how to calculate time value of money

Present Value	How much you got now.
Future Value	How much what you got now grows to when compounded at a given rate

I give you 100 dollars. You take it to the bank. They will give you 10% interest per year for 2 year.

• The Present Value = $ 100
• Future Value = $121.

FV= PV (1 + i )N

• FV = Future Value
• PV = Present Value
• i = the interest rate per period
• n= the number of compounding periods

Determine Future Value Compounded Annually
What is the future value of $34 in 5 years if the interest rate is 5%? (i=.05)

• FV= PV ( 1 + i ) ^N
• FV= $ 34 ( 1+ .05 ) ⁵
• FV= $ 34 (1.2762815)
• FV= $43.39.

Determine Future Value Compounded Monthly
What is the future value of $34 in 5 years if the interest rate is 5%? (i equals .05 divided by 12, because there are 12 months per year. So 0.05/12=.004166, so i=.004166)

• FV= PV ( 1 + i ) ^N
• FV= $ 34 ( 1+ .004166 ) ⁶⁰
• FV= $ 34 (1.283307)
• FV= $43.63.

Determine Present Value Compounded Annually
You can go backwards too. I will give you $1000 in 5 years. How much money should you give me now to make it fair to me. You think a good interest rate would be 6% ( You just made that number up). (i=.06)

• FV= PV ( 1 + i ) ^N
• $1000 = PV ( 1 + .06) ⁵
• $1000 = PV (1.338)
• $1000 / 1.338 = PV
• $ 747.38 = PV

O.K. so you give me $ 747.38 today and in 5 years I'll give you $1000. Sound fair?? You will get 6% interest on your money.

Determine Present Value Compounded Monthly
Here's that last one again, but with monthly compounding instead of annual compouding. (i equals .06 divided by 12, because there are 12 months per year so 0.06/12=.005 so i=.005)

• FV= PV ( 1 + i ) ^N
• $1000 = PV ( 1 + .005) ⁶⁰
• $1000 = PV (1.348)
• $1000 / 1.348= PV
• $741.37 = PV

An Annuity is a bunch of structured payments or equal payments made regularly, like every month or every week.

·         You win the lottery. The lottery guy comes to your house and says you have to choose between getting $ 1,000,000 now in one lump sum, or getting structured payments of $ 50,000 a year for the next 22 years. Which do you take?? Or, similarly, let's say you were injured on the job or whatever and were awarded an annuity of structured payments of $50,000 a year for the next 22 years. Perhaps you want to sell your annuity (the payments) to someone and get a lump sum of cash today. Is it worth $1,000,000?

First you have to choose an interest rate. Money is generally worth less in the future, right? So that $50,000 payment you get in 22 years is not going to be worth as much as it is today? You know, stuff will be more expensive then, right? So guess an interest rate, in this case, the rate of inflation for the next 22 years. Lets say 4%. Now, you have to figure out what is the present value of the $50,000 times 22 years discounted by 4% and then compare it with the million bucks. There are basically 2 ways to do this.

• Use a financial calculator.
• Use an annuity table.

Use a financial calculator - The PV of an Annuity.

1. Enter n (the number of compounding periods - in this case the number of years). Press 22 and then push the N button.
2. Enter i (the interest rate per period - in this case the number of years). Press 4 and then push the i button.
3. Enter FV (the future value). It is zero. You want to know the Present Value, not the future value, right? Push 0 and then push the FV button.
4. Enter PMT (the payment). You are not making a payment, you are getting one. So you have to show a negative number. Press 50000, then the CHS (change sign button), then push the PMT button.
5. Push the PV (present value) button.
6. Answer = $722,555. This means 22 annual structured payments of 50,000 each is worth only $722,555 of today's dollars. So you should take the million bucks from the lottery guy in one lump sum.

Use an annuity table - The PV of an Annuity.

