Time Value of Money – Present Value of Lump Sums

On the previous page, you learned everything that you will ever need to know in order to solve time value of money problems! That’s quite a bold statement, but it is true. The key is that we derived the basic time value of money formula:

$$F{V_N}=PV(1+i)^N$$

From that formula, we can easily derive other formulas for solving for any of the variables in a time value problem. So, if you learn nothing else from these pages, that formula will serve you well. Of course, sometimes a little more knowledge is helpful. Giving you that knowledge is the purpose of these other pages.

Calculating Present Value

The present value is the current value of a stream of cash flows. Thus the name: Present value means “what is it worth right now?” To solve for the present value, we merely rearrange our basic time value formula:

$$PV=\frac{F{V_N}}{(1+i)^N}$$

So, to find the present value we take the future value and divide it by $(1+i)^N$ where $N$ is the period where the future value is located. Note that since we are dividing the future value, the present value will always be less than the future value.

Let’s look at an example:

Suppose that you want to purchase a house in two years. You expect that you will need to have $20,000 at that time to use as a down payment. If you can purchase a two-year certificate of deposit that pays 5% per year, how much will you need to invest today in order to meet your goal?

In this case we know that the future value is \$20,000. Furthermore, the problem has given us the time frame (2 years) and the interest rate (5% per year). Therefore, we can find the present value with:

$$PV = \frac{\text{20,000}}{(1.05)^2 }= \text{18,140.59}$$

So, in order to have \$20,000 in two years while earning 5% per year, you must invest \$18,140.59 today.

How PV is Related to FV, N, and i

That is quite a lot of money, so you might wonder if you can somehow get away with saving less money and still meet your goal. There are several potential ways that can be deduced by studying the present value formula. Let’s look at the relationships between the variables:

• PV is positively related to FV — This means that to achieve a higher future value you must invest more today, all other things being equal. Similarly, if FV is lower, then so will be the PV.
• PV is inversely related to the interest rate — Higher interest rates mean that your money grows more quickly. Therefore, if you can earn a higher interest rate, you can invest less today (lower PV) to reach a particular FV. Similarly, the lower the interest rate, the more you will have to invest (higher PV) to meet your goal.
• PV is inversely related to the number of periods — The longer your time frame, the less you need to invest today. Alternatively, shorter time periods require larger initial investments.

Therefore, you can reduce the investment needed today by reducing your goal (FV), by increasing the holding period, or by increasing the interest rate.

So, what if you only need to save \$15,000 instead of \$20,000? Obviously, all other things being equal, the present value should be lower. Let’s see:

$$PV = \frac{\text{15,000}}{(1.05)^2 }= \text{13,605.44}$$

There you have it: lower the FV and you also lower the PV. However, take note of the fact that we reduced the FV by \$5,000, but the PV declined by only \$4,535.15. The relationship between PV and FV is nonlinear because of the way that compounding works.

Alright, what if you can’t reduce the FV? Well, you could raise the interest rate. Be careful here, though, because in order to earn higher interest rates you have to take more risk. In any case, suppose that you have another investment alternative that will pay you 8% per year for the next two years. How much do you need to save now?

$$PV = \frac{\text{20,000}}{(1.08)^2 }= \text{17,146.78}$$

So, as was noted above, earning a higher interest rate does lower the PV.

Finally, you could also delay the purchase of the house. If you could wait four years instead of two, then you would only have to save:

$$PV = \frac{\text{20,000}}{(1.05)^4 }= \text{16,454.05}$$

We have looked at each of these variables in isolation, but they can also work together. So, if you could lower the FV, raise the interest rate, and increase the time horizon then you could significantly reduce the PV.

Dealing with Non-Annual Periods and Rates

In the above examples the interest rates were annual and the number of periods were measured in years. Very often, however, that is not the case. Fortunately, non-annual interest rates and periods are easy to deal with. Just remember this rule: Interest rates and the number of periods must always agree as to the length of a time period.

So, if you are told that you will be earning interest that is compounded monthly, then your number of periods must be measured in months. Similarly, if you are told that you will hold an investment for some number of weeks, then your interest rate must be a weekly rate. This cannot be emphasized enough, and it applies whether you are using formulas (as we are here), financial calculators, or spreadsheets.

How much would you need to invest today if you wish to have $1,500 in three years if the interest rate is 7% per year compounded monthly?

In this problem we want to solve for the PV, and we know that the FV is \$1,500. However, $N$ is not 3 years and $i$ is not 7%. Instead, since we are told that interest is compounded on a monthly basis, we must convert both $N$ and $i$ to monthly values. We know that there are 12 months in a year, so $N = 36 \text{ months}$ ($=3 \times 12$), and $i = 0.58333\% \text{ per month}$ ($=0.07/12$).

Now, using the converted values, we can properly solve the problem:

$$PV = \frac{\text{1,500}}{(1.0058333)^{36}}= \text{1,216.62}$$

Note that if we had ignored the monthly compounding and just used the numbers as given, then we would have gotten the wrong answer:

$$PV = \frac{\text{1,500}}{(1.07)^3}= \text{1,224.45}$$

The wrong answer is nearly $8 too high, and it would be off by even more if the interest rate was higher or the number of periods was greater.

Please continue to the next page to learn about solving for the interest rate and/or the number of periods.