## Required Rate of Return:

The required rate of return is the minimum return on investment that is investor seeks on his investment. The required rate of return is also used while evaluating long term investment opportunities to discounted cash flow generator from it.

Values given:

Textile purchased = $25 million Market value =$20 million

Estimated revenue = \$5 million

Forecast = -20%

MARR = 6%

Solving:

{eq}PV \ = \ \dfrac{Estimated \ revenue}{MARR \ - \ Forecast}\left [ 1 \ - \ \left ( \dfrac{1 \ + \ Forecast}{1 \ + \ MARR} \right )^N \right ] \\ PV \ = \ \dfrac{5}{6\% \ + \ 20\%}\left [ 1 \ - \ \left ( \dfrac{1 \ - \ 20\%}{1 \ + \ 6\%} \right )^N \right ] \\ PV \ = \ \dfrac{5}{0.06 \ + \ 0.20}\left [ 1 \ - \ \left ( \dfrac{1 \ - \ 0.20}{1 \ + \ 0.06} \right )^N \right ] \\ PV \ = \ \dfrac{5}{0.26}\left [ 1 \ - \ \left ( \dfrac{0.80}{1.06} \right )^N \right ] \\ PV \ = \ 20 \ \left [ 1 \ - \ \left ( \dfrac{0.80}{1.06} \right )^N \right ] \\ PV \ = \ 20 \ - \ \dfrac{20e^{-0.2N}}{1.06^{N}} {/eq}

{eq}NPV \ = \ \left ( \dfrac{Estimated \ revenue}{MARR \ - \ Forecast}\left [ 1 \ - \ \left ( \dfrac{1 \ + \ Forecast}{1 \ + \ MARR} \right )^N \right ] \right ) \ - \ \left [ 20 \ - \ \dfrac{20e^{-0.2N}}{1.06^{N}} \right ] \\ NPV \ = \ \left ( \dfrac{5}{6\% \ + \ 20\%}\left [ 1 \ - \ \left ( \dfrac{1 \ - \ 20\%}{1 \ + \ 6\%} \right )^N \right ] \right ) \ - \ \left [ 20 \ - \ \dfrac{20e^{-0.2N}}{1.06^{N}} \right ] \\ NPV \ = \ \left ( \dfrac{5}{0.06 \ + \ 0.20}\left [ 1 \ - \ \left ( \dfrac{1 \ - \ 0.20}{1 \ + \ 0.06} \right )^N \right ] \right ) \ - \ \left [ 20 \ - \ \dfrac{20e^{-0.2N}}{1.06^{N}} \right ] \\ NPV \ = \ \left ( \dfrac{5}{0.26}\left [ 1 \ - \ \left ( \dfrac{0.80}{1.06} \right )^N \right ] \right ) \ - \ \left [ 20 \ - \ \dfrac{20e^{-0.2N}}{1.06^{N}} \right ] \\ NPV \ = \ \left ( 20 \ \left [ 1 \ - \ \left ( \dfrac{0.80}{1.06} \right )^N \right ] \right ) \ - \ \left [ 20 \ - \ \dfrac{20e^{-0.2N}}{1.06^{N}} \right ] \\ NPV \ = \ \left ( 20 \ - \ \dfrac{20e^{-0.2N}}{1.06^{N}} \right ) \ - \ \left [ 20 \ - \ \dfrac{20e^{-0.2N}}{1.06^{N}} \right ] \\ NPV \ = \ 0 {/eq}

N PV of revenue (A) Difference in PV market value of mill (B) NPV ((C) = (A) - (B))
3 10.963749 10.78414335 0.179605246
4 12.991508 12.8817875 0.109720879
5.208 14.78961 14.78954179 0.00006821
5.209636667 14.791665 14.79174379 -0.00008879

So, N is 5.208 years