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Present Value
The time value of money principle says that future dollars are not worth as much as dollars today.
You should be able to explain why! It is extremely important and influences almost everything we do from now on.
In the vernacular (I love that word) what this means is that you are unwilling to make an interest-free loan. Fortunately we can compare present and future values with a rather simple equation.
(1) 
This will give you the present value of a single future cash flow (FV) . In fact for ease down the road we will generally use CF instead of FV. Future Value (FV) will be reserved for when we are actually solving for a future value. (For example how much will we have in 5 years). A simple Present Value example follows:
What is the present value of $8,000 to be paid at the end of three years if the correct (risk adjusted interest rate) is 11%?
(2) 
= 8,000/(1.11)3
= 8,000/1.36
=$5,849
Note that if you had so desired you could write this equation as
(3) PV = CF * (1/(1+r)t )
Which would be:
PV = 8,000 * (1/1.11)3)
=8,000 * .7312
= $5,849
The second term in equation 3, (1/(1+r)t ), is known as the present value discount factor or present value interest factor. It is usually abbreviated PVIF(r%, N periods). You can find this number either mathematically or from present value tables. Specifically this is the present value of a dollar and can be found on table A-1. (Note the higher the required interest rate, i.e. the more risk, the lower the present value.)
Continuing our example, suppose that you were willing to make a loan where you would get $8,000 back at the end of the third year, and $10,000 at the end of the fourth year. What is the present value of this?
Correct, you find the present value of each cash flow and then add the present values. Thus,
PV = 8,000/(1.11)3 + 10000/(1.11)4
= $12,436.84
Generically, we can thus rewrite equation #2 as:
(4) 
Whereby we calculate the present value of each cash flow and then sum the present values. As you can imagine this can get quite cumbersome if we had many future cash flows. As a result many short cuts have been devised. Chief among these is when all of the cash flows are identical. This we call an annuity. When we have an annuity we do not need to add up each individual value but can use the present value (and later future value) tables.
Annuity: A series of equal cash flows that begin at the end of one period.
Examples of annuities include loan payments and certain long term contracts such as pensions, leases, and certain sports contracts.
Example what is the present value of an annuity of $250 a year at the end of year for 6 years if interest rates are 12%?
To solve this we could add each individual present value up
(so calculate the Present Value of each cash flow individually)
or
can use the following discount factor and then multiply by the cash flow.
(5) PVIFA(r,n)=PVAF(r,n)= 
Thus if interest rates are 12% and you will receive 6 payments, the discount factor is 4.114. Thus the Present Value (PV) of 6 payments of $250 if interest rates are 12% is
PV = PVAF(r,n) * CF
= 4.114 * $250
= $1,028.50
This answer will be the same whether you solve the problem mathematically, as we just did, or using the table A2. (try it!)
An important assumption in using the annuity discount factors is that the cash flows occur at the END of each year. If the cash flows are occurring at the beginning of each year, the cash flows are called an annuity-due.
Stop and think for a second what we are doing in an annuity due. The first cash flow occurs today. Thus the present value of the first cash flow is equal to the cash flow. One year from now you will receive another cash flow. This second cash flow occurs at the same time (or technically 1 day later) than the first cash flow of a regular annuity. To the present value of an annuity due is
(6) PV = CF + PVAF(r,n-1) * CF
Using the above example but assuming the first payment is made today (rather than in one year). we can value the cash flows using the annuity-due equation.
PV = 250 + PVAF(12%, 5)* CF
= 250 + 3.6048 * 250
= $1,151.20
Note that the present value is greater than before. Why? Because the payments were all shifted up one year, thus allowing you (the lender) to reinvest sooner and make more money. Alternatively if you were the borrower you are paying earlier so you lose interest that you could have earned by keeping your money invested.
Another shortcut that you will be responsible for is a perpetuity. A perpetuity is a stream of cash flows that goes on forever. Or at least we assume it does. This may or may not be a good assumption (why?-forever is a very long time!) but is very easy to use.
(7) PV of a perpetuity = CF/r
Note the cash flow does not have a subscript. Why? Because the definition of a perpetuity says that all cash flows are identical.
Example: To pay for a new highway the local government sells a perpetuity that promises to pay $1000 a year from now until the end of time. If interest rates are 10%, what is the most that you would be willing to pay to get these future cash flows?
PV = CF / r
= $1000 / (.1)
=$10,000
An interesting (this is interesting, right?!?!?!) extension of this is that you can also value a growing perpetuity. This is a series of cash flows that grows at a constant rate. For example suppose that the above perpetuity were promised to grow by 4% per year. Thus this year you get $1000, the next year you get 1000(1.04)=$1,040 etc.
PV=CF1 / (r-g) where CF1 is the cash flow you will receive in one year.
Example: suppose you JUST received $1000 and now want to sell your growing perpetuity. What is the least you should accept for the claim on these future cash flows if the growth rate is 4% and the correct risk adjusted interest rate is 10%.
PV =
= $1,040/(.1-.04)
= $17,333
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