Somewhere in your book, I bet there is a table that looks something like this:

	1%	2%	3%	4%
1	00.9901	00.9804	00.9703	00.9615
2	01.9704	01.9416	01.9135	01.8861
3	02.9410	02.8839	02.8286	02.7751
4	03.9020	03.8077	03.7171	03.6299
5	04.8534	04.7135	04.5797	04.4518
6	05.7955	05.6014	05.4172	05.2421
7	06.7282	06.4720	06.2302	06.0021
8	07.6517	07.3255	07.0197	06.7327
9	08.5660	08.1622	07.7861	07.4353
10	09.4713	08.9826	08.5302	08.1109
11	10.3676	09.7868	09.2526	08.7605
12	11.2551	10.5753	09.9450	09.3851
13	12.1337	11.3484	10.6350	09.9856
14	13.0037	12.1062	11.2961	10.5631
15	13.8651	12.8493	11.9379	11.1184
16	14.7179	13.5777	12.5611	11.6523
17	15.5623	14.2919	13.1661	12.1657
18	16.3983	14.9920	13.7535	12.6593
19	17.2260	15.6785	14.3238	13.1339
20	18.0456	16.3541	14.8775	13.5903
21	18.8570	17.0112	15.4150	14.0292
22	19.6604	17.6580	15.9369	14.4511

1. Find this table.
2. On the left, find the number of compounding periods (in this case years) - 22
3. On the top, find the interest rate - 4%
4. Find below where they meet. It says 14.4511
5. Multiply 14.4511 times the Payment - $50,000
6. Answer = $722,555. This means 22 annual structured payments of 50,000 each is worth only $722,555 of today's dollars. So you should take the million bucks from the lottery guy in one lump sum

Perpetuities - are equal payments made regularly, like every month or every year, that go on foreve

The question is....How much money will it cost you. In today's dollars. What is the present value of this perpetuity. (Hint: starting now and going on forever and ever, you assume the interest rate at your bank is going to be 3%).

PV (of a perpetuity) = payment / interest rate

Every year the interest you earn is used to pay for the scholarship. The principal in your bank account doesn't really change year to year.

• PV (of a perpetuity) = payment / interest rate
• PV = $ 1000 / .03
• PV = $ 33,333

So, you put $ 33,333 into the bank. Each year the money earns $1000 interest. That interest becomes the scholarship.

Kinds of Interest Rates

let's say I give you a credit card and the interest rate on the card is 3% per month. What is the annual rate that you are actually charged?? 36%?? Well, no. It's actually 42.57%.

Nominal Rate	Nominal means "in name only". This is sometimes called the quoted rate.
Periodic Rate	The amount of interest you are charged each period, like every month.
Effective Annual Rate	The rate that you actually get charged on an annual basis. Remember you are paying interest on interest.

In the example

• The Nominal Rate is 36%.
• The Periodic Rate is 3% (you are charged 3% interest on your balance every month)
• The Effective Annual Rate is 42.57%

Nominal Rate = Periodic Rate X Number of Compounding Periods

Effective Annual Rate = (1+ i / m)m -1

• m = the number of compounding periods
• i = the nominal interest rate

O.K., so let's try the example again.

• Effective Annual Rate = (1+ i / m)^m -1
• Effective Annual Rate = ( 1 + .36 / 12 )¹² -1
• Effective Annual Rate = (1.03)¹² - 1
• Effective Annual Rate = (1.4257) -1
• Effective Annual Rate = .4257
• Effective Annual Rate = 42.57 %

cash Flow

Cash Flow is money you get a little at a time.

·         Lets say, for example that for the next 4 years you will get the following cash flow.

Cash Flow
in 1 year	$ 320
in 2 years	$ 400
in 3 years	$ 650
in 4 years	$ 300

If you assume that the interest rate is 6.5% (which means that after you get the money, it will be invested and you will get 6.5% interest from it), compounded monthly, how much money will you have in 4 years? In other words, what will the future value of this cash flow be?

Compounding Formula

FV=PV ( 1 + i / m)mn

• FV = Future Value
• PV = Present Value
• i = Interest rate (annual)
• m = number of compounding periods per year
• n = number of years

So you have to figure out the future value of each payment and then add them together.

