Introduction
Time Value of Money is an important concept in the financial and business world. The concept can be a challenge to some students. Through this interactive tutorial, you will learn
To make things easier, we are going to walk through the use of these concepts using a real world example.
Jack’s Story
(Jack’s voice) Hi, my name is Jack Wilson. I am 55 years old and am planning to retire in 5 years. My lifelong dream is to have my own fishing boat to use in my retirement. I know exactly the kind of boat I want and I have the money for it --- I recently come into an inheritance of $400,000---- but I don’t want to buy it right now mainly because I don’t have the time for fishing. Five years from now when I retire and plan to buy the boat, it will cost me $350,000. I want to put aside the money for my boat in this Sunrise fund, which will earn an annual compound interest rate of 12%. I need to figure out how much of my inheritance must I invest in the Sunrise Fund today so I can buy my dream boat at retirement. Please help me. Thanks!
Concept of Time Value of Money
Before we go on, let’s discuss some basic information about the time value of money. Why does Jack want to invest some money in the Sunrise fund? Why doesn’t he simply lock $350,000 in a closet now and use it 5 years later to buy his $350,000 boat?
The simple answer is that his investment in the Sunrise fund will earn him interest!
A dollar you invest today is worth more than a dollar promised at some time in the future. When Jack invests, for example, $350,000 today in the Sunrise fund and leaves it there for five years, his $350,000 will earn him 12% interest for 5 years. At the end of the five years, Jack will have his principal, $350,000, which he can spend for his boat along with the additional earned interest. This describes the concept of Time Value of Money.
To summarize, Time Value of Money indicates a relationship between time and money --- a dollar received today is worth more than a dollar promised at some time in the future.
Simple Interest
Jack said that the Sunrise fund will earn him an annual compound interest of 12%. What does “compound interest” mean and how much interest will he earn?
Well, let’s study Compound Interest by comparing it to “simple interest.”
Simple Interest is computed on the amount of principal and principal only. If Jack invests $350,000 for 5 years at a simple annual interest of 12%, he would earn an interest of $210,000. This is calculated using the formula:
Principal * Rate (Interest Rate) * Time (number of periods) $350,000 * 12% * 5
Wow, that’s a lot of interest earnings.
So simple interest = principal * interest rate per period * number of periods.
Compound Interest
Unlike simple Interest which is computed on principal only, compound interest is computed on principal and on any interest earned that has not been paid or withdrawn.
If Jack invested $350,000 in the Sunrise fund today for five years at an annual compound interest rate of 12%, by the end of the first year, Jack will have earned .
For the second year, Jack will earn a compound interest of 12% * $350,000 + $42,000 (interest for year 1) + $47,040 (interest for year 2) = 0.12 * (350,000 + 42,000 + $47,050) = $52,684.80. Notice that Jack earns more interest in his 2nd year than in the first year.
For the third year, Jack will earn a compound interest of 12% * $350,000 + $42,000 (interest for year 1) + $47,040 (interest for year 2) = 0.12 * (350,000 + 42,000 + $47,040) = $52,684.80.
Now, based on what you’ve learned about Compound Interest, find out how much of interest Jack will get during his fourth year.
For the fourth year, Jack will earn a compound interest of …the Interest Rate
12% * (Principal $350,000 + year 1 interest $42,000 + year 2 interest $47,040 + year 3 interest $52,648.80) The total = 0.12 * (350,000 + 42,000 – 47,040 + 52,684.80) = 59,006.98.
Now based on everything you’ve learned so far, use a pen, calculator and a piece of paper to figure out how much interest Jack will earn during his fifth year?
Jack’s fifth year’s interest earning = 12% * (350,000 + 42,000 + 47,040 + 52,684.80 + 59,006.98) = $66,087.81.
The reality is that by the end of the five years, Jack will still have his $350,000 principal, plus all the interest he has earned throughout the five years, which is ($42,000 year 1 interest + $47,040, year 2 interest + $52,684.80, year 3 interest + $59,006.98, year 4 interest + $66,087.81, year 5 interest) = $350,000 + $266,819.60, = By the end of the fifth year, Jack will have a total of $616,819.60 in return from his original $350,000 investment.
