Skip to main content
Loading Explore guides…
Future Value of Single Cash Flow · Acedra
Home
Explore
Economics
Future Value of Single Cash Flow
Sign in to save
Chat
If YouTube asks you to sign in, open the video on YouTube.
Watch on YouTube
Reload player
Transcript
41 segments
Copy
Hide video
0:04
Moving now to the concept of Future Value, and we will start with a simple case where we look at the future value of a single cash flow. The future value at time, N, is equal to the present value, into 1 + r, which is the interest rate, to the power of N. Say the present value is 100, so at time=0, the value is 100, the interest rate is 10%. What is the future value after one year? If you plug into this formula, what you see is the following the present value is 100, r is 10%, and in this
0:47
formula, we will use the decimal so (1 + 0.1) ^ 1. This is equal to 110. So after 1 year 100 will compound to 110. What about at the end of your 2? Again from the formula, we have 100 into 1.1 ^ 2 and this is 121. So at the end of two years, we have a hundred and Twenty-One. We can say that FV2, the future value at the end of year 2, is 121. Notice the following, if we only had simple interest where a 100 is invested, then in year 1, the interest rate would be 10, in year
1:36
2, the interest would be 10, and at the end of two years we would have a 120. So why is there a difference between 120 and the future value of 121 that we have calculated using this formula? The answer is because of compounding, what has happened is at the end of year 1. We have 110, which is the original 100 plus the $10 interest, that entire amount $100 and the $10 interest is being reinvested for the 2nd so in our compounding scenario even the $10 is receiving interest and
2:15
it is the $1 interest on these 10 dollars that is giving us the extra $1 over here. Let's see if you can do this practice problem now. This is fairly straightforward. And here's what you should have done. So recognize the formula that we just saw. Future value is present value into (1 + r) ^ n. So N here is 2.5, you plug in the numbers and that is what you should get. Let us now learn how to use the financial calculator, and then we will do a future value
3:02
calculation using the calculator. The first time you use your calculator, and we'll talk here about the Texas Instruments ba II plus, you need to set the floating decimal and what that means is when you set to floating decimal calculator automatically gives you the appropriate number of decimal places. So you hit the following keystrokes; 2nd, which is right here, Format, which is over here, and then, Enter, which is here, and then you will see the option to
3:43
enter the number of decimal places, you enter 9, and the display is going to be 9. Then you hit 2nd again and QUIT, that will return the calculator to standard mode. What you have now done is set decimal places to nine which represents a floating decimal.
4:05
Let's work through this problem now. You invest $100 today at 10% compounded annually. How much will you have in five years? You have learned how to do this using the formula, but now we'll see how to solve this problem using the calculator, and we are going to use this set of functions which represents the time value of money functions on the calculator. Here is what you do. 2nd quit, just to set the calculator in your standard mode, and then 2nd CLR time value of money. This
4:42
clears the time value of money worksheet. If you don't do this than the calculator remembers what you did last time. Then you hit 5 N. N stands for the number of periods, in our particular case, we have five years, which means that N is equal to 5. 5N and the display then we'll be N = 5. Then you say 10 I/Y, so this is saying that the interest rate per period is 10%. Notice that here you are entering the percent as opposed to the formal earlier where you
5:22
entered a decimal, then you enter 100 and PV, this is saying that the initial investment is a 100. Zero payment, that means that over the five-year period time 0 to time 5 in between you are not making any payments. So this is zero, and then you compute (CPT) the future value that will give you this number. We have a minus because the calculator assumes that the future value will have the opposite sign relative to the present value. If you put in plus 100 here, then at the end of
6:03
five years, the calculator assumes that you are taking the money out. So it will give you -161. If during entry you had put in -100 then the calculator would give you + 161. We'll now talk about Frequency of Compounding. What we have done so far has talked about compounding once a year, but compounding of interest can happen more than once a year. Say you invest 80,000 in a 3-year CD. This CD offers a stated annual interest rate of 10% compounded quarterly.
6:43
How much will you have at the end of three years? How many periods do we have in this case? Three years and in each year, we have four quarters. So the number of periods is equal to 3 * 4 which is equal to 12. What is our initial investment of present value? That present value is 80,000. What about the interest rate? If we are using 12 periods then we need to look at the interest rate for each period. The stated annual rate is 10% which means that the interest rate per period must be 10 over
7:28
4, which is 2.5. The payments here are 0. You can then simply compute the future value. The answer you should get using your calculator is 107,591. You could also solve this problem using the formula and the way you think of that is as follows; your future value at the end of 12 periods. Remember three years means 12 periods, in this example, is equal to the present value, which is 80,000 into 1 plus the interest rate. Now, the interest rate has to be the interest rate per period which
8:15
is 0.1, that's the 10%, divided by 4 because we have 4 periods in a year, and then raised to the power of the number of periods, which is 3 years * 4 periods per year. And when you do this calculation, you should get the same answer.