First Payment

• FV = PV ( 1 + i / m)^mn
• FV = $320 (1 + .065 / 12 )^{12 X 3} (three years)
• FV = $320 (1.0054167)³⁶
• FV = $320 (1.2146716)
• FV = $388.69

Second Payment

• FV = PV ( 1 + i / m)^mn
• FV = $400 (1 + .065 / 12 )^{12 X 2} (two years)
• FV = $400 (1.0054167)²⁴
• FV = $400 (1.1384289)
• FV = $455.37

Third Payment

• FV = PV ( 1 + i / m)^mn
• FV = $650 (1 + .065 / 12 )^{12 X 1} (one year)
• FV = $650 (1.0054167)¹²
• FV = $650 (1.0669719)
• FV = $693.53

Fourth Payment - ( The payment is not compounded. There no time to earn interest)

• FV = PV ( 1 + i / m)^mn
• FV = $300 (1 + .065 / 12 )^{12 X 0}(0 years.)
• FV = $300 (1.0054167)⁰
• FV = $300 (1) (remember anything to the power of zero is 1)
• FV = $300

Finally, add up all the numbers

$ 388.69
$ 455.37
$ 693.53
$ 300.00
----------
$1,837.59

So after 4 years, you will have $1,837.59. That is the future value of your uneven cash flow.

Cash Flow- We are going to assume that the project we are considering approving has the following cash flow. Right now, in year zero we will spend 15,000 dollars on the project. Then for 5 years we will get money back as shown below.

Year	Cash flow
0	-15,000
1	+7,000
2	+6,000
3	+3,000
4	+2,000
5	+1,000

Payback - When exactly do we get our money back, when does our project break even. Figuring this is easy. Take your calculator.

Year	Cash flow	Running Total
0	-15,000	-15,000
1	+7,000	-8,000	(so after the 1st year, the project has not yet broken even)
2	+6,000	-2,000	(so after the 2nd year, the project has not yet broken even)
3	+3,000	+1,000	(so the project breaks even sometime in the 3rd year)

But when, exactly? Well, at the beginning of the year we had still had a -2,000 balance, right? So do this.

Negative Balance / Cash flow from the Break Even Year	=	When in the final year we break even
-2,000 / 3,000	=	.666

So we broke even 2/3 of the way through the 3rd year. So the total time required to payback the money we borrowed was 2.66 years.

Discounted Payback - is almost the same as payback, but before you figure it, you first discount your cash flows. You reduce the future payments by your cost of capital. Why? Because it is money you will get in the future, and will be less valuable than money today. (See Time Value of Money if you don't understand). For this example, let's say the cost of capital is 10%.

Year	Cash flow	Discounted Cash flow	Running Total
0	-15,000	-15,000	-15,000
1	7,000	6,363	-8,637
2	6,000	4,959	-3,678
3	3,000	2,254	-1,424
4	2,000	1,366	-58
5	1,000	621	563

So we break even sometime in the 5th year. When?

Negative Balance / Cash flow from the Break Even Year	=	When in the final year we break even
-58 / 621	=	.093

So using the Discounted Payback Method we break even after 4.093 years.

Net Present Value (NPV) - Once you understand discounted payback, NPV is so easy! NPV is the final running total number. That's it. In the example above the NPV is 563. That's all. You're done, baby. Basically NPV and Discounted Payback are the same idea, with slightly different answers. Discounted Payback is a period of time, and NPV is the final dollar amount you get by adding all the discounted cash flows together. If the NPV is positive, then approve the project. It shows that you are making more money on the investment than you are spending on your cost of capital. If NPV is negative, then do not approve the project because you are paying more in interest on the borrowed money than you are making from the project.

Profitability Index

Profitability Index	equals	NPV	divided by	Total Investment	plus	1
PI	=	563	/	15,000	+	1

So in our example, the PI = 1.0375. For every dollar borrowed and invested we get back $1.0375, or one dollar and 3 and one third cents. This profit is above and beyond our cost of capital.

Internal Rate of Return - IRR is the amount of profit you get by investing in a certain project. It is a percentage. An IRR of 10% means you make 10% profit per year on the money invested in the project. To determine the IRR, you need your good buddy, the financial calculator.

Year	Cash flow
0	-15,000
1	+7,000
2	+6,000
3	+3,000
4	+2,000
5	+1,000

Enter these numbers and press these buttons.

-15000	g	CFo
7000	g	CFj
6000	g	CFj
3000	g	CFj
2000	g	CFj
1000	g	CFj
f	IRR

After you enter these numbers the calculator will entertain you by blinking for a few seconds as it determines the IRR, in this case 12.02%. It's fun, isn't it!

