Question

In this​ problem, we consider replacing an existing electrical water heater with an array of solar panels. The net installed investment cost of the panels is $1,365 ​($2,100 less a 35​% tax credit from the​ government). Based on an energy​ audit, the existing water heater uses 220 kilowatt hours​ (kWh) of electricity per​ month, so at $0.12 per​ kWh, the cost of operating the water heater is $26.4 per month. Assuming the solar panels can save the entire cost of heating water with​ electricity, answer the following questions.

b. What is the IRR of this investment if the solar panels have a life of 12 ​years?

a. The simple payback period is 52 months. ​(Round to the nearest whole​ number.)

b. The IRR of the investment is ​% ______per month

Answer 1

(a) Payback period (PBP) is the time by when cumulative cash flows equal zero.

If PBP be M months, then

$26.4 x M = $1,365

M = $1,365 / $26.4 = 51.7

M ~ 52 months.

(b) IRR is found using interpolation.

IRR = RL + [NPVL / (NPVL - NPVH)] x (RH - RL) where

RL: Lower discount rate (assumed 1% per month)

RH: Higher discount rate (assumed 2% per month)

NPVL: NPV at 1%

NPVH: NPV at 2%

Number of months = 12 x 12 = 144

NPVL ($) = - 1,365 + 26.4 x P/A(1%, 144) = - 1,365 + 26.4 x 76.1372** = - 1,365 + 1,872.97 = 507.97

NPVH ($) = - 1,365 + 26.4 x P/A(2%, 144) = - 1,365 + 26.4 x 47.1123** = - 1,365 + 1243.77 = - 121.23

IRR = 1% + [507.97 / (507.97 + 121.23)] x (2 - 1)%

= 1% + (507.97 / 629.2) x 1%

= 1% + 0.81 x 1%

= 1% + 0.81%

= 1.81% per month

**PVIFA(r%, N) = [1 - (1 + r)^-N] / r

PVIFA(1%, 144) = [1 - (1.01)^-144] / 0.01 = (1 - 0.2386) / 0.01 = 0.7614 / 0.01 = 76.1372

PVIFA(2%, 144) = [1 - (1.02)^-144] / 0.02 = (1 - 0.0578) / 0.02 = 0.9422 / 0.02 = 47.1123