8:37
Let's see if you can solve this problem. The correct answer is B. What we have done here is shown the working using the formula, but you could have done this also using your calculator functions and just to make sure you've understood the calculator function well, let us do it together. The number of periods here is equal to 365, why? Because we just have 1 year, but we are assuming daily compounded. So everyday is a period. The interest rate that you will have to plug in
9:15
is equal to 4% which is the stated annual rate. We put this in as 4%/365, that's your return per day. And then the present value is equal to 3 million. The payments are equal to zero, and the future value is what you have to compute. Here is what you should get. Moving now to Continuous Compounding. We have seen an example with annual compounding, then we talked about quarterly compounding. We have also seen daily compounding. If we keep increasing the number of compounding periods until we have
10:06
an infinite number of compounding periods per year, then we say that we have continuous compounding.
10:15
The formula for continuous compounding is given right here. The future value at the end of N years. Now, keep in mind, this is the number of years is equal to present value into e, this is the mathematical number, e^r, where r is the stated annual rate. It is also sometimes called the continuously compounded rate multiplied by N which is again the number of years. An investment worth 50,000 earns interest that is compounded continuously. The stated annual interest rate
10:53
is 3%. What is the future value of the investment after 3 years? Using the formula the future value at the end of three years is equal to 50,000 which is the present value multiplied by e, which is right here on your calculator, that is really e^x so into e^r, r is the stated annual rate expressed as a decimal so 0.036 * the number of years which is 3, so the way you do this on your calculator is as follows. First thing just multiply .036 into 3, then
11:42
hit the 2nd button, and e^x that will give you an expression for this and then multiply by 50,000. When you do the calculation, here is what you should get.
12:01
Here is an important concept building exercise. Assume that the stated annual interest rate is 12%. What is the future value of $100 at different compounding frequency? And then what is the corresponding return? I'll do the first one for you. If you have annual compounding then at the end of one year $100 must become 112, which means that the return is 12%. What if you have semi- annual compounding? Will return still be 12% or more than 12%? Hopefully you recognize intuitively
12:40
that the return will be slightly more than 12%, and given what you've learned over the last few minutes. I want you to fill this table out before you move on.
12:54
If you've done things correct here is what you should get. Notice that as we compound more often, the future value becomes a little bit higher. When you get to daily versus continuous, the number is almost the same. If you go to the third decimal place, then you will notice that the return using continuous compounding is a little bit higher, but at two decimal places it is essentially the same. So the return keeps getting better as we compound more often, but it
13:26
doesn't become a lot better. Note also that if you have two Banks A and B Bank A tells you that they will give you a return of 12.5% but there is only annual compounding, whereas, Bank B gives you a return of 12% and the return is compounded daily, what is better? Even though 12.5 looks higher, but you know that if your return is stated as 12%, but then is compounding continuously, effectively you are going to get a return of 12.75%. So, the offer from Bank B is actually better than
14:12
the offer from Bank A, and this brings us to the concept of stated versus effective rates. In the example that we just saw. The stated rate was 12% across the board, but the effective rate depends on the number of compounding periods. When we have discrete compounding then the effective rate is calculated using this formula. Let us do this for one of the scenarios. The effective annual rate is equal to 1 + the period rate. Let's say that we are doing quarterly compounding,
14:50
then the periodic interest rate is equal to 12, which is 0.12 / 4. So that would mean that the periodic rate is .03 or 3%, we raise this to the power of m. So that's the number of periods in a year. If you have quarterly compounding that would be 4, and then minus 1. A lot of people get confused about this -1, and the idea is actually fairly straightforward. If you look at this part of the expression, essentially we have 1.03 ^ 4 which is telling us, how much $1 will become at the end of
15:30
4 periods. Each period being a quarter. $1 is going to become 1.1255, but this is not a rate. To figure out what your return is you need to subtract the original $1 that you invested and what you are then left with is 0.1255 so that is your effective return and expressed as a percent, this is 12.55%. If we do continuous compounding, then the effective annual rate is e^r r is the stated rate of 12%. e ^ 0.12 -1. And when we do this we should get 1.1275. That's the
16:22
e^.12 and in case you're not familiar with this on the calculator, you plug in .12 and then 2nd and then e^x, that will give you this number, -1, which will give you this number which is the return expressed as a percentage that is 12.75. Moving now to the future value of a series of cash flows and first let's just understand some terminology. An annuity is a finite set of level sequential cash flows. Level means equal and sequential means that the cash flow is happening at equally spaced points
17:10
in time. An ordinary annuity is one type of annuity where the first cash flow occurs one period from today.
