Future and Present Values

Future value is the amount a single sum, deposited today, will grow to over a period of time given a positive interest rate with compound interest.  Present value is the amount a single sum due at some point in the future is worth today given a positive discount rate with compound discounting.   Future value of an annuity is the amount a series of equal deposits at equal intervals will grow to over a period of time given a positive interest rate with compound interest.  Present value of an annuity is the amount a series of equal payments at equal intervals due in the future is worth today given a positive discount rate with compound discounting.

Future value

Restating the above definition – future value is the amount a single sum, deposited today, will grow to over a period of time given a positive interest rate with compound interest.  The determination of future value assumes the single amount will be left undisturbed except for the addition of the compound interest it earns.  It also assumes that the interest rate stays the same over the entire time period.   The most straightforward examples of this are probably a deposit to an interest bearing savings account or an investment in a multiple period certificate of deposit.

If \$1,000 is put into a savings account today what will be the balance in that account after five years if the interest rate is 5% compounded annually?  To solve this problem one needs to calculate future value.  Mathematically this is done by FV = PV (1 + i)ⁿ where: FV = Future value at the end of n periods.  PV = Amount of the single sum invested today (time zero).  i = Interest rate per compounding period stated as a decimal.  n = number of compounding periods.  So for this situation it is FV = \$1,000 x 1.05 = \$1,000 x 1.276282 = \$1,276.28.  This can also be done using a compound interest table such as Table A on page 547 of the text.  When this method is used the problem is usually stated FV = PV x FVIF where FV and PV are as defined above and FVIFis the future value interest factor from a compound interest factor table at i interest rate for n periods.  For this problem we go to the 5% column and the 5 periods row and the factor at that spot is 1.276 so we solve the problem by FV = \$1,000 x 1.276 = \$1,276.  As you can see the table merely does the mathematical calculation of (1 + i)ⁿ and rounds a little bit sooner.

With the more and more common usage of calculators and computers tables like these are becoming less and less common.  On a financial calculator such as the Hewlett Packard 10B or 10BII the problem is solved by setting -

I/yr = 5

N = 5

PV = 1,000

PMT = 0

And solving for FV.

The answer one gets is \$1,276.28.  As you can see all three of these approaches are really doing the same thing, which is solving \$1,000 x 1.05 x 1.05 x 1.05 x 1.05 x 1.05.

# Present value

Present value is the amount a single sum due at some point in the future is worth today given a positive discount rate with compound discounting.  In other words, what one is doing here is going just the opposite direction in time from what was done with future value.  Here one goes from a known value at a point in the future to an unknown value at the present assuming there is a given interest rate during the entire period.  There is also the assumption that no amounts other than compound interest are added or deducted during the period.  Another way of looking at this is to ask “What amount is needed today to grow to \$X in n periods if interest at i interest rate is compounded each period?”  This is represented mathematically as PV = FV x (1 / (1+i)ⁿ) where all notations are defined the same as they were above.

What amount needs to be deposited today so that it can grow to \$1,276.28 in five years if the discount (interest) rate is 5%?  Using the above formula one gets PV = \$1,276.28 x (1/ 1.05) = \$1,276.28 x .783526 = \$1,000.  What this calculation shows is that \$1,000 today is the economic equivalent of \$1,276.28 to be received in five years if the going rate of interest is 5%.  It can be said that a person who feels that 5% is the appropriate interest rate per year for a 5-year investment will be indifferent between \$1,000 today or \$1,276.28 to be received in five years.

As with future value problems present value problems can also be solved using a compound interest (discount) table.  In the text the table is shown as Table B on pages 549 and 550 and it is referred to as the present value interest (or present value of a single sume) factor table.  A present value problem solved using the table is usually presented as PV = FV x PVIF where PV and FVare as previously defined and PVIF is the present value interest factor at i interest rate for n periods of time.  For this problem the factor in the 5% column at the 5 period row is .78353 so we have PV = \$1,276.28 x .78353 = \$1,000.  Again one can see that the use of this table merely does some mathematical calculation for you.  It solves 1 / (1 + i)ⁿ.