FAF and FV = FVF * PV
Wow…that’s a lot of calculation. What if Jack is investing 15 rather than just 5 years! Do you have to add everything 15 times? Well, there is a short cut. And it’s called Future Value Factor (FVF).
The FVF determines the future value for 1. For every $1 of investment, the FVF for n period at i interest rate = (1 + i) powered by n <formula>
For every dollar Jack invests in the Sunrise Fund, the FVF for 5 years at an annual compound interest rate of 12% = (1+12%) powered by n = (1+0.12) powered by n. This means that Jack will get $1.762341683 at the end of the 5 year period.
Since Jack invested $350,000, he will receive a total future value of FVF $1.726341683 per dollar investment * $350,000 = $616,819.60. This brings us to another formula: FV always = PV * FVF
With these two formulas, the calculation is much easier, isn’t it?
Tips! There are different types of compound interest tables that have calculated FVF values for you. Click here to see how to find the FVF using “Future Value of 1 at Compound Interest”.
Since Jack’s investment is 5 years, at a compound interest rate of 12%, the table tells you that the FVF value is $1.76234, exactly as what we have calculated using the formula.
PV
So if Jack invests $350,000 in the Sunrise Fund now, he will get a total return of $616,819.60 five years later. That’s far more than what he will need for his boat. So how much exactly does he need to invest today so he will get a total return of $350,000 for his boat?
Using the formula, we just learned FV = FVF * PV
The FV is $350,000 = (FVF) $1.762341683 * Present Value (unknown X). Therefore, Present Value (X) = Future Value / FVF = Future Value * 1 / FVF = $290,000.
Because 1 / FVF is so frequently used, we give it a name called Present Value Factor = 1 / (1+l)n.
Therefore, Present Value = Future Value * PVF
Tips! Click here to see how to find the Present Value Factor using “Present Value of 1 at Compound Interest”.
Since Jack’s investment is 5 years, at a compound interest rate of 12%, the table tells you that the FVF value is $0.56743, exactly as what we have calculated using the formula.
PV formula
The Present Value formula is the information that Jack needs to calculate the unknown amount.
Jack wants to know how much he should invest today (a present value) so he can get a return of $350,000 (a future value) five years later. To apply our formula, PV = FV * PVF = $350,000 * PVF
PVF (n,i) = 1 / (1+l)n
In Jack’s case, n = 5 (years), and I = 12% per year
= 1 / (1 + 12%)5
= 1 / (1 + 0.12)5
= 1 / 1.125
= 0.567426855
Therefore PV = $350,000 * 0.567426855 = $198,599.40
Jack should invest $198,599.40 out of his $400,000 for his dream boat.
Jack’s question is now resolved.
PV forumlar
Now let’s review everything we’ve learned from this tutorial.
We learned the concept of Time Value of Money--- that a dollar we received today is worth more than a dollar promised at some time in the future.
We learned Simple Interest Rate = Principal * Interest Rate * Number of Period
We learned Compound Interest Rate is calculated not only on principal but also on interest earned.
For a complex compound interest calculation, we can use the following formulas. If we know the present value, and want to figure out a future value, we will first figure out the FVF for the investment. A FVF for n periods at a compound interest rate of I per period equals 1 + I the interest rate powered by n, number of periods.
< FVF (nl) = (1 + l)n >
Then we will calculate the future value using the formula, FV = PV * FVF
If we know a future value, and want to figure out the present value, we will first figure out the PVF for the investment. A PVF for n periods at a compound interest rate of I per period always equals 1 divided by ….1 + I, the interest rate powered by n, number of periods.
< PVF (nl) = 1 / (1 + l)n >
Then we will calculate the present value using the formula, PV = FV * PVF
Exercise Questions
Using the formulas you’ve learned, answer the following questions.
[End of Audio]