# Mathematics of Lump Sums

Post on: 29 Сентябрь, 2016 No Comment

Future Value of Lump Sums

The most basic type of cash flow is a lump sum. That is, a single cash flow that occurs at a single point in time. Despite its simplicity, the lump sum cash flow is the bedrock upon which all other types of cash flows are built. According to the principal of value additivity. every type of cash flow stream can be decomposed into a series of lump sums. Therefore, it is vital to understand the mathematics of lump sums if you are to understand more complex types of cash flows.

## Calculating the Future Value

If you were to invest a certain dollar amount today, it would grow to a larger (hopefully!) value at some point in the future. The value at the future point in time is called the future value. In order to calculate the future value we must know the rate at which the investment will grow (called the interest rate or discount rate ) and the length of time that the investment will be held (number of periods ).

*Suppose that you invest $100 today at an interest rate of 8% per year and expect to hold the investment for one year. How much will the investment be worth at the end of this period? In other words, what is the future value?*

In this problem, the $100 that is invested today is known as the present value. and your investment will grow at the rate of 8% per year for a period of one year. So, the future value is equal to the present value plus the interest earned over the course of the year. In other words:

100 + 100 * 0.08 = 108

So, the future value will be $108 at the end of the year. We can express the above equation algebraically as:

FV_{1} = PV + PV * i

where FV_{1} is the future value at the end of period 1, and i is the interest rate per period. Note that we can simplify that formula by factoring out the PV so that it becomes:

FV_{1} = PV(1 + i)

Or, in numbers:

108 = 100(1.08)

Now, suppose that at the end of period 1 you decide to hold the investment for another year. At that point, your principal (present value) will be $108 and it will grow for another year at 8%, so after two years you will have:

108 + 108 * 0.08 = 116.64

Take note of the fact that in year 2 we are earning interest on our original investment (the $100) and also on the interest that was earned in the first year (the $8). So, in year 2 and all subsequent years, we will be earning interest on top of previously earned interest. This is known as compounding. and is a very powerful idea as we will see.

Continuing our example, mathematically we can say:

or, if we substitute PV(1 + i) for FV_{1}.

FV_{2} = PV(1 + i)(1 + i)

We can simplify the formula even further:

FV_{2} = PV(1 + i) 2

In numbers, this is:

FV_{2} = 100(1 + 0.08) 2 = 116.64

which is exactly what we found above. So, $100 invested at 8% per year for 2 years will grow to $116.64. Notice that in the second year, you earned $8.64 in interest as compared to only $8 in the first year. Where did the extra $0.64 come from? That is the interest that you earned on the $8 in interest from year 1 (8 * 0.08 = 0.64). In other words, the extra interest came from the process of compounding.

Have you noticed the beginning of a pattern in the formula? Let’s extend the investment for one more year:

FV_{3} = 100(1 + 0.08) 2 (1 + 0.08) = 100(1 + 0.08) 3 = 125.97

Notice that the future value after three years is equal to:

FV_{3} = PV(1 + i) 3

So, the pattern should now be clear. To calculate the future value of a lump sum we simply take the present value times 1 plus the interest rate raised to a power that is equal to the number of periods. Mathematically:

FV_{N} = PV(1 + i) N

Now, let’s see what the future value would be if we held the investment for 10 years:

FV_{10} = 100(1 + 0.08) 10 = 215.89

## Simple vs. Compound Interest

You can see the power of compound interest in that last example. If we were dealing with simple interest (i.e. you do not earn interest on top of interest) then you would have earned only $80 in interest over the 10-year period. However, with compound interest, you have actually earned $115.89 in interest. So, compounding added an extra $35.89 over 10 years. It gets even better over longer time frames and/or with higher interest rates.

To see this even better take a look at the following chart, which shows the difference between compound and simple interest over long periods. Notice how the future value grows exponentially with compound interest. The dotted line shows the difference in future values over 30 years.

You can download an Excel spreadsheet that contains the above chart so that you can experiment with the numbers.

## The Rule of 72

The Rule of 72 is an often useful tool that can be used to approximate how long it will take to double your money at a particular interest rate:

Years to double money = 72 interest rate

So, using the rule we can see that at 8% it will take about 9 years to double your money:

Years to double money = 72 8% = 9 years

Alternatively, the rule can tell us what interest rate is needed to double your money in a particular number of years:

Interest Rate to Double Money in N Years = 72 N

So, to double your money every 15 years you would need to earn about 4.80% per year:

Interest Rate to Double Money in 15 Years = 72 15 = 4.80% per year

This rule can often be used to mentally check answers or to quickly see if a statement makes sense.

Please continue to the next page to learn about calculating the present value of a lump sum .