## Present Value Of A Single Amount

Present Value is an amount today that is equivalent to a future payment, or series of payments, that has been discounted by an appropriate interest rate.  Since money has time value, the present value of a promised future amount  is worth less the longer you have to wait to receive it.   The difference between the two depends on the number of compounding periods involved and the interest (discount)  rate.

The relationship between the present value and future value can be expressed as:

 PV = FV [ 1 / (1 + i)n ]

Where:

PV = Present Value
FV = Future Value
i = Interest Rate Per Period
n = Number of Compounding Periods

Example 1: You want to buy a house 5 years from now for \$150,000. Assuming a 6% interest rate compounded annually, how much should you invest today to yield \$150,000 in 5 years?

FV = 150,000
i =.06
n = 5

 PV = 150,000 [ 1 / (1 + .06)5 ] =  150,000 (1 / 1.3382255776) = 112,088.73

 End of Year 1 2 3 4 5 Principal 112,088.73 118,814.05 125,942.89 133,499.46 141,509.43 Interest 6,725.32 7,128.84 7556.57 8,009.97 8,490.57 Total 118,814.05 125,942.89 133,499.46 141,509.43 150,000.00

Example 2: You find another financial institution that offers an interest rate of 6% compounded semiannually.  How much would you deposit today to yield \$150,000 in five years?

Interest is compounded twice per year so you must divide the annual interest rate by two to obtain a rate per period of 3%.  Since there are two compounding periods per year, you must  multiply the number of years by two to obtain the total number of periods.

FV = 150,000
i = .06 / 2 = .03
n = 5 * 2 = 10

 PV = 150,000 [ 1 / (1 + .03)10] = 150,000 (1 / 1.343916379) = 111,614.09

Exercise: Find PV for the above data if compounding is monthly.

##### Present Value of Annuities

An annuity is a series of equal payments or receipts that occur at evenly spaced intervals. Leases and rental payments are examples.  The payments or receipts occur at the end of each period for an ordinary annuity   while they occur at the beginning of each period for an annuity due.

##### Present Value of an Ordinary Annuity

The Present Value of an Ordinary Annuity (PVoa) is the value of a stream of expected or promised future payments that have been discounted to a single equivalent value today.  It is extremely useful for comparing two separate cash flows that differ in some way.

PV-oa can also be thought of as the amount you must invest today at a specific interest rate so that when you withdraw an equal amount each period, the original principal and all accumulated interest will be completely exhausted at the end of the annuity.

The Present Value of an Ordinary Annuity could be solved by calculating the present value of each payment in the series using the present value formula and then summing the results. A more direct formula is:

 PVoa = PMT [(1 – (1 / (1 + i)n)) / i]

Where:

PVoa = Present Value of an Ordinary Annuity
PMT = Amount of each payment
i = Discount Rate Per Period
n = Number of Periods

Example 1: What amount must you invest today at 6% compounded annually so that you can withdraw \$5,000 at the end of each year for the next 5 years?

PMT = 5,000
i = .06
n = 5

 PVoa = 5,000 [(1 – (1/(1 + .06)5)) / .06] = 5,000 (4.212364) = 21,061.82

 Year 1 2 3 4 5 Begin 21,061.82 17,325.53 13,365.06 9,166.96 4,716.98 Interest 1,263.71 1,039.53 801.90 550.02 283.02 Withdraw -5,000 -5,000 -5,000 -5,000 -5,000 End 17,325.53 13,365.06 9,166.96 4,716.98 .00

Example 2: In practical problems, you  may need to calculate both the present value of an annuity (a stream of future periodic payments) and the present value of a single future amount:

For example, a computer dealer offers to lease a system to you for \$50 per month for two years.  At the end of two years, you have the option to buy the system for \$500.  You will pay at the end of each month.  He will sell the same system to you for \$1,200 cash.  If the going interest rate is 12%, which is the better offer?

You can treat this as the sum of two separate calculations:

1. the present value of an ordinary annuity of 24 payments at \$25 per monthly period Plus
2. the present value of \$500 paid as a single amount in two years.

PMT = 50 per period
i = .12 /12 =  .01    Interest per period
(12% annual rate / 12 payments per year)
n = 24 number of periods

PVoa = 50 [ (1 – ( 1/(1.01)24)) / .01] = 50 [(1- ( 1 / 1.26973)) /.01] = 1,062.17

+

FV = 500 Future value (the lease buy out)
i = .01 Interest per period
n = 24 Number of periods

PV = FV [ 1 / (1 + i)n ]   = 500 ( 1 / 1.26973 ) =  393.78

The present value (cost) of the lease is \$1,455.95 (1,062.17 + 393.78). So if taxes are not considered, you would be \$255.95 better off paying cash right now if you have it.