Question

A new machine will cost $220,000 and generate after-tax cash inflows of $30,000 for 10 years. Find the NPV if the firm uses a 10% opportunity cost of capital. What is the IRR? What is the payback period? (no Excel)

Answer 1

NPV

In case of equal annual cash flows, their present value can be computed by multiplying them with the present value interest factor annuity of $1 (PVIFA) which can be computed as follows -

where, r is the cost of capital or discount rate, n is no. of years

At 10% for 10 years -

NPV = (-)Initial investment + Annual cash inflows x PVIFA (10%, 10)

or, NPV = (-)$220,000 + $30,000 x 6.144567106 = (-)$35,662.98682 or (-)$35,662.99

IRR

IRR is the rate at which present value of cash inflows is equal to the initial investment or the rate at which NPV is zero.

So, Present value of cash inflows = Initial investment

or, Annual Cash Inflows x PVIFA (IRR, 10) = Initial investment

or, $30,000 x PVIFA (IRR, 10) = $220,000

or, PVIFA (IRR, 10) = $220,000 / $30,000 = 7.33333333333

Now, we need to refer this value in the PVIFA table in year 10, and see in which rate does it fall -

At 6%, PVIFA = 7.36008705133

At 7%, PVIFA = 7.02358154085

Our value lies in between these two and as such the rate also lies between 6% and 7%. So, we need to interpolate -

Difference required = 7.36008705133 - 7.33333333333 = 0.026753718

Total Difference = 7.36008705133 - 7.02358154085 = 0.33650551048

IRR = Lower rate + Difference in rates x (Difference required / Total difference)

or, IRR = 6% + 1% x (0.026753718 / 0.33650551048) = 6.0795045% or 6.08%

Payback Period

Payback period is the period within which the initial investment is recovered by the cash inflows of the project. In case of equal annual cash inflows, payback period can be computed as -

Payback period = Initial investment / Annual cash inflows = $220,000 / $30,000 = 7.333333 years or 7.33 years