How to Think About Time Value of Money Problems
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One of the biggest obstacles to correctly solving time value of money problems is identifying the cash flows and their timing. On this page I will offer some tips that I hope will be helpful.
There are Always Five Variables
Every time value of money problem has five variables: Present value (PV), future value (FV), number of periods (N), interest rate (i), and a payment amount (PMT). In many cases, one of these variables will be equal to zero, so the problem will effectively have only four variables. You will always know the values of all but one of these, and it is that missing value for which you will be solving.
It may seem that problems involving uneven cash flow streams don't fit the description above, but they do. Thanks to the principle of value additivity, we can think of uneven cash flow streams as a series of lump sum cash flows. If necessary, you can always deal with each cash flow separately, and then add up your results in the end.
Identify The Variables
The most important thing is to be able to identify the variables that you have been given, and the one that you are looking for. What makes this difficult, sometimes, is that you are given a problem (sometimes a long one) and you have to identify the variables amidst a bunch of words. Furthermore, the words that are used and the order in which the variables are given is never the same. That leads us to the first tip:
|Tip:||When reading through a time value of money problem you should always stop when you come to a number. Write down (and label) that number to the side of the problem. That way, you will have separated the values from the text.|
As noted above, there are up to five variables in every problem. Here are some general ideas about how to identify them:
- Present Value
- Any value that occurs at the beginning of the problem (or the beginning of a part of the problem) is a present value. The key is that the present value occurs before any other cash flows. Usually, when a present value is given, it will be surrounded by words indicating that an investment happens today.
- Future Value
- The future value is usually the last cash flow. Obviously, it is a cash flow that occurs at some time period in the future. The future value is a single cash flow. If it occurs more than once, then it is probably an annuity payment.
- Annuity Payment
- An annuity payment is a series of two or more equal payments that occur at regular time periods. Each payment, if taken alone, is a future value, but the key point is that the annuity payment is a recurring payment. That is, there are more than one of them in a row.
- Interest Rate
- The interest rate is the growth rate of your money over the life of the investment. It is usually the only percentage value that is given. However, some problems will have different interest rates for different time frames. For example, problems involving retirement planning will often give pre-retirement and post-retirement interest rates. Frequently, when you are being asked to solve for the interest rate, you will be asked to find the compound average annual growth rate (CAGR).
- Number of Periods
- The number of periods is the total length of time that the investment will be held. Typically, it is given as a number of years, though it will often need to be adjusted to some other time scale. For example, if you are told that the investment pays interest quarterly (4 times per year) then you must adjust N so that it reflects the total number of quarterly (not annual) time periods.
|Tip:||The interest rate, number of periods, and annuity payment variables must all agree on the length of a time period (a day, a week, a month, a year, etc). That is, i is always the interest rate per period, N is always the total number of periods, and PMT is always the amount of the payment per period. Very often, it is necessary to make adjustments to the values given in a problem. For example, interest rates are usually given as annual rates. However, if payments occur monthly, then the interest rate must be adjusted to a monthly rate (typically by dividing the annual rate by 12). Similarly, the number of years would have to be changed to the number of months.|
Draw a Time Line
Even after you have successfully identified the cash flows, it can be difficult to track the timing of the cash flows in your head. This is where time lines are so important. A time line is a graphical representation of the size and timing of the cash flows.
When you are first learning to solve time value problems, drawing time lines is a very good idea. In the picture above, you can easily see that the problem consists of a five-year $100 annuity (PMT), and a $1,000 cash flow (FV) that occurs at the end of the investment. The time line helps you to see exactly when each cash flow occurs, and therefore how many periods it needs to be moved (either forward or backwards in time). As the problems that you are solving become more complex, the importance of drawing time lines increases.
Break the Problem into Smaller Pieces
Very often time value problems are pretty straightforward. It may involve only a single lump sum cash flow, or a simple annuity. For more complicated problems, it can be very helpful to break the problem into several pieces and solve them separately. In the end, just add up the answers from each piece of the problem (this is known as the Principle of Value Additivity). Keep in mind that we almost always want to know the answer as of some point in time, so all of the cash flows need to be moved to that time period before they can be added together.
The time line shown above is a good example of a problem that can be solved in two (or six, if you want) pieces. To find the present value of that stream of cash flows, we would find the present value of the five-year $100 annuity first. Then, find the present value of the $1,000 lump sum. The final step would be to add the two present values to get the present value of the entire stream of cash flows.
Sometimes you have no choice but to break the problem into pieces. For example, when solving problems relating to future retirement income needs. Very often this type of problem involves two time periods (before retirement and after retirement), and perhaps also more than one interest rate (usually lower during retirement). In this situation, you need to treat it as two problems. First solve the "after retirement" problem, and then solve the "before retirement" problem using the results from the first part.
Try to Estimate the Answer
One final important tip is to always try to estimate what your final answer should be. Your estimate doesn't have to be precise. You just want to be able to identify obviously incorrect answers. This is especially true when using financial calculators or spreadsheets. Always remember the old saying, "garbage in, garbage out." If you enter incorrect numbers, or you get the timing of the cash flows wrong, you will always get the wrong answer.
There are relationships between the variables that you should understand, and that can help you when estimating the answers. Here is an analogy that may help:
Think of the problem as a road trip. The present value is your starting point, and the future value is your destination. The number of periods is the distance to be traveled, and the interest rate is the average speed that you will be traveling.
Using that analogy, however imperfect it may be, we can identify several important relationships between the variables:
- The future value is always bigger than the present value.
- From any given present value (starting point), the longer you drive (N) or the faster you go (i), the bigger the future value will be.
- If you slow down (use a lower interest rate), it will take longer (larger N) to get from the present value to the future value. If you speed up (higher interest rate), you will get there faster (lower N).
- If you drive for less time (lower N), you will have to go faster (higher i) to reach the same destination (FV).
And so on. There are endless variations, and annuity payments add to them. The point is that you need to understand how the variables interrelate if you want to be able to estimate answers. Ultimately, the best way to get good at solving problems is to solve a lot of them, so practice is very important.
Please continue on to the mathematics of lump sums.