Present Value of a Single Sum
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Calculate it here!

 V0 = Vn / (1+i)n
where:
n = number of years in projection
i = compound interest rate
Vn = Value at year n
V0 = Value at year 0
Vn (Future Value) $
i (0.00 format) 
n (number of years) 

V0 (Present Value) $


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The Present Value of a Single Sum
The Introduction to Interest

We have discussed the general concept of interest (i) and know that it represents the time value of money; our preference for current consumption versus future consumption. How is it used in practice? To demonstrate how this works, we will actually solve for the Future Value of a Single Sum and then convert it into a Present Value.

We can start with a very simple exercise of considering $10,000 placed in a Bond Fund and receiving 10% interest.

To determine the value of the Bond at the end of a year we would multiply the principal (V0) by the interest rate (i)

 $10,000 x 10% = $1,000 (the amount of interest paid in one year)
Now we add the principle back to the interest and we get our total
 $10,000 + $1,000 = $11,000
Could we do this all in one step, calculate the interest and add the principle back to the total? YES. We would combine the steps to look like this:
= ($10,000) + ($10,000 x i)
= $10,000 x (1 + i)
= $10,000 x (1.10)
= $11,000
Great, so we know how much it is worth after one year in the Bond. How about the next year? We could do the same thing.
= $11,000 x (1 + i)
= $11,000 x (1.10)
= $12,100
And the third year?
$12,100 x (1.10) = $13,310
Notice that we are dealing with compound interest. That means that the interest we generate on the investment generates interest in future years. Simple interest is generated but not used in future years to generate more interest. This might be seen in instances when the interest is pulled out of the account each year, like in a retirement account that has matured. I retire and live off the interest. Therefore from one year to the next the "basis" of the account stays the same. In most problems we deal with we will consider compound interest. Now, back to our flow of revenue from our account. This is good, but we need another way to calculate the income stream from our investment because 20 years becomes tiresome to calculate this way. Look closer at the two year calculation:
Vn = (Principle) x (interest 1st year) x (Interest 2nd Year)
We say, "The value at year n" for Vn
Vn = ($10,000) x (1 + 0.10) * (1 + 0.10)
Vn = ($10,000) x (1 + 0.10)2
Vn = ($12,100)


If we looked at the 3rd year we would see that the function is the same:

Vn = ($10,000) x (1 + i)3
So to extrapolate from our observations, we could say the following:
V1 = V0 + V0 x i    factor out V0
V1 = V0 (1 + i)
In the second year:
V2 = V1 + (V1 x i)
V2 = V0 (1+i) + V0(1+i) x i   : (next, factor out V-0(1+i) and we get)
V2 = V0 (1+i) (1+i)
V2 = V0 (1+i)2


In the third Year:

V3 = V0 (1+i) (1+i) (1+i)
V3 = V0 (1+i)3


So now we can generalize to the following:

 Vn= V0(1+i)n
where:
n = number of years in projection
i = compound interest rate
Vn = Value at year n
V0 = Value at year 0


This formulae forms the basis of all financial decision making formulae we encounter in finance. With this basic proof of the calculation, we can begin to manipulate it and use it to determine other calculations. We call this formulae the "Future Value of a Single Sum" it is also sometimes called "Net Future Value". We might use this formulae to determine the value of an investment at some time in the future where we know the interest rate.
 

Example

What is the value of my savings account at Bank of America that started at $5,000 and earns 5% interest after 7 years?

Vn = V0(1+i)n
Vn = $5,000 (1.05)7
Vn = $5,000 (1.4071)
Vn = $7,035.50

We can use this formulae which solves for the future value and solve instead for the present value. Simply move the variables around…
 

Vn = V0(1+i)n
Divide both sides by (1 + i)n
V0 = Vn / (1+i)n
The Present value of a Future Single Sum
Example

At 7% interest, what is the present value of a $10,000 bond that will mature in 5 years?

V0 = Vn / (1+i)n
V0 = $10,000 / (1+0.07)5
V0 = $10,000 / (1.07)5
V0 = $10,000 / (1.4025)
V0 = $7,129.86

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Kamiak Econometrics, a Division of Kamiak Ridge, LLC