Midterm 1 Time Shifting

Consider an asset that pays $30 in month one and increases by $30 per month until month 20, when you receive $600 per month. The asset continues to pay $600 per month until month 35. Your monthly discount rate is 2%.

How, using factor notation, would you represent the present value of
benefits in months one through 20?

(a) 30(P|G, i=2%,21)/(1+.02)

(b) 30(P|G, i = 2%, 21)(1 + .02)

(c) 30(P|G, i = 2%, 21)

(d) 30(P|G, i = 2%, 20)

(e) None of the above.

The answer to this question is (b) because we have $30 in month one and we find the present worth knowing the gradient is increasing by $30 a month. So it looks like this 30(P|G, i = 2%, 21). This alone describes a linear gradient series that has non-negative values from time period 2 through 21. We have to time shift to make the values go from 1 to 20. Option b makes the proper shift. Note that adding the N, 21, and the exponent for the denominator, -1, recalling that $(1+.02)$ is equal to $\frac{1}{(1+.02)^{-1}}$, yields that last time period you observe the cash flow, 20.

Option A describes a benefit that starts out with $30 in period 3 and increases by $30 until period 22.

Option C describes a benefit that starts out with $30 in period 2 and increases by $30 until period 21.

Option D describes a benefit that starts out with $30 in period 2 and increases by $30 until period 20.

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