__Present and future value of streams__

__Present and future value of streams__

Previously we discussed the impact of interest on a single cash deposit or loan; that is, in a single cash flow. We now extend that discussion to the case where cash flows occur at several time periods, and hence constitute a cash flow stream or sequence.

**The Ideal Bank**

An ideal bank applies the same rate of interest to both deposits and loans, and it has no service charge or transaction fees. Its interest rate applies equally to any size of principal, from 1 cent (or fraction thereof) to $1 million (or even more). Furthermore, separate transactions in an account are completely additive in their effect on future balances.

Note that the distinction of an ideal bank *doesn’t *imply that interest rates for all the transactions are identical. For example, a 2-year certificate of deposit (CD) might offer a higher rate than 1-year CD. However, the 2-year (CD) must offer the same rate as a loan that is payable in 2 years.

If an ideal bank has an interest rate that is independent of the length of time for which it applies, and that interest is compounded according to normal rules, it is said to be a **constant ideal bank.** For simplicity we will assume that interest rates are indeed constant.

**Future Value**

Now we return to the study of cash flow streams. Let us decide on a fixed time cycle of compounding ( for example, yearly ) and let a period be the length of this cycle. We assume that cash flows occur at the end of each period (although some flows might be zero). We shall take each cash flow and deposit it in a constant ideal bank as it arrives. (If the flow is negative, we cover it by taking out a loan. ) Under the terms of a constant ideal bank, the final balance in our account can be found by combining the results of the individual flows. Explicitly, consider the cash flow stream \(\left( x_{0},x_{1},x_{2}………x_{n} \right)\). at the end of *n* periods, the initial cash flow \(x_{0}\) will have grown to \(x_{0}\left( 1+r \right)^{n}\), where \(r\) is the interest rate *per period* (which is the yearly rate divided by the number of periods per year). The next cash flow, \(x_{1}\), received after the first period, will at the final time have been in the account for only *n-1 *periods, and have a value of \(x_{1}\left( 1+r \right)^{n-1}\). Likewise, the next flow \(x_{2}\)_{ }will collect interest during *n-2 *periods and have value \(x_{2}\left( 1+r \right)^{n-2}\). The final flow \(x_{n}\) will not collect any interest, so will remain \(x_{n}\). The total value at the end of *n* periods is therefore

\(FV\; =\; x_{0}\left( 1+r \right)^{n}\; +\; x_{1}\left( 1+r \right)^{n-1}\; +…….+x_{n}\)

**Future Value of a stream:***Given a cash flow stream \(\left( x_{0},x_{1},x_{2}………x_{n} \right)\) and interest rate r each period, the future value of the stream is*

\(FV\; =\; x_{0}\left( 1+r \right)^{n}\; +\; x_{1}\left( 1+r \right)^{n-1}\; +…….+x_{n}\)

**Present Value **

The present value of a general cash flow stream like the future value can also be calculated by considering each flow element separately. Again consider the stream \(\left( x_{0},x_{1},x_{2}………x_{n} \right)\). the present value of the first element \(x_{0}\) is just that value itself since no discounting is necessary. The present value of the flow \(x_{1}\) is \(\frac{x_{1}}{\left( 1+r \right)}\), because that flow must be discounted by one period (again the interest rate \(r\) is the per-period rate.). Continuing this way, we find that the present value of the entire stream is

\(PV\; =\; x_{0}+\frac{x_{1}}{\left( 1+r \right)}+\frac{x_{2}}{\left( 1+r \right)^{2}}+……….\frac{x_{n}}{\left( 1+r \right)^{n}}\; \)

We can summarize this important result as follows:

**Present Value of a stream:***Given a cash flow stream \(\left( x_{0},x_{1},x_{2}………x_{n} \right)\) and an interest rate r per period, the present value of this cash flow stream is*

\(PV\; =\; x_{_{0}}\; +\; \frac{x_{_{1}}}{\left( 1+r \right)}\; +\; \frac{x_{_{2}}}{\left( 1+r \right)^{2}}\; +……………\frac{x_{_{n}}}{\left( 1+r \right)^{n}}\; \)

The present value of a cash flow stream can be regarded as the present payment amount that is equivalent to the entire stream. Thus we can think of the entire stream as being replaced by a single flow at the initial time.

There is another way to interpret the formula for the present value that is based on transforming the formula for future value. Future value is the amount of future payment that is equivalent to the entire stream. We can think of the stream as being transformed into that single cash flow at period *n*. The present value of this single equivalent flow is obtained by discounting it by \(\left( 1+r \right)^{n}\). that is, the present value and the future value are related by

\(PV\; =\; \frac{FV}{\left( 1+r \right)^{n}}\)

**Present Value and an Ideal Bank**

In general, if an ideal bank can transform the stream \(\left( x_{0},x_{1},x_{2}………x_{n}\right)\) into the stream \(\left( y_{0},y_{1},y_{2}………y_{n}\right)\), it can also transform in the reverse direction. Two streams that can be transformed into each other are said to be __equivalent streams. __

How can we tell whether two given streams are equivalent? The answer is the main theorem on present value.

*Main theorem on present value**The cash flow streams \(\left( x = x_{0},x_{1},x_{2}………x_{n} \right)\) and \(\left( y = y_{0},y_{1},y_{2}………y_{n}\right)\) are equivalent for a constant ideal bank with interest rate r if and only if the present values of the two stream, evaluated at the bank’s interest rate, are equal.*

This is important because it implies that present value is the only number needed to characterize a cash flow stream when an ideal bank is available. The stream can be transformed in a variety of ways by the bank, but the present value remains the same. So if someone offers you a cash flow stream, you only need to evaluate its corresponding present value, because you can then use the bank to tailor the stream with that present value to any shape you desire.

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