Question

Splash Nation is considering purchasing a water park in Atlanta, Georgia, for \(1,910,000. The new facility will generate annual net cash inflows of \)483,000 foreight years. Engineers estimate that the facility will remain useful for eight years andhave no residual value. The company uses straight-line depreciation, and its stockholdersdemand an annual return of 10% on investments of this nature.

Requirements

1. Compute the payback, the ARR, the NPV, the IRR, and the profitability index ofthis investment.

2. Recommend whether the company should invest in this project.

Short answer

Payback:3.95 years

ARR:25.6%

NPV:$666,757

IRR:19%

PI:1.35

Step-by-step solution

Computation of CB ratios

Computation of payback period

$\begin{array}{c}\mathbf{P}\mathbf{a}\mathbf{y}\mathbf{b}\mathbf{a}\mathbf{c}\mathbf{k}\mathbf{=}\frac{\mathbf{A}\mathbf{m}\mathbf{o}\mathbf{u}\mathbf{n}\mathbf{t}\mathbf{I}\mathbf{n}\mathbf{v}\mathbf{e}\mathbf{s}\mathbf{t}\mathbf{e}\mathbf{d}}{\mathbf{A}\mathbf{n}\mathbf{n}\mathbf{u}\mathbf{a}\mathbf{l}\mathbf{n}\mathbf{e}\mathbf{t}\mathbf{c}\mathbf{a}\mathbf{s}\mathbf{h}\mathbf{i}\mathbf{n}\mathbf{f}\mathbf{l}\mathbf{o}\mathbf{w}}\\ \mathbf{=}\frac{\mathbf{\$}\mathbf{1}\mathbf{,}\mathbf{910}\mathbf{,}\mathbf{000}}{\mathbf{\$}\mathbf{483}\mathbf{,}\mathbf{000}}\\ \mathbf{=}\mathbf{3}\mathbf{.}\mathbf{954}\mathbf{y}\mathbf{e}\mathbf{a}\mathbf{r}\mathbf{s}\end{array}$

Calculation of ARR

role="math" localid="1656049007785" $\begin{array}{c}\mathbf{AnnualDepreciation}\mathbf{=}\frac{\mathbf{Cost}\mathbf{-}\mathbf{ResidualValue}}{\mathbf{UsefulLife}}\\ \mathbf{=}\frac{\mathbf{\$}\mathbf{1}\mathbf{,}\mathbf{910}\mathbf{,}\mathbf{000}\mathbf{-}\mathbf{\$}\mathbf{0}}{\mathbf{8}}\\ \mathbf{=}\mathbf{\$}\mathbf{238}\mathbf{,}\mathbf{750}\end{array}$

$\begin{array}{c}\mathbf{A}\mathbf{v}\mathbf{e}\mathbf{r}\mathbf{a}\mathbf{g}\mathbf{e}\mathbf{a}\mathbf{n}\mathbf{n}\mathbf{u}\mathbf{a}\mathbf{l}\mathbf{o}\mathbf{p}\mathbf{e}\mathbf{r}\mathbf{a}\mathbf{t}\mathbf{i}\mathbf{n}\mathbf{g}\mathbf{i}\mathbf{n}\mathbf{c}\mathbf{o}\mathbf{m}\mathbf{e}\mathbf{=}\mathbf{A}\mathbf{n}\mathbf{n}\mathbf{u}\mathbf{a}\mathbf{l}\mathbf{n}\mathbf{e}\mathbf{t}\mathbf{c}\mathbf{a}\mathbf{s}\mathbf{h}\mathbf{i}\mathbf{n}\mathbf{f}\mathbf{l}\mathbf{o}\mathbf{w}\mathbf{-}\mathbf{A}\mathbf{n}\mathbf{n}\mathbf{u}\mathbf{a}\mathbf{l}\mathbf{D}\mathbf{e}\mathbf{p}\mathbf{r}\mathbf{e}\mathbf{c}\mathbf{i}\mathbf{a}\mathbf{t}\mathbf{i}\mathbf{o}\mathbf{n}\\ \mathbf{=}\mathbf{\$}\mathbf{483}\mathbf{,}\mathbf{000}\mathbf{-}\mathbf{\$}\mathbf{238}\mathbf{,}\mathbf{750}\\ \mathbf{=}\mathbf{\$}\mathbf{244}\mathbf{,}\mathbf{250}\end{array}$

$\begin{array}{c}\mathbf{A}\mathbf{v}\mathbf{e}\mathbf{r}\mathbf{a}\mathbf{g}\mathbf{e}\mathbf{i}\mathbf{n}\mathbf{v}\mathbf{e}\mathbf{s}\mathbf{t}\mathbf{e}\mathbf{d}\mathbf{a}\mathbf{m}\mathbf{o}\mathbf{u}\mathbf{n}\mathbf{t}\mathbf{=}\frac{\mathbf{T}\mathbf{o}\mathbf{t}\mathbf{a}\mathbf{l}\mathbf{I}\mathbf{n}\mathbf{v}\mathbf{e}\mathbf{s}\mathbf{t}\mathbf{m}\mathbf{e}\mathbf{n}\mathbf{t}}{\mathbf{2}}\\ \mathbf{=}\frac{\mathbf{\$}\mathbf{1}\mathbf{,}\mathbf{910}\mathbf{,}\mathbf{000}}{\mathbf{2}}\\ \mathbf{=}\mathbf{\$}\mathbf{955}\mathbf{,}\mathbf{000}\end{array}$

