Q10SE - Horngren'S Financial And Managerial Accounting

Question

Your grandfather would like to share some of his fortune with you. He offers to give you money under one of the following scenarios (you get to choose): 

1. \(7,250 per year at the end of each of the next eight years 

2. \)49,650 (lump sum) now 

3. $98,650 (lump sum) eight years from now

Requirements 

1. Calculate the present value of each scenario using an 8% discount rate. Which scenario yields the highest present value? Round to nearest whole dollar. 

2. Would your preference change if you used a 10% discount rate?

Step-by-Step Solution

Answer

1. Present value in Scenario 1 equals $41,658.50, in Scenario 2 equals  $49,650 and in Scenario 3 equals $53,271. In a situation where the discount rate is 8% scenario 3 is preferable because it yields the highest present value.
2. In a situation where the discount rate is 10% scenario 2 is preferable because it yields the highest present value

Step 1: Definition of Present Value

The current value of any sum of money that will be received in the future time period is known as the present value. This concept states that the value of money today is higher than the value of the same amount of money in future.

Step 2: Present value of each scenario

1. $7,250 per year at the end of each of the next eight years

Year	Future value	X	Present value factor	=	Present value
1	$7,250	X	0.926	=	$6,713.5
2	$7,250	X	0.857	=	$6,213.25
3	$7,250	X	0.794	=	$5,756.5
4	$7,250	X	0.735	=	$5,328.75
5	$7,250	X	0.681	=	$4,937.25
6	$7,250	X	0.630	=	$4,567.5
7	$7,250	X	0.583	=	$4,226.75
8	$7,250	X	0.540	=	$3,915
Total	$41,658.5

2. 49,650 (lump sum) now: The present value of this amount will be the same because it is now received in a lump sum.

3. $98,650 (lump sum) eight years from now

$\begin{array}{rcl}\mathit{P}\mathit{r}\mathit{e}\mathit{s}\mathit{e}\mathit{n}\mathit{t}\mathbf{ }\mathit{v}\mathit{a}\mathit{l}\mathit{u}\mathit{e}\mathbf{ }\mathit{o}\mathit{f}\mathbf{ }\mathit{l}\mathit{u}\mathit{m}\mathit{p}\mathbf{-}\mathit{s}\mathit{u}\mathit{m}\mathbf{ }\mathit{p}\mathit{a}\mathit{u}\mathit{m}\mathit{e}\mathit{n}\mathit{t}& \mathbf{=}& \mathit{F}\mathit{u}\mathit{t}\mathit{u}\mathit{r}\mathit{e}\mathbf{ }\mathit{v}\mathit{a}\mathit{l}\mathit{u}\mathit{e}\mathbf{\times }\mathit{P}\mathit{r}\mathit{e}\mathit{s}\mathit{e}\mathit{n}\mathit{t}\mathbf{ }\mathit{v}\mathit{a}\mathit{l}\mathit{t}\mathit{u}\mathit{e}\mathbf{ }\mathit{f}\mathit{a}\mathit{c}\mathit{t}\mathit{o}\mathit{r}\\ & \mathbf{=}& \mathbf{\$}\mathbf{98}\mathbf{,}\mathbf{650}\mathbf{\times }\mathbf{\left(}\frac{\mathbf{1}}{{\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{8}\mathbf{)}}^{\mathbf{8}}}\mathbf{\right)}\\ & \mathbf{=}& \mathbf{\$}\mathbf{98}\mathbf{,}\mathbf{650}\mathbf{\times }\mathbf{0}\mathbf{.}\mathbf{540}\\ & \mathbf{=}& \mathbf{\$}\mathbf{53}\mathbf{,}\mathbf{271}\end{array}$

Scenario 3 yields highest present value.

Step 3: Change in preference if discount rate is 10%

1. $7,250 per year at the end of each of the next eight years 

Year	Future value	X	Present value factor	=	Present value
1	$7,250	X	0.909	=	$6,590.25
2	$7,250	X	0.826	=	$5,988.5
3	$7,250	X	0.751	=	$5,444.75
4	$7,250	X	0.683	=	$4,951.75
5	$7,250	X	0.621	=	$4,502.25
6	$7,250	X	0.564	=	$4,089
7	$7,250	X	0.513	=	$3,719.25
8	$7,250	X	0.467	=	$3,385.75
Total	$38,671.5

2. 49,650 (lump sum) now: The present value of this amount will be the same because it is now received in a lump sum.

3. $98,650 (lump sum) eight years from now

$\begin{array}{rcl}\mathit{P}\mathit{r}\mathit{e}\mathit{s}\mathit{e}\mathit{n}\mathit{t}\mathbf{ }\mathit{v}\mathit{a}\mathit{l}\mathit{u}\mathit{e}\mathbf{ }\mathit{o}\mathit{f}\mathbf{ }\mathit{l}\mathit{u}\mathit{m}\mathit{p}\mathbf{-}\mathit{s}\mathit{u}\mathit{m}\mathbf{ }\mathit{p}\mathit{a}\mathit{u}\mathit{m}\mathit{e}\mathit{n}\mathit{t}& \mathbf{=}& \mathit{F}\mathit{u}\mathit{t}\mathit{u}\mathit{r}\mathit{e}\mathbf{ }\mathit{v}\mathit{a}\mathit{l}\mathit{u}\mathit{e}\mathbf{\times }\mathit{P}\mathit{r}\mathit{e}\mathit{s}\mathit{e}\mathit{n}\mathit{t}\mathbf{ }\mathit{v}\mathit{a}\mathit{l}\mathit{t}\mathit{u}\mathit{e}\mathbf{ }\mathit{f}\mathit{a}\mathit{c}\mathit{t}\mathit{o}\mathit{r}\\ & \mathbf{=}& \mathbf{\$}\mathbf{98}\mathbf{,}\mathbf{650}\mathbf{\times }\mathbf{\left(}\frac{\mathbf{1}}{{\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{8}\mathbf{)}}^{\mathbf{8}}}\mathbf{\right)}\\ & \mathbf{=}& \mathbf{\$}\mathbf{98}\mathbf{,}\mathbf{650}\mathbf{\times }\mathbf{0}\mathbf{.}\mathbf{467}\\ & \mathbf{=}& \mathbf{\$}\mathbf{46}\mathbf{,}\mathbf{069}\mathbf{.}\mathbf{55}\end{array}$

Note: In this case scenario 2 will be preferred.