Ah, yes, but there are problems.

• Sometimes it gets confusing putting all the numbers in, especially if you have alternate between a lot of negative and positive numbers.
• IRR assumes that the all cash flows from the project are invested back into the project. Sometimes, that simply isn't possible. Let's say you have a sailboat that you give rides on, and you charge people money for it. Well you have a large initial expense (the cost of the boat) but after that, you have almost no expenses, so there is no way to re-invest the money back into the project. Fortunately for you, there is the MIRR.

Modified Internal Rate of Return - MIRR - Is basically the same as the IRR, except it assumes that the revenue (cash flows) from the project are reinvested back into the company, and are compounded by the company's cost of capital, but are not directly invested back into the project from which they came.

WHAT?

OK, MIRR assumes that the revenue is not invested back into the same project, but is put back into the general "money fund" for the company, where it earns interest. We don't know exactly how much interest it will earn, so we use the company's cost of capital as a good guess.

Why use the Cost of Capital?

Because we know the company wouldn't do a project which earned profits below the cost of capital. That would be stupid. The company would lose money. Hopefully the company would do projects which earn much more than the cost of capital, but, to play it safe, we just use the cost of capital instead. (We also use this number because sometimes the cash flows in some years might be negative, and we would need to 'borrow'. That would be done at our cost of capital.)

How to get MIRR - OK, we've got these cash flows coming in, right? The money is going to be invested back into the company, and we assume it will then get at least the company's-cost-of-capital's interest on it. So we have to figure out the future value (not the present value) of the sum of all the cash flows. This, by the way is called the Terminal Value. Assume, again, that the company's cost of capital is 10%. Here goes...

Cash Flow	Times		=	Future Value of that years cash flow.	Note
7000	X	(1+.1) 4	=	10249	compounded for 4 years
6000	X	(1+.1) 3	=	7986	compounded for 3 years
3000	X	(1+.1) 2	=	3630	compounded for 2 years
2000	X	(1+.1) 1	=	2200	compounded for 1 years
1000	X	(1+.1)0	=	1000	not compounded at all because this is the final cash flow
TOTAL			=	25065	this is the Terminal Value

OK, now get our your financial calculator again. Do this.

-15000	g	CFo
0	g	CFj
0	g	CFj
0	g	CFj
0	g	CFj
25065	g	CFj
f	IRR

Why all those zeros? Because the calculator needs to know how many years go by. But you don't enter the money from the sum of the cash flows until the end, until the last year. Is MIRR kind of weird? Yep. You have to understand that the cash flows are received from the project, and then get used by the company, and increase because the company makes profit on them, and then, in the end, all that money gets 'credited' back to the project. Anyhow, the final MIRR is 10.81%.

Decision Time- Do we approve the project? Well, let's review.

Decision Method	Result	Approve?	Why?
Payback	2.66 years	Yes	well, cause we get our money back
Discounted Payback	4.195 years	Yes	because we get our money back, even after discounting our cost of capital.
NPV	$500	Yes	because NPV is positive (reject the project if NPV is negative)
Profitability Index	1.003	Yes	cause we make money
IRR	12.02%	Yes	because the IRR is more than the cost of capital
MIRR	10.81%	Yes	because the MIRR is more than the cost of capital

CAPM - The Capital Asset Pricing Model

"Cap-M" looks at risk and rates of return and compares them to the overall stock market. If you use CAPM you have to assume that most investors want to avoid risk, (risk averse), and those who do take risks, expect to be rewarded. It also assumes that investors are "price takers" who can't influence the price of assets or markets. With CAPM you assume that there are no transactional costs or taxation and assets and securities are divisible into small little packets. Had enough with the assumptions yet? One more. CAPM assumes that investors are not limited in their borrowing and lending under the risk free rate of interest. By now you likely have a healthy feeling of skepticism. We'll deal with that below, but first, let's work the CAPM formula.

Beta - Now, you gotta know about Beta. Beta is the overall risk in investing in a large market, like the New York Stock Exchange. Beta, by definition equals 1.0000. 1 exactly. Each company also has a beta. You can find a company's beta at the Yahoo!! Stock quote page. A company's beta is that company's risk compared to the risk of the overall market. If the company has a beta of 3.0, then it is said to be 3 times more risky than the overall market.

