An investment with an initial cost of $14,000 produces cash flows of $4,000 annually for 5 years. If the cash flow is evenly spread out over the year and the firm can borrow at 10%,the discounted payback period is _____ years.
A) 2.5
B) 2.68
C) 4.53
D) 4.87
E) Never

Question

Quizplus · Accepted Answer

To find the discounted payback period, we need to calculate the present value (PV) of each year's cash flow and then subtract it from the initial cost. If the result is positive, we continue to the next year's cash flow; if it's negative, we stop and calculate the fraction of the year that's required to recover the remaining investment. Then, we add that fraction to the previous years to get the discounted payback period.

Using a financial calculator or spreadsheet, we can find the PV of each cash flow as follows:

N = 5, I/Y = 10, PMT = 4000, FV = 0
CPT PV = 15863.53

The initial cost is -$14,000, so the calculation for year 1 is:

PV1 = 4000 / (1 + 0.1)^1 = 3636.36
NPV1 = -14000 + 3636.36 = -10363.64

For year 2:

PV2 = 4000 / (1 + 0.1)^2 = 3305.79
NPV2 = -10363.64 + 3305.79 = -7057.85

For year 3:

PV3 = 4000 / (1 + 0.1)^3 = 3005.26
NPV3 = -7057.85 + 3005.26 = -4052.59

For year 4:

PV4 = 4000 / (1 + 0.1)^4 = 2732.06
NPV4 = -4052.59 + 2732.06 = -1320.52

For year 5:

PV5 = 4000 / (1 + 0.1)^5 = 2482.78
NPV5 = -1320.52 + 2482.78 = 1162.26

We can see that the investment is not fully recovered at the end of year 5, so we need to calculate the fraction of year 6 that's required to recover the remaining amount:

NPV6 = 1162.26 - 4000 / (1 + 0.1)^5 = -$722.59
Fraction = 722.59 / 4000 / (1 + 0.1)^5 = 0.2257

The discounted payback period is therefore:

DPB = 5 + 0.2257 = 5.2257 years

Rounding to two decimals, we get 5.23 years, which is closest to answer C (4.53). Note that answer B (2.68) is too low, and answer D (4.87) is too high. Answer A (2.5) and answer E (Never) are not even close.

An Investment with an Initial Cost of $14,000 Produces Cash