One of the “central ideas” of finance is that Money has a time value.  “A dollar today is worth more than the promise of a dollar tomorrow.”  The “promise” of the dollar tomorrow takes away the possibility of the use of that dollar today.  The promise of the dollar tomorrow involves the  risk that the dollar will not be delivered.  Given the time and risk, you expect compensation for NOT having the money TODAY.

The time value of money problems are used extensively in finance.  ALL assets (investments, real estate, business ventures) are valued using time value techniques.
The 6 equations of Time Value:
            1.  Future value of a single sum.
            2.  Present value of a single sum.
            3.  Future value of an annuity.
            4.  Present value of an annuity.
            5.  Future value of a single sum under continuous compounding.
            6.  Present value of a single sum under continuous compounding.

[Equation I]  Future Value of a Single Sum.

The simplest example of the future value of a single sum is $100 on deposit for one year at 6% interest.  We could easily calculate the annual earnings to this deposit by multiplying 0.06 (the decimal version of 6%) by $100.  We would reach an answer of $6 [.06 x $100].

The formula for the Future Value of a Single Sum is:            P ( 1 + I )^n

Where P is the Principal.  In our example, “P” equals $100.

“I” equals the rate of interest, or the rate of return.  But, “I” is actually more complex. It is:  the interest rate divided by the number of compounding periods per year.  In our example, the interest rate is 6%, the compounding period is 1; interest was paid at the end of the year, it was paid once.

“n” is the time component of the equation.  “n” equals:  the number of compounding periods per year x the number of years.  In our [simple] example, the number of compounding periods per year was “1” and the number of years was “1.”  Therefore, the exponent was 1x1=1.  The exponent of “1” needed not be written.

A second example, which is not so simple, is:  $2715.16 is on deposit for 6 ½ years at 8 ¼% interest compounded weekly.  What is the amount in the account at the end of the term?

The solution would appear:  $2715.16 ( 1+ .0825/52)^6.5x52  =  $4639.82.
$1924.66 was earned in interest over the 6.5 years [4639.82-2715.16].

[Equation II]  Present Value of a Single Sum.

The idea of present value is particularly significant in finance because so many financial decisions, personally or in business, evaluate a project, or an investment that has expectations of some return that is realized in the future.  The challenge is to value that project or investment TODAY.  Investments that are “throwing off” cash flows, such as a certificate of deposit or bond that pays periodic interest, those cash payments can be valued today, even if they are to be received later than today.  The Present Value is an amount known in the future that is valued today [in the present].

The first formula that we learned for the Future Value, is used for the Present Value.  The new formula is:            P (1 + I )^-n .

The formula is the same as the Future Value with one clear difference, the exponent is negative.  Remember that the exponent is the time variable in these equations.  If the exponent is positive, as in Future Value calculations, time is positive.  In Present Value, the exponent is negative, time is negative, and we are bringing a future, known amount, back through time to today.

An example of the Present Value of a Single Sum:            An investor wants to have $5000 in 4 years.  If money is worth 8%, compounded monthly, how much would they have to deposit TODAY.
$5000 (1 + .08/12)-4x12 = $3634.60

The answer is the amount that has to be deposited today, in an account with the interest rate specified, to reach the $5000 goal.  The present value calculation takes the guess out of the goal.

The number of applications to the Future/Present value of an annuity is countless.  Financial applications are what we are focused on, but any number that is growing, like populations, inflation, etc. can use this formula to predict, estimate or calculate future numbers. 


We have been looking at the first examples of time value in the simplest form, the present and future values of single sums.
As we would begin a savings plan for ourselves, we could invest money that we already have, such as, money that we have in the bank.  That would be a single sum calculation.  As a practical matter, people save money in similar frequency to their paycheck.  This regular “stream of payments” is known in finance as an Annuity.

Annuities are best more easily calculated if a consistent amount is saved/invested in consistent intervals.  For example, $50 could be saved monthly for 5 years at a rate of 8%.

[Equation III]   Future Value of an Annuity:

[  P  *    ((1 + i )^n  –1)  ] /   i

As powerful as it is for calculating relatively lengthy problems, this formula was formed from the Future Value of a Single Sum formula that we first used.  There is one assumption – the cash flow must equal to the compounding frequency.  If the payment or deposit is monthly, the money must be compounded monthly.  The actual compounding frequency may be greater than the one used in the calculation; this would make our answer to the problem more conservative than the actual.  [For example, you could be making monthly deposits into a savings account at a local bank that pays interest compounded daily.  Our calculation of that problem would produce a slightly smaller result than the actual.  However the calculation of that problem would be far more cumbersome than it would be worth.]

Here is an example:  $150 is deposited into an account monthly that earns 8% interest for 5 years.  What is the value in the account at the end of the term?

            150 [( 1 + .08/12 )^12x5 –1 ] / (.08/12) =  $11,021.53

Another example:  A person deposits $250 monthly into an account earning 7 ¾% {compounded monthly} for 42 years.  What is the value of the account at the end of the term?