Ks = Krf + B ( Km - Krf)

• Ks = The Required Rate of Return, (or just the rate of return).
• Krf = The Risk Free Rate (the rate of return on a "risk free investment", like U.S. Government Treasury Bonds - Read our Disclaimer)
• B = Beta (see above)
• Km = The expected return on the overall stock market. (You have to guess what rate of return you think the overall stock market will produce.)

As an example, let's assume that the risk free rate is 5%, and the overall stock market will produce a rate of return of 12.5% next year. You see that XYZ company (Read our Disclaimer) has a beta of 1.7.

What rate of return should you get from this company in order to be rewarded for the risk you are taking? Remember investing in XYZ company (beta =1.7) is more risky than investing in the overall stock market (beta = 1.0). So you want to get more than 12.5%, right?

• Ks = Krf + B ( Km - Krf)
• Ks = 5% + 1.7 ( 12.5% - 5%)
• Ks = 5% + 1.7 ( 7.5%)
• Ks = 5% + 12.75%
• Ks = 17.75%

So, if you invest in XYZ Company, you should get at least 17.75% return from your investment. If you don't think that XYZ Company will produce those kinds of returns for you, then you would probably consider investing in a different stock

Who introduced the CAPM - Capital Asset Pricing Model?
Harry Markowitz worked on diversification and modern portfolio theory. Jack Treynor, John Lintner, Jan Mossin and William Sharpe all contributed to the theory of CAPM. William Sharpe, Harry Markowitz and Merton Miller jointly got a Nobel Prize in Economics for contributing to financial economics. See, if you study hard and think up new stuff maybe you can get a Nobel Prize too.

Ah, but CAPM has some flaws. (don't we all)

• If you go to a casino, you basically pay for risk. It's possible that the folks on Wall Street sometimes have the same mindset as well. Now remember that CAPM assumes that given "X%" expected return investors will prefer lower risk (in other words lower variance) to higher risk. And the opposite would be true as well - given a certain level of risk investors would prefer higher returns to lower ones. OK, but maybe the Wall Street people get a kick out of "gambling" their investment. Not saying it's been proven to be the case, just saying it could be. CAPM doesn't allow for investors who will accept lower returns for higher risk.
• CAPM assumes that asset returns are jointly normally distributed random variables. But often returns are not normally distributed. So large swings, swings as big as 3 to 6 standard deviations from the mean, occur in the market more frequently than you would expect in a normal distribution.
• CAPM assumes that the variance of returns adequately measures risk. This might be true if returns were distributed normally. However other risk measurements are probably better for showing investors' preferences. Coherent risk measures comes to mind.
• With CAPM you assume that all investors have equal access to information and they all agree about the risk and expected return of the assets. This idea, by the way is called "homogeneous expectations assumption". Be ready for your professor to ask, "What's the Homogeneous Expectations Assumption and do you believe it's valid". Good luck with that one.
• CAPM can't quite explain the variation in stock returns. Back in 1969, Myron Scholes, Michael Jensen and Fisher Black presented a paper suggesting that low beta stocks may offer higher returns than the model would predict.
• CAPM kind of skips over taxes and transaction costs. Some of the more complex versions of CAPM try to take this into consideration.
• CAPM assumes that all assets can be divided infinitely and that those small assets can be held and transacted.
• Roll's Critique: Back in 1977, Richard Roll offered the idea that using stock indexes as a proxy for the true market portfolio can lead to CAPM being invalid. The true market portfolio should include stuff like real estate, human capital, works of art and so on, basically anything that anyone holds as an investment. However, the markets for those assets are often non-transparent and unobservable. So often financial people will use a stock index instead. Does it kind of seem like they are fudging a little bit. You might argue they are.
• CAPM assumes that individual investors have no preference for markets or assets other than their risk-return profile. But is that really the case? Say a guy loves drinking Coke. Only Coke. He's collects old Coke bottles and stuff. OK, now, is that guy going to buy stock in Pepsi based only on its risk-return profile, or is he going to buy stock in Coke so he can brag to everyone about how many shares he has?