Considering the time value of money and assuming annual compounding with i = 10% per year, what is the payback period for the following project?

Question

Considering [latex]the time value of money[/latex] and assuming annual compounding with [latex]i = 10\%[/latex] per year, what is the payback period for the following project?

4	3	2	1	0	Year
$2,928	$2,662	$2,420	$2,200	-$5,000	Cashflow

Accepted Answer

Since the $5,000 investment is made now, its time-value with reference to year 0 remains unaffected. As required, the cash inflows should account for the time value of money. The first-year income of $2,200 gets modified as:

[latex]P=2,200({P}/{F},10\%,1)[/latex]

[latex]=2,200 imes 0.9091[/latex]

=2,000

Thus, by the end of the first year, $2,000 of the investment is recovered. The investment yet to be recovered = $5,000 – $2,000 = $3,000.

For the second year, the $2,420 income gets modified as:

[latex]P=2,420({P}/{F},10\%,2)[/latex]

[latex]=2,420 imes 0.8264[/latex]

=2,000

Thus, by the end of the second year, the remaining investment yet to be recovered = $3,000 – $2,000 = $1,000.

For the third year, the $2,662 income gets modified as

[latex]P=2,662({P}/{F},10\%,3)[/latex]

[latex]=2,662 imes 0.7513[/latex]

=2,000

Since the third-year time-valued income of $2,000 is more than the amount to be recovered ($1,000), the payback period is under 3 years. The interpolation between years 2 and 3 is straightforward since the amount to be recovered at the end of second year ($1,000) is exactly half of the recovery during the third year ($2,000). Thus, the payback period is 2.5 years. Alternatively.

Payback period [latex]=2+{1000}/{2000}[/latex]

=2+0.5

=2.5 years

Note that in this example the given incomes were variable (increasing at 10% each year—the same as the interest rate i). The consideration of time value changed these incomes, rendering them uniform at $2,000 each year. We could therefore have used the formula, Equation (5.1), to evaluate the payback period as:

Payback period [latex]={Investment}/{Income per year}[/latex]

[latex]={\$5,000}/{(\$2,000 per year)}[/latex]

=2.5 years

Question 5.7: Considering the time value of money and assuming annual com...

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