Compound Interest, Present Value, Depreciation – Formulas and Examples, Part Two

Question

Introduction:

In finance and economics, present value and depreciation are key concepts that help in evaluating investments, assets, and future cash flows. Present value helps us determine the value of money today versus its worth in the future, while depreciation accounts for the loss in value of assets over time due to factors like wear and tear.

Present Value

The present value of a sum is the amount of money that, when invested at a certain interest rate for a specified period, will grow to equal a future value. In other words, the present value is what a future sum is worth today, considering the time value of money.

For example,
if you have $100 today and invest it at an interest rate of 5% per annum, after 2 years, the amount will grow to $110.25. In this case, we say that $100 is the present value of $110.25, which will be received in 2 years.

If P is the present value of an investment and it is invested at an interest rate r per period, then the future amount A after n periods is given by the formula:

A=P 1+ r 100 n

Here:
- A = Future amount
- P = Present value (initial investment)
- r = Rate of interest per period (expressed as a percentage)
- n = Number of periods

Depreciation

Depreciation refers to the reduction in the value of an asset over time, typically due to wear and tear or obsolescence. Assets such as machinery, vehicles, and electronic items decrease in value each year as they are used. This reduction is quantified by the depreciation rate.

If the value of an asset is initially P and it depreciates at a rate r% per year, then the value of the asset after n years is given by the formula:

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V=P 1&minus; r 100 n

Where:
- V = Value of the asset after n years
- P = Initial value of the asset
- r = Depreciation rate (as a percentage)
- n = Number of years

Example 1: Present Value of a Depreciated Asset

Ben purchased a car 4 years ago for $20,000. The value of the car depreciates by 25% per annum. What is the current (present) value of the car?

Using the depreciation formula,  
- P = $20,000 (initial value)  
- r = 25% (depreciation rate)  
- n = 4 years

V=200001&minus;251004=20000&times;0.3164=6,328

So, the present value of the car after 4 years is $6,328.

Population Growth or Decline Formula

The principles of compound interest and depreciation can also be applied to population growth or decline. If the population of a town is P and it increases by r% annually, then the population after n years is given by the formula:

P n =P 1+ r 100 n

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If the population decreases by r% annually, the formula becomes:

P n =P 1&minus; r 100 n

Where:
- Pn = Population after n years
- P = Initial population
- r = Growth or decrease rate (percentage)
- n = Number of years

Example 2: Population Growth

A town has a current population of 50,000 people, and the population is growing at a rate of 3% per year. What will be the population after 5 years?

Using the population growth formula:  
- P = 50,000  
- r = 3%  
- n = 5 years

Pn=500001+31005=50000&times;1.1593=57,965

So, the population after 5 years will be 57,965.

Calculation of Present Population

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To calculate the present population of a town with an annual growth rate of 12%, we use the formula for population growth based on compound interest. The formula is:

$P_n = P \times \left(1 + \frac{r}{100}\right)^n$

Where:
- Pn = Population after n years
- P = Initial population
- r = Growth rate per year (as a percentage)
- n = Number of years

Given Data:
- 3 years ago, the population was 1 million (i.e., P = 1,000,000)
- Growth rate r = 12%
- Time n = 3 years

Substitute the values into the formula:

Pn=1,000,000&times;1+121003=1,000,000+1.123

P n =1,000,000&times;1.404928 =1,404,928

So, the present population is approximately 1,404,928.

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Summary

- The present population of the town is 1,404,928 after 3 years of 12% annual growth.
- Different interest rates for different years require the multiplication of individual yearly growth factors.
- For fractional years, split the period into full and partial years.
- The difference between compound and simple interest can be calculated using a simple formula for small periods like 2 or 3 years.

Financial planning, investment analysis, and economic projection will not be possible without a proper understanding of the concept of present value, depreciation, and population growth or decline.

The present value will assist in measuring the value of future sums in their present value; depreciation allows us to trace assets that, over some time, lose their value.

Using the same logic applied to calculate population, one can portray the future population showing growth or decline. These formulas are derived to make an individual or a business investor better informed on various investment, asset management, and resource allocation decisions.

Part 1

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