### Multi-period Future Value and Present Value

#### Multi-period Future Value

Building on the single-period case, it is easy to find the future value of a cash flow several periods away. We need to apply the interest factor (1 + r) for every period that interest is accrued.

One-period case: Future Value = C0 * (1 + r)
If we want to find the value after two periods, we just plug in the right side of the equation above for C0:
FV = [C0 * (1 + r)] * (1 + r)
This can be rewritten as:
FV = C0 * (1 + r) * (1 + r)
or
FV = C0 * (1 + r)^2
Note: I use carrots "^" to show a power. In other words, x^2 would be read as "x squared".

Now that we have a two period case, it is easy to see how this formula can be carried out several periods:

FV = C0 * (1 + r)1 * (1 + r)2 * ... * (1 + r)T
or
FV = C0 * (1 + r)^T
Where r is the interest rate,
C0 is the cash flow at time 0 (now),
FV is the future value, and
T is the number of periods.

#### Example

Problem: You are given \$10,000 and you want to invest it in a 5 year CD that yields 4% interest per year. How much money will you have in 5 years?

FV = C0 * (1 + r)^5
FV = \$10,000 * (1 + .04)^5
FV = \$12,166.53

#### Multi-period Present Value

We use the same methodology to find a multi-period present value, beginning with the single-period present value formula, then adding a discount for each additional period:

Present Value = C1 / (1 + r)1
PV = C2 / [(1 + r)1 * (1 + r)2 PV = CT / [(1 + r)1 * (1 + r)2 * ... * (1 + r)T]
PV = CT / (1 + r)^T

#### Example

Problem: You are offered a savings bond that will yield 6% interest for 20 years. The bond will not make any payments until the 20 years are up, then it will give a lump sum of \$100,000. How much will it cost today?

PV = CT / (1 + r)^T
PV = \$100,000 / (1 + .06)^20
PV = \$31,180.47

#### Note on Variable Interest

In the formulas above we use constant interest and constant discounting for all the time periods. It is easy to change this assumption, but we cannot simplify the formulas as much. Consider interest rates that differ each year: r1 for period 1, r2 for period 2, etc. This would be like getting 4% interest in the first year, then 7% interest the next year.

Our new formulas are:
FV = C0 * (1 + r1) * (1 + r2) * ... * (1 + rT)
PV = CT / [(1 + r1) * (1 + r2) * ... * (1 + rT)]

With this simple knowledge of interest and discounts and how they relate to future value and present value, we can move on to more difficult problems and formulas.

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