The problem describes a retail company considering two expansion strategies: Project A (Physical store) and Project B (Online store). The company's cost of capital is 13% (0.13). We need to calculate the payback period, net present value (NPV), and profitability index for both projects.

1. Problem Description

The problem describes a retail company considering two expansion strategies: Project A (Physical store) and Project B (Online store). The company's cost of capital is 13% (0.13). We need to calculate the payback period, net present value (NPV), and profitability index for both projects.

2. Solution Steps

(a) Payback Period

Project A:

* Year 0: -$1,150,000

* Year 1: $300,

0

0

0. Cumulative: -$1,150,000 + $300,000 = -$850,000

* Year 2: $300,

0

0

0. Cumulative: -$850,000 + $300,000 = -$550,000

* Year 3: $300,

0

0

0. Cumulative: -$550,000 + $300,000 = -$250,000

* Year 4: $300,

0

0

0. Cumulative: -$250,000 + $300,000 = $50,000

Payback occurs during Year

4. Calculate the fraction of the year needed:

$250,000/$300,000 = 0.8333 years.

Payback Period A = 3 + 0.8333 = 3.8333 years

Project B:

* Year 0: -$1,470,000

* Year 1: $418,

0

0

0. Cumulative: -$1,470,000 + $418,000 = -$1,052,000

* Year 2: $418,

0

0

0. Cumulative: -$1,052,000 + $418,000 = -$634,000

* Year 3: $418,

0

0

0. Cumulative: -$634,000 + $418,000 = -$216,000

* Year 4: $418,

0

0

0. Cumulative: -$216,000 + $418,000 = $202,000

Payback occurs during Year

4. Calculate the fraction of the year needed:

$216,000/$418,000 = 0.5167 years.

Payback Period B = 3 + 0.5167 = 3.5167 years

(b) Net Present Value (NPV)

The formula for NPV is:

$NPV = \sum_{t=0}^{n} \frac{CF_t}{(1+r)^t}$

Where:

$CF_t$ = Cash flow at time t

$r$ = Discount rate (cost of capital)

$n