When using the HP 10B or 10BII financial calculator one can solve the problem by setting

N = 5

I/yr = 5

PMT = 0

FV = \$1,276.28

and solving for the PV.  The answer one gets is \$1,000.  As you can see all three of these approaches are really doing the same thing, which is solving \$1,276.28 x (1/1.05) x

(1/1.05)  x  (1/1.05)  x  (1/1.05) x  (1/1.05).

As another example of a present value problem determine what amount you would need today to be willing to forgo receiving \$1,500 three years from now if you can earn 7.5% interest per year.  From the information we know that the future value in three year’s time is \$1,500 so FV = \$1,500, N = 3, I/yr = 7.5, and Pmt = 0.  Using a calculator and solving for PV we get \$1,207.44.  In other word \$1,207.44 to be received today is the economic equivalent of \$1,500 to be received in three years if the interest rate is 7.5% per annum.

If you try to solve this problem using the PVIF from a present value table you’ll notice there is no 7.5% interest column.  There is a 7% and an 8% factor in the three period row so what you need to do is approximate the 7.5% factor for the three period row by a process called interpolation.  With interpolation you first determine the difference between the closest interest rate below and the next interest rate above the interest rate that you are looking for.  In this case the closest below is 7% and the next above is 8% so the difference between the two is 1%.  Then you determine what percentage of that difference needs to be added to the lower of the two rates to bring it to the rate whose factor you are trying to calculate.  In this problem we want 50% of the 1% difference.  Next you subtract the table factor for 7% from the table factor for 8%.  For this we get .79383 (the 8% factor)- .8163 (the 7% factor) = -.02247.  The difference between the two factors is multiplied by the percentage of the difference between the two rates that needs to be added to the lower of the rates.  In this case it is -.02247 x .5 =

-.011235.  The final step is to add this adjustment to the factor for the lower interest rate.  So finally we get .8163 + (-.011235) = .805065 which becomes the approximation of the factor for 7.5%.  We are now in a place to solve the problem.  We can say PV = FV x PVIF or PV = \$1,500 x .805065 = \$1,207.60.  After all this work notice that the answer is not the same as the one determined by using the calculator.  That is because interpolation is just a close approximation of the actual factor.  The actual factor in point of fact is .804961 and can be derived by using a financial calculator or by solving the equation PV = FV x 1 /(1+i)ⁿ.  Because using tables can be so time consuming if you must interpolate and because if you interpolate you get only an approximation you will be best served to have a financial calculator that you know how to use.

# Future value of an annuity

The future value of an annuity is the amount a series of equal deposits (or payments) at equal intervals will grow to over a period of time given a positive interest rate with compound interest.  If the deposits are made at the end of each period it is referred to as an ordinary annuity and if the deposits are made at the beginning of the period it is known as an annuity due.  We will look first at ordinary annuities.  As an example assume someone decides he/she is going to put a \$2,000 payment into an RRSP every year at the same time starting a year from now.  Further assume that the expected growth rate on the investment is 7%.  If one wants to know how many dollars the individual will have after a certain number of years then it is a future value of an ordinary annuity problem.  The formula for a problem like this is FV = A x [((1+i)ⁿ -1) / i] where FV = Future value at the end of n periods, A = amount of the annuity payment or deposit, i = interest rate per period, and n = the number of periods.  For the example presented earlier if we want to know the value in 25 years it becomes FV = \$2,000 x ((1.07) - 1) / .07 and then solving this it becomes \$2,000 x (5.427433 – 1) / .07 =  \$2,000 x 63.249038 = \$126,498.08.

A future value of an annuity problem that is solved using a table is presented as FV = A x FVIFA where FV, A, i, and n are all as previously defined and FVIFA is the future value interest factor for an annuity at i interest rate for n periods.   The appropriate table in the text is Table C on pages 551 and 552.  The use of the table is the same as was described above for the tables of future and present value.  The factor from the table for 25 periods at 7% interest is 63.249.  The future value of the \$2,000 annuity for 25 years at 7% interest is \$126,498 (\$2,000 x 63.249).  The slight difference from above is due to rounding.