17:20
Let's say that you receive $100 one year from today, another $100 2 years from today, and then another $100 3 years from today. So this is an annuity, the payments are level because you are getting $100 each time. They are sequential you are getting payments at 1 year periods, and the first payment is one year from today, which means this is a ordinary annuity. With an annuity due the first cash flow occurs immediately. So if you have 100, 100 and 100, this is a annuity due. A perpetuity is a
18:00
set of level never ending sequential cash flows with the first cash flow occurring one period from today. Here is a example of a perpetuity. The first cash flow is happening one period from today, and then every year you receive $100 and this goes on forever. Let us now calculate the future value of ordinary annuity. Say you have an ordinary annuity with A equal to 1,000. A refers to the annuity payments. In this particular case we have 1000, 1000,
18:38
over a five-year period. The interest rate is 5%. We have five years and what we are asked to calculate is the future value at the end of year 5. The Brute-force way of doing this is to simply take each cash flow and see how much each of these will be worth at the end of five years. This cash flow of thousand at the end of five years is obviously 100 at this point in time, the 100 that you receive over here will be compounded 1 period so, this would be 1000 into 1.05 compounded
19:23
once, this is going to be thousand into 1.05 squared. This is going to be compounded 3 periods. This is going to be compounded for 4 periods. You take all these numbers, this, this, this, the number that is compounded for three periods and four periods, and you add them up and what you should get is 5,525.63. There is a formula that allows you to calculate the future value and actually on the next slide I'll show you how to do this on the calculator
20:05
which is what you should be doing on the exam, but here I'm just introducing the fact that there is a formula and at times you can get something called a future value annuity factor. So let's understand what this means. We have the same annuity. Each payment is 1000, interest rate is 5, N is, number of periods is 5, so same scenario. The future value at the end of year five can be expressed using this Formula. A is 1000 and the 1 + r, 1.05, this is the interest rate raised to the power of 5,
20:42
minus 1, divided by the Interest rate, which is 0.05, this number in here is called the future value annuity factor. If you do the calculation, you should get 5.5256. So this is the future value annuity factor. To come up with a future value, you take the annuity payment 100, and multiply it by the future value annuity factor and you will get 5525.6 which is the same answer that we got on the previous slide. On the exam, if you are given the annuity number,
21:29
and the future value annuity factor, you simply need to multiply the two numbers to get the future value. And finally, let's see how we do this using a calculator and this is the method that you should use on your exam. We plug in N equal to 5, that's the number of periods. The interest rate per period is 5% So plug that in. Present value is zero, that's because you are not making any investment at Time=0, the first cash flow is happening 1 period from today. The
22:06
payment is 1000, that's the equal payments happening at the end of every period, and then you compute future value and the answer that you should get is the same as what we've seen on the previous two slides. You should get 5525.6, and again, this is what you should do on the exam because this the fastest method, and doesn't require you to memorize the annuity formula. No,w I want you to work through practice question 6. so either you can use the formula and
22:47
get this answer or what you can also do and the recommended method is to use the calculator. So let's look at what people plug in the calculator. Notice that she is is depositing 24,000 a year for 15 years. So N is equal to 15, we are told that the interest rate is 12%. So I is equal to 12%. The present value is 0, because nothing is being invested at time 0, the payments are equal to 24,000, and then we compute the future value. When you do this computation what you should
23:32
get is 894.713. Try to solve this practice problem now. The correct answer is C and if you think about it, we are given the annuity so that the A is given, which is essentially the payment, and if we know the future value annuity factor, which is 21.664, then to come up with a future value, all you need to do is multiply 120,000 by this. You don't really need to use the interest rate. So in terms of least likely the best answer is C. How do we deal with unequal cash
24:21
flows? And this is a fairly straightforward concept. We want the future value at the end of year 5. Let's say that the interest rate is 10% the way you need to do this is individually compound every cash flow, and this connects with the brute-force method that we talked about at the start of this segment. So, if you draw your cash flow, this is time 0, 1, 2, 3, 4, 5. This is your timeline. We have $1,000 at the end of year 1. So this is going to be compounded for 4 periods so this will
25:00
become 1000 into 1.1 ^ 4. Then you have 2,000 at the end of the 2nd period. This will compound three periods. So 2000 into 1.1 ^ 3, and then 3000 from here.
25:20
This is compounded 2 periods. So 3000 into 1.1 ^ 2. 4000 is going to be compounded just one period. And the final 5000 at the end of your 5 is what you have. So at all these numbers, that is your future value at the end of year 5. When you do the calculation what you should get a 17156.1
Here to help you learn
Sign in to keep a private conversation with your course tutor.
Ask me anything about the material…
Sign in to chat