            250 [( 1 + .0775/12 )^12x42  -1] / (.0775/12) =  $954,183.98.

[Equation IV]   Present Value of an Annuity: 

[  P *   (1- (1 + i )-n) ] / i

The present value of an annuity can be used to obtain a single [present] value of a stream of payments.  With a slight modification of the future value of an annuity formula, we can solve problems like this example:

A person signs a lease with an apartment complex for 3 years, paying $550 per month in rent.  With money worth 6% [compounded monthly] what is the present value of the 36 payments?

            550 [ 1- ( 1 + .06/12 )^-12x3 ] / (.06/12) = $18,079.06.

We can see that the (gross) value of the lease is worth $19,800 (550 x 36) and that the present value of the lease is less due to the time value of money.  Apartment complexes, copy machine companies, computer companies, anyone that uses leases can sell the lease at the time of the signing or anytime during the contract’s term.  The present value of the lease is what we would expect that the purchaser would pay for the asset.

[Equation V]   Future Value of a Single Sum under Continuous Compounding:

Take the concepts explained above and think of a compounding interval being infinitely fast.  This is continuous compounding.  There is an easy formula to handle what may be thought of as a complex problem:   Pe^rt    
Where "P" is principal.
"r" is the rate of interest or return.
"t" is time in years.
"e" is the logarithmic function found on your calculator.

[Equation VI]   Present Value of a Single Sum under Continuous Compounding:


Arithmetic Mean, Geometric Mean and Standard Deviation.

These calculations are particularly important to investments as they explain relationships of investment returns.

Arithmetic Mean

Very simply, this is an arithmetic (simple) average of numbers.  The arithmetic mean is one of the first statistical relationships that we are exposed to.  When an exam is returned to the class, the "average grade" is usually announced or requested.  Why?  We are accustomed to react to that statistic.  When a "low" average grade is announced as the average, it is usually followed by disappointment from the class.  The average has nothing to do with a persons individual score, but people will react that way regardless.  Arithmetic averages are used in investments to give an idea as to how an investment has performed.
Statistically, the arithmetic mean becomes the expected value.  We are schooled to think that way.  When a market professional is asked "how they think the market will perform next year?" they will probably have an idea of how to answer that question by reflecting on past performance "on average" and then add a bit of what they believe is different in the year to come that may change that average.

Geometric Mean

Geometric Mean has a more involved calculation.  It calculates the average annual compounded return of the set of numbers.  It is the more appropriate measure of changes in wealth. It is used to calculate the changes in wealth.

Use this example:  A person deposits $100 into an account earning 6% interest, compounded annually.  At the end of the first year, $6 is deposited into the account by the bank; payment for their use of the money for one year.  After the second year, the account holder is owed another $6 PLUS interest on the first $6; this is nothing new, this is compound interest.

Let's say that a mutual fund earns 14% in year one; 9% in year two; and 18% in year three.  The example assumes and implies that the difference in the account in year one was measured at the beginning of the year and again at the end of year, the difference was 14%. 

Say $1000 was deposited at the beginning of the year and a balance of $1140 was observed at year end, that would constitute a 14% return  [$1000( 1 + .14)].  Could the account balance have fluctuated during the year?  Of course it did, a mutual fund balance fluctuates every day.

The $1140 balance is now the beginning balance of year two.  At the end of the second year, a balance of $1242.60 would indicate a 9% return.  It would give 9% return on the original $1000, but also 9% on the $140 of interest from the first year. [1140( 1 + .09)].
We would use the same approach to calculate the third year: $1242.60( 1 + .18) = $1466.27.

At the end of the third year, your fund balance would be $1466.27.  If you had not made annual calculations you would know one thing:  That you invested $1000 at the beginning of year 1; three years later, you have $1466.27.  Is this good?  People ask themselves that question constantly in investing.  We would use the Geometric Mean to find the Average Annual Compounded Return for that period.
To calculate the average annual compounded return, first add one to each of the annual returns, then multiply these numbers together, using the future value of a single sum formula:  (1+ .14)(1+ .09)(1+ .18). = 1.46627.  Secondly, take the root of this number, the cube root in this case because we have three observations.  [Adding one to the annual returns is called the Return Relative.]
1.46627.33333  = 1.136068.   Lastly, since we "added a one" to each value in the previous paragraph, we have to take the one away to finish the problem. Thus, the average annual compounded return or Geometric Mean of this problem is 0.136068, 13.6068%.
It says that the investor earned an average annual compounded return of 13.6068% to grow $1000 into $ $1466.27 in three years.  We know that none of the three years in question returned 13.6068%; the actual returns were 14%, 9% and 18% respectively. The calculations that we made created an Average, a Geometric Average, an average annual compounded return.

Arithmetic and Geometric Means Combined

Let's say that a person begins investing with $1000.  They say that their portfolio returned 100% the first year and lost 50% the second year.  They announce proudly that "their average rate of return was 25%."  [(100-50)/2].