On a financial calculator one can determine the answer to a future value of an annuity problem by setting

N = 25

I/yr = 7

PV = 0

PMT = \$2,000

and solving for FV.  The answer that results is \$126,498.08, which is just the same as the result obtained by using the mathematical equation.  Note: Financial calculators have a BEG/END function key.  When working with an ordinary annuity the function needs to be set on END.  That makes the calculator assume the annuity payment comes at the end of the periods.

Looking at the future value of an annuity problem slightly differently if you know the desired future value, the interest rate, and the time period then you can solve for the amount of the annuity necessary to get that desired future value.  If you ‘re 25 years old and you want to have \$250,000 set aside by the time you are age 40 so you can start your own business.  If you invest wisely you think a 9% return is possible.  Plugging this into the calculator we get

N = 15 (determined by 40 – 25)

I/yr = 9

PV = 0

FV = \$250,000

and then solving for PMT one gets \$8,514.72.  If your rate of return estimate is correct then if you invest just over \$8,500 each year for 15 years starting a year from now you’ll have \$250,000 at the end of the 15 years.

So far what has been presented relates to an ordinary annuity (one in which the deposits or payments are made at the end of the period).  The future value of an ordinary annuity can be converted to the future value of an annuity due (one where the deposits or payments are made at the beginning of the periods) by multiplying it by (1+i).  If we look at the above example where \$2,000 was invested each year for 25 years at an interest rate of 7% but now assume the deposits are made at the beginning of each period instead of at the end of each period then the amount of the future value becomes \$126,498.07 x 1.07 = \$135,352.95.  The difference is because each deposit earns one more year’s interest. If one is using the FVIFA from a table then just multiple that factor by (1+i) to convert it to an annuity due factor or if one is using a financial calculator make sure its BEG/END function is set on BEG (beginning).

Present value of an annuity

The present value of an annuity is the amount a series of equal payments (or deposits) made at equal intervals for a certain number of periods in the future are worth today given a positive discount rate with compound discounting.  What single dollar amount today is the economic equivalent of the series of equal payments to be received at equal intervals in the future is another way to view this problem.  If the payments (deposits) are made at the end of each period it is an ordinary annuity and if the payments are made at the beginning of each period it is an annuity due.  We first will deal with an ordinary annuity.  As an example assume an individual would like to have an amount today that will provide her/him with a \$20,000 payment starting a year from now and continuing each year for the next 15 years.  The interest rate during the entire time period will be 7%.  We solve for the amount by using the present value of an annuity formula, which is PV = A x [(1 - (1+ i)) / i].  Based on the above information the calculation becomes PV = \$20,000 x [(1 – (1.07)) / .07)] which then becomes \$20,000 x [(1 - .362446) / .07] or \$20,000 x 9.107914 = \$182,158.28.  Thus an individual needs to set aside \$182,158.28 today in an investment that earns 7% per year if he/she wants to receive \$20,000 per year starting a year from now.

Note: Taking something to the minus exponent (-n) means you are taking (1/(1+i)) x (1 / (1+i)) x (1 / (1+i)) for n times.  If there is an annuity of \$1 for three periods at 10% interest per period then its present value is \$1 x [(1 – (1 / 1.1) x (1 / 1.1) x (1 / 1.1)) / .1] or \$1 x [(1- (.909091 x .909091 x.909091)) / .1] =  \$1 x [(1-.751315) /.1] = \$1 x 2.486850 = \$2.486850.

Present value of annuity problems like future value, present value, and future value of annuity problems can be solved using interest factor tables or a financial calculator.  A problem such as this to be solved using interest factor tables is presented as PV = A x PVIFA.  The present value interest factor of an annuity at i interest rate for n periods (PVIFA) is taken from a Present Value of an Annuity table (in the text this is Table D on pages 553 and 554), using the same procedures as was described earlier.  Using the \$20,000 annuity example above one gets PV = \$20,000 x 9.1079 = \$182,158.  The factor 9.1079 is found by taking the number at the intersection of the 7% column and the 15 period row.