$\begin{array}{c}\mathbf{A}\mathbf{R}\mathbf{R}\mathbf{=}\frac{\mathbf{A}\mathbf{v}\mathbf{e}\mathbf{r}\mathbf{a}\mathbf{g}\mathbf{e}\mathbf{a}\mathbf{n}\mathbf{n}\mathbf{u}\mathbf{a}\mathbf{l}\mathbf{o}\mathbf{p}\mathbf{e}\mathbf{r}\mathbf{a}\mathbf{t}\mathbf{i}\mathbf{n}\mathbf{g}\mathbf{i}\mathbf{n}\mathbf{c}\mathbf{o}\mathbf{m}\mathbf{e}}{\mathbf{A}\mathbf{v}\mathbf{e}\mathbf{r}\mathbf{a}\mathbf{g}\mathbf{e}\mathbf{a}\mathbf{m}\mathbf{o}\mathbf{u}\mathbf{n}\mathbf{t}\mathbf{i}\mathbf{n}\mathbf{v}\mathbf{e}\mathbf{s}\mathbf{t}\mathbf{e}\mathbf{d}}\\ \mathbf{=}\frac{\mathbf{\$}\mathbf{244}\mathbf{,}\mathbf{250}}{\mathbf{\$}\mathbf{955}\mathbf{,}\mathbf{000}}\\ \mathbf{=}\mathbf{0}\mathbf{.}\mathbf{256}\mathbf{o}\mathbf{r}\mathbf{25}\mathbf{.}\mathbf{6}\mathbf{\%}\end{array}$

Computation of NPV

$\begin{array}{c}\mathbf{P}\mathbf{r}\mathbf{e}\mathbf{s}\mathbf{e}\mathbf{n}\mathbf{t}\mathbf{v}\mathbf{a}\mathbf{l}\mathbf{u}\mathbf{e}\mathbf{o}\mathbf{f}\mathbf{a}\mathbf{n}\mathbf{n}\mathbf{u}\mathbf{a}\mathbf{l}\mathbf{n}\mathbf{e}\mathbf{t}\mathbf{c}\mathbf{a}\mathbf{s}\mathbf{h}\mathbf{i}\mathbf{n}\mathbf{f}\mathbf{l}\mathbf{o}\mathbf{w}\mathbf{=}\mathbf{A}\mathbf{n}\mathbf{n}\mathbf{u}\mathbf{a}\mathbf{l}\mathbf{c}\mathbf{a}\mathbf{s}\mathbf{h}\mathbf{I}\mathbf{n}\mathbf{f}\mathbf{l}\mathbf{o}\mathbf{w}\mathbf{\times }\left[\frac{1-\frac{1}{{\left(1+r\right)}^n}}{r}\right]\\ \mathbf{=}\mathbf{\$}\mathbf{483}\mathbf{,}\mathbf{000}\mathbf{\times }\left[\frac{1-\frac{1}{{\left(1+0.1\right)}^8}}{0.1}\right]\\ \mathbf{=}\mathbf{\$}\mathbf{483}\mathbf{,}\mathbf{000}\mathbf{\times }\mathbf{5}\mathbf{.}\mathbf{3349}\\ \mathbf{=}\mathbf{\$}\mathbf{2}\mathbf{,}\mathbf{576}\mathbf{,}\mathbf{757}\end{array}$

$\begin{array}{c}\mathbf{N}\mathbf{e}\mathbf{t}\mathbf{P}\mathbf{r}\mathbf{e}\mathbf{s}\mathbf{e}\mathbf{n}\mathbf{t}\mathbf{V}\mathbf{a}\mathbf{l}\mathbf{u}\mathbf{e}\mathbf{=}\mathbf{P}\mathbf{r}\mathbf{e}\mathbf{s}\mathbf{e}\mathbf{n}\mathbf{t}\mathbf{V}\mathbf{a}\mathbf{l}\mathbf{u}\mathbf{e}\mathbf{o}\mathbf{f}\mathbf{i}\mathbf{n}\mathbf{f}\mathbf{l}\mathbf{o}\mathbf{w}\mathbf{s}\mathbf{-}\mathbf{C}\mathbf{o}\mathbf{s}\mathbf{t}\mathbf{o}\mathbf{f}\mathbf{i}\mathbf{n}\mathbf{v}\mathbf{e}\mathbf{s}\mathbf{t}\mathbf{m}\mathbf{e}\mathbf{n}\mathbf{t}\\ \mathbf{=}\mathbf{\$}\mathbf{2}\mathbf{,}\mathbf{576}\mathbf{,}\mathbf{757}\mathbf{-}\mathbf{\$}\mathbf{1}\mathbf{,}\mathbf{910}\mathbf{,}\mathbf{000}\\ \mathbf{=}\mathbf{\$}\mathbf{666}\mathbf{,}\mathbf{757}\end{array}$

Computation of IRR

IRR is the rate at which Present value of cash inflow equals initial investment.

