Present Value of a Single Amount

I. Present Value Formula

The present value of a single future payment is the amount that the payment is worth today. Finding the present value is called discounting, which is the reverse of compounding. 

Whenever we need to find the present value of a future amount, we can use the future value formula, just rearranged. Take our future value formula,

FV=PVx(1+i)ⁿ

We can rearrange this future value formula into the present value formula

PV=FV/(1+i)ⁿ

Where:

PV = Present value amount. 

FV = Future value amount

n = Number of time periods that interest will be added and compounded over the life of the deposit, cost, etc. 

i = Periodic interest rate aka discount rate (annual rate adjusted for number of compounding periods per year. For example, if interest is to be compounded monthly, then a rate of 12% per year will be restated to be 1% per month.)

Example 1

You buy a refrigerator for $800 but you don’t have to make the payment until next year (that is, one year later). The opportunity cost of money is 3%. Opportunity cost represents what you could earn with the money – in this case, the return on your best investment opportunity. What price are you paying for the refrigerator in present dollars?

PV=$800/(1+0.03)¹

PV=$800/1.03¹

PV=$800/1.03

PV=$776.70

Example 2

Lets say you can defer payment on the refrigerator for an additional year.

PV=$800/(1+0.03)²

PV=$800/1.03²

PV=$800/1.0609

PV=$754.08

Multiple Compounding Periods In A Year

As we discussed in the compound interest/future value of a single amount section, make sure your i and n amounts are correct. 

Example 3

What is the present value of receiving a single amount of $10,000 at the end of five years, if the time value of money is 6% per year, compounded semiannually?

Here n = 10, because there are 10 six-month (or semiannual) periods in five-years time (5 years x 2 semiannual payments per year). Because the compounding occurs semiannually, the rate for discounting is i = 3% per six-month period (the annual rate of 6% divided by the two semiannual periods in each year).

PV=$10,000/(1+0.03)¹⁰

PV=$10,000/(1.03)¹⁰

PV=$10,000/1.3439164

PV=$7,440.94

Here's another version of the same present value formula with slightly different variables. This one is designed to be fool proof since it figures the periodic interest rate for you (r/n) as well as the correct exponent (nt). 

Formula for Present Value

The present value formula is:

where	P = present value
	A = desired future amount
	r = nominal interest rate (as a decimal fraction)
	n = number of times interest is calculated in one year
	t = times (in years)

EXAMPLE 4

A house painting company is planning to expand its operations in three years time. It will require $24,000 in order to expand. How much must it invest now, at 4.6% interest compounded annually?

Solution

P = ? A = $24000 r = 4.6% = 0.046 n = 1
t = 3 years		Replace the variables with their values
		Add
		Raise

The present value is $20,970.86 so the company must invest that amount now to have $24,000 in three years.

II. Discount Factor

A discount factor is a decimal number multiplied by a cash flow value to discount it back to its present value. The discount factor can be derived from the net present value formula above. 

1. Net Present Value Formula

PV = FV/(1 + r)ⁿ

2. Make FV = 1, and change PV to DF (Discount Factor)

DF = 1/(1 + r)ⁿ

Example: 

We will use example 2 from above. The discount rate for that problem is 3%.

DF = 1/(1 + 0.03)²

DF = 0.9426

Now multiply the future value of $800 by the discount rate.

$800 x 0.9426 = $754.08

This matches the answer from example 2.

https://efficientminds.com/wp-content/uploads/2012/08/web_chapter_28_tvm.pdf

Microsoft Word - 4 - The Time Value of Money

https://www.accountingcoach.com/present-value-of-a-single-amount/explanation/3

https://opentextbc.ca/businesstechnicalmath/chapter/9-2-compound-interest-2/

https://www.wallstreetprep.com/knowledge/discount-factor/

Corporatefinanceinstitute: Discount Factor