On a financial calculator one can determine the answer to a present value of an annuity problem by setting

N = 15

I/yr = 7

PMT = -\$20,000

FV = 0

and then solving for PV.  The answer that results is \$182,158.28, which is just the same as the result obtained by using the mathematical equation.

Notice when using most financial calculators the payment is set to a minus amount because it will be flowing from the present value sum.  If you don’t show the payment as a negative then the calculator will automatically show the present value as being negative instead of positive but the absolute amount will be okay.

1.      Given the market rate of interest is 6.5% per year which of the following is the economically superior?   \$12,000 to be received today, \$16,000 to be received 5 years from now, or \$3,000 per year for 5 years starting a year from now.  What is the value of each?

2.      You borrow \$25,000 to buy a vehicle.  Your loan carries a 9% per year compounded monthly interest rate (.75% per month).  You have 48 monthly payments starting a month from now.  How much is your monthly payment?  How much would it be if the payments started now rather than at the end of the month?

3.      You’re treasurer of an organization that needs to borrow \$50,000.  You’ve found three different financial institutions willing to lend you the funds.  A will lend you the money for three years but then wants a \$65,000 repayment.  B wants you to pay \$21,000 per year at the end of each of the next three years.  C wants you to pay \$4,500 per year interest at the end of each year and repay the principal at the end of three years.  Which loan is the financially better deal?

4.      Samantha Chen is 30 years old and she decides she is going to put \$2,000 per year from age 31 through age 65 inclusive into a retirement savings plan.  Past history shows that the plan has averaged a 7.5% per year return and Samantha feels it will continue to do so.  If the plan does as she expects how much will Samantha have in 35 years?  If she then wants to withdraw \$26,000 per year how long can she do so?  Assume the interest rate continues at 7.5%.

Solutions to the exercises are shown on the next page.  Try solving these on your own before checking the solutions.

Solutions to exercises

1.      The way to answer this one is to put all the alternatives on the same basis and then compare them to see which one is best.  The basis that is used for comparison is present value.  The present value of the first alternative is \$12,000 because it is that amount that will be received today.  The second alternative’s present value is \$11,678.09 which we get on the calculator by: PMT = 0; N = 5; I = 6.5; FV = 16,000 then solving for PV.  Alternative number three has a present value of \$12,467.04 determined on the calculator by: PMT = 3,000; N = 5; I = 6.5; FV = 0 then solving for PV.  The best choice is alternative three because it has the highest present value.

2.      What you are trying to find out here is the amount of the monthly payment you will have to make so on the calculator you will be solving for Pmt.  If the payments are made starting a month from now then that is an ordinary annuity and the calculator is set on END; N = 48; I = .75; PV = -25,000; FV = 0; then PMT becomes \$622.13.  If one changes END to BEG and leaves the rest the same then PMT is \$617.49, which is the amount of the payment if they start now rather than a month from now.

3.      To find out which is the financially best deal one can find the effective annual interest rate you are being charged under each of the alternatives.  With alternative A the interest rate you are being charged is found by: PV = -50,000; FV = 65,000; PMT = 0; N= 3; then solve for I and get 9.139288%.  For B it is PMT = 21,000; PV = -50,000; FV = 0; N = 3; then I becomes 12.509636%.  For alternative C it is PMT = 4,500; PV = -50,000; FV = 50,000; N = 3; then I = 9%.  Alternative C is the one where you are charged the lowest interest rate.

4.      In this exercise you are first trying to determine the future value of \$2,000 a year for 35 year if it is being invested at 7.5%.  To determine that you have I = 7.5; N = 35; PMT = 2,000; PV = 0: then solve for FV, which equals \$308,503.21.  Then you’re asked to given that amount how many years can she withdraw \$26,000 per year.  That is calculated by setting PV = -308,503.21; I = 7.5; FV = 0; PMT = 26,000; then solving for N and you get 30.5 years.