large, the shortcut is more convenient.

Example 2

The government of Ghana is to receive an amount of GH¢10,000 for the next 100 years. If

interest rate is assumed to remain at 10% per year throughout the 100 years, find the net present value of all the cash flows.

*( )

( ) +

Example 3

Find the net present value of the cash flow shown above when =5%.

First, we can find the present value of the uniform cash flows in period 2.

[

( )

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then becomes the future value at period 2 whose present is sought at period 0. Thus ( )

Example 4

You borrowed GH¢10,000 from EcoBank at an interest rate of 10% to be repaid in 5 years. What is the equal amount to be saved each year in order to repay the loan in full at the end of 5 years?

*( ) ( ) + ( )

( )

( ) ( )

Example 5

Find the net present value of the cash flow shown above. Assume = 5%.

*

( )

( ) +

*

( )

( ) +

( )

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Let and respectively be the present value of the uniform and gradient part. Let also be the total present value of the arithmetic cash flow. Then:

We want a shortcut to compute .

can be found using equation (3). Therefore, our focus is on .

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

Form equation 2 by doing: ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Do ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( ) ( ) ( ) ] ( )

The first part of the right-hand-side is similar to the equation that led to the shortcut of equation (3) when is replaced with . Therefore, the equation above reduces to:

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Note:

1. In general, . Where the sign changes to ( ) when the gradient part is decreasing. Also, once we have the present value, the future value can be obtained through ( )

2. occurs two period to the first gradient series.

Example 1

Find the present worth of a cash flow that begins with GH¢3000 in the first period and increases by GH¢500 per period for 10 years with interest of 5%.

[( ) ( ) ]

*

( )

( ) +

( ) [( ) ]

( ) [( )

( ) ]

Example 2

The income from Precious Metal Mining Corporation has been decreasing uniformly for four years.

If income in year 1 was GH¢300,000 and it decreased by GH¢30,000 per year to year 4, what is the present worth of income stream at 5% per year?

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[

( )

( ) ]

( ) [( ) ( ) ]

This is similar to that of the arithmetic gradient cash flow except that payment at period is always increased by of the payment at period . That is:

( )[ ]

The total present value of such cash flows (uniform plus gradient part) is given as:

[ (

)

]

*

+

Where; = growth rate (rate of increase) = interest rate

Example

The academic fees of the University of Ghana (UG) is set to increase by 10% per year for the next 4 years. You are a first year student and the fee is GH¢1,000 this year. Your guardian would be travelling for a while and would like to deposit an amount today in a fund that would be sufficient to pay the total of your tuition for the four years. How much should your guardian deposit today if the fund pays interest rate of: a. 10% and b. 5%

Solution

a.

[

]

b.

[ (

)

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### Uniform Cash Flow in Perpetuity

When is forever (e.g. and endowment fund), equation (3) which is the present value of all future cash flows reduces to

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