( 1 + i)-n-(1+ni) i[(1+i)"n-1]


Gradient Series, Uniform Series Factor

Analysis Approach 1: Compare worths at t=0 (present worth)

PW(1) = 1,322.50*(PIF,15,2) = 1322.50*0.756147 = 1,000 PW(2) = 1,000

Answer: Cash Flow 1 and Cash Flow 2 are equivalent

Analysis Approach 2: Compare worths at t=2 (future worth) FW(1) = 1,322.50

FW(2) = 1,000*(FIP,15,2) = 1,000*1.3225 = 1,322.50 Answer: Cash Flow 1 and Cash Flow 2 are equivalent

Generally the comparison (hence the determination of equivalence) for the two cash flow series in this example would be made as present worths (t=0) or future worths (t=2), but the equivalence definition holds regardless of the point in time chosen. For example:

Analysis Approach 3: Compare worths at t=1 W1(1) = 1,322.50*(P I F,15,1)

= 1,322.50*0.869565 = 1,150.00 W1(2) = 1,000*(FIP,15,1) = 1,000*1.15 = 1,150.00

Answer: Cash Flow 1 and Cash Flow 2 are equivalent

Thus, the selection of the point in time, t, at which to make the comparison is completely arbitrary. Clearly however, some choices are more intuitively appealing than others (t= 0 and t=2 in the above example).

In economic analysis, "indifference" means "to have no preference" The concept is primarily applied in the comparison of two or more cash flow profiles. Specifically, a potential investor is indifferent between two (or more) cash flow profiles if they are equivalent.

Question: Given the following two cash flows at 15%/yr which do you prefer? Cash Flow 1: Receive $1,322.50 two years from today Cash Flow 2: Receive $1,000.00 today Answer: Based on the equivalence calculations above, given these two choices, an investor is indifferent.

The concept of equivalence can be used to break a large, complex problem into a series of smaller more manageable ones. This is done by taking advantage of the fact that, in calculating the economic worth of a cash flow profile, any part of the profile can be replaced by an equivalent representation without altering the worth of the profile at an arbitrary point in time.

Question: You are given a choice between (1) receiving P dollars today or (2) receiving the cash flow series illustrated in Figure 4.5. What must the value of P be for you to be indifferent between the two choices if i=12%/yr?

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