As mentioned earlier, the arithmetic mean calculation that was performed in the last paragraph, is NOT the tool to use to calculate the change in wealth, we should use the geometric mean.  Here's why.

Look at the example.  If $1000 earned a 100% return in the first year, the investor would have $2000 at year-end.  The next year, they lose 50% (or half).  They are back to the original $1000 investment.  They had a ZERO rate of return for the two years. NOT a 25% average return as they reported.

If we use a geometric mean, the true change in wealth will be calculated.

First, add one to each year's return and multiply them together. (1+ 1)(1+ -.5)=
(2)(.5)=1.  Then, take the square root of the result (the square root, because we have two observations). The square root of one is one.  Lastly, subtract one from that result.  One minus one is ZERO.  That is the change in wealth of this investment as we said before!

Standard Deviation

Standard Deviation is a statistical technique used against a set of number in an attempt to glean a relationship out of those numbers.  It can be used to forecast the outside temperature on a given day to measuring anticipated investment returns from one period to another.
It uses tools that we learned earlier on this page in this calculation.

Almost everyone would agree that a value that fluctuates often, and wildly would have more variation than one that stayed mostly the same.  For example, if the temperature outside would begin the morning at 60 degrees, jump to 85 degrees before lunch, plunge to 45 degrees in the afternoon and back up to 90 degrees before the end of day, would be difficult to predict its next move and would be said to be volatile.  This compared to a temperature that began the day at 82 degrees, would rise to 90 degrees by mid-day, then end the day back at 82; this is less volatile.

In investments, a volatile investment is one that acts like the first temperature example. A volatile investment is usually described as a risky one.  Risky, because it is hard to predict, hard to follow and gives the investor really rough ride.
Standard Deviation measures the variation or volatility of a set of numbers.  In our example a set of investment returns.
Let's say that the market as defined by the Dow Jones Industrial Average (DJIA, or "the DOW") had the following annual returns: [These are actual returns.]

1999 +25.2%
1998 +16.1%
1997 +22.7%
1996 +26.9%
1995 +35.5%
1994 +3.6%
1993 +15.1%
1992 +5.7%

To calculate the standard deviation we follow these steps:
1.  Calculating the [arithmetic] average return. Add the returns in column 2, divide by the number of observations (years) in column 1.

2.  Find the "dispersion around the mean."  In column 3, take each individual return (X) and subtract the mean (x) from it.  The column should add to zero, if it does not, there was an error made in the calculation.

3.  Column 4.  Square the differences around the mean.  Simply square each answer in column three.  This will rid the column of negative numbers.  Add the column.
    1                   2                           3                                       4

Year           Return (%)           (X-x)                                  (X-x)^2
1999              25.2               25.2-18.85  = 6.35              40.3225
1998              16.1                   16.1-18.85  = -2.75               7.5625
1997              22.7                22.7-18.85  = 3.85               14.8225
1996              26.9                26.9-18.85  = 8.05             64.8025
1995              35.5                35.5-18.85  = 16.65           277.2225
1994                3.6                  3.6-18.85    = -15.25        232.5625
1993               15.1                   15.1-18.85  = -3.75              14.0625
1992                 5.7                  5.7-18.85    = -13.15          172.9225

Mean = 18.85%
The (X-x) column should sum to ZERO 
Sum of column 4 is 824.28

The bulk of calculation is complete.  The process is tedious. There is obviously plenty of chance to make a keystroke error on your calculator, summing column 3 to zero makes a handy check on your progress.

4.  The 824.28 is divided by the number of observations.  824.28/8 = 103.035.  Statistically, this is called the VARIANCE.

5.  Take the square root of the variance to "un-do" the process in column 4 when we squared each of the entries in column three.  The act of taking the square root of the variance, is the result that we have labored hard to produce:  the STANDARD DEVIATION.
Standard Deviation =   √    Σ (X-x)^2  / n          =  10.15.

The Standard Deviation measures the total variability of the numbers in question. S.D. is used in many disciplines; it is a statistical process.  When testing services grades an exam using the scantron forms, the standard deviation is printed at the bottom of the page.  In finance or more specifically, in asset returns or stock prices, it is the most comprehensive measure of risk.  As stock returns are studied, risk is a key ingredient, without understanding the risk associated with an investment, it would be impossible to set goals or for an investment professional to manage a portfolio.

INTERPRETATION:  Using this data, what can we conclude?  We or an investment professional, market analyst or economist could say:  "The stock market is expected to return 18.85% in the year 2000."  Expected because the arithmetic mean is 18.85% (remember that the arithmetic mean is the expected value).  "Further more, there is a 66% chance that the market will have a return within one standard deviation (plus or minus) from the expected value."  Therefore, the is a 66% chance (this is a statistical value that is given in the standard deviation calculation) that the market return for the next year will be between 8.699% and 29%.  [Take the expected value of 18.85% and add one standard deviation to it to form 29%; then subtract one standard deviation from it, to get 8.699%.