Let’s say IRR = R%

Then,

$\begin{array}{c}\mathbf{I}\mathbf{n}\mathbf{i}\mathbf{t}\mathbf{i}\mathbf{a}\mathbf{l}\mathbf{I}\mathbf{n}\mathbf{v}\mathbf{e}\mathbf{s}\mathbf{t}\mathbf{m}\mathbf{e}\mathbf{n}\mathbf{t}\mathbf{=}\mathbf{P}\mathbf{r}\mathbf{e}\mathbf{s}\mathbf{e}\mathbf{n}\mathbf{t}\mathbf{V}\mathbf{a}\mathbf{l}\mathbf{u}\mathbf{e}\mathbf{o}\mathbf{f}\mathbf{n}\mathbf{e}\mathbf{t}\mathbf{c}\mathbf{a}\mathbf{s}\mathbf{h}\mathbf{i}\mathbf{n}\mathbf{f}\mathbf{l}\mathbf{o}\mathbf{w}\mathbf{s}\\ \mathbf{\$}\mathbf{1}\mathbf{,}\mathbf{910}\mathbf{,}\mathbf{000}\mathbf{=}\mathbf{A}\mathbf{n}\mathbf{n}\mathbf{u}\mathbf{a}\mathbf{l}\mathbf{c}\mathbf{a}\mathbf{s}\mathbf{h}\mathbf{I}\mathbf{n}\mathbf{f}\mathbf{l}\mathbf{o}\mathbf{w}\mathbf{\times }\left[\frac{1-\frac{1}{{\left(1+r\right)}^n}}{r}\right]\\ \mathbf{\$}\mathbf{1}\mathbf{,}\mathbf{910}\mathbf{,}\mathbf{000}\mathbf{=}\mathbf{\$}\mathbf{483}\mathbf{,}\mathbf{000}\mathbf{\times }\left[\frac{1-\frac{1}{{\left(1+R\right)}^8}}{R}\right]\\ \mathbf{3}\mathbf{.}\mathbf{9545}\mathbf{=}\left[\frac{1-\frac{1}{{\left(1+R\right)}^8}}{R}\right]\end{array}$

By hit and trial method if R is taken 19% for 8 years then,

$\begin{array}{c}\mathbf{3}\mathbf{.}\mathbf{9545}\mathbf{=}\left[\frac{1-\frac{1}{{\left(1+0.19\right)}^8}}{0.19}\right]\\ \mathbf{3}\mathbf{.}\mathbf{9545}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{9545}\end{array}$

So the IRR = 19%

Computation of profitability index

$\begin{array}{c}\mathbf{P}\mathbf{r}\mathbf{o}\mathbf{f}\mathbf{i}\mathbf{t}\mathbf{a}\mathbf{b}\mathbf{i}\mathbf{l}\mathbf{i}\mathbf{t}\mathbf{y}\mathbf{I}\mathbf{n}\mathbf{d}\mathbf{e}\mathbf{x}\mathbf{=}\frac{\mathbf{P}\mathbf{r}\mathbf{e}\mathbf{s}\mathbf{e}\mathbf{n}\mathbf{t}\mathbf{v}\mathbf{a}\mathbf{l}\mathbf{u}\mathbf{e}\mathbf{o}\mathbf{f}\mathbf{n}\mathbf{e}\mathbf{t}\mathbf{c}\mathbf{a}\mathbf{s}\mathbf{h}\mathbf{i}\mathbf{n}\mathbf{f}\mathbf{l}\mathbf{o}\mathbf{w}}{\mathbf{I}\mathbf{n}\mathbf{i}\mathbf{t}\mathbf{i}\mathbf{a}\mathbf{I}\mathbf{n}\mathbf{v}\mathbf{e}\mathbf{s}\mathbf{t}\mathbf{m}\mathbf{e}\mathbf{n}\mathbf{t}}\\ \mathbf{=}\frac{\mathbf{\$}\mathbf{2}\mathbf{,}\mathbf{576}\mathbf{,}\mathbf{757}}{\mathbf{\$}\mathbf{1}\mathbf{,}\mathbf{910}\mathbf{,}\mathbf{000}}\\ \mathbf{=}\mathbf{1}\mathbf{.}\mathbf{35}\end{array}$

Recommendation

By looking at the above analysis it can be seen that the payback period of the investment is around 4 years having IRR greater than the required return. The profitability index of the investment project is also 135%. So the project is investable.