Calculate it here!
n = number of years in projection i = compound interest rate V_{n} = Value at year n V_{0} = Value at year 0 

The Present Value of
a
Single Sum
The Introduction to
Interest
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 (V_{0}) 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 = $11,000Could 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)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.
= $10,000 x (1 + i)
= $10,000 x (1.10)
= $11,000
= $11,000 x (1 + i)And the third year?
= $11,000 x (1.10)
= $12,100
$12,100 x (1.10) = $13,310Notice 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:
V_{n} = (Principle) x (interest 1st year) x (Interest 2nd Year)
We say, "The value at year n" for V_{n}
V_{n} = ($10,000) x (1 + 0.10) * (1 + 0.10)
V_{n} = ($10,000) x (1 + 0.10)^{2}
V_{n} = ($12,100)
If we looked at the 3rd year we would
see that the function is the same:
V_{n} = ($10,000) x (1 + i)^{3}So to extrapolate from our observations, we could say the following:
V_{1} = V_{0} + V_{0} x i factor out V_{0}In the second year:
V_{1} = V_{0} (1 + i)
V_{2} = V_{1} + (V_{1} x i)
V_{2} = V_{0} (1+i) + V_{0}(1+i) x i : (next, factor out V0(1+i) and we get)
V_{2} = V_{0} (1+i) (1+i)
V_{2} = V_{0} (1+i)^{2}
In the third Year:
V_{3} = V_{0} (1+i) (1+i) (1+i)
V_{3} = V_{0} (1+i)^{3}
So now we can generalize to the following:
V_{n}= V_{0}(1+i)^{n}where:
n = number of years in projection
i = compound interest rate
V_{n} = Value at year n
V_{0} = 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.
What is the value of my savings account at Bank of America that started at $5,000 and earns 5% interest after 7 years? V_{n} = V_{0}(1+i)^{n} 
We can use this formulae which solves
for
the future value and solve instead for the present value. Simply move
the
variables around…
V_{n} = V_{0}(1+i)^{n}
Divide both sides by (1 + i)^{n}
V_{0} = V_{n }/ (1+i)^{n}
The Present value of a Future Single Sum
At 7% interest, what is the present value of a $10,000 bond that will mature in 5 years? V_{0} = V_{n }/ (1+i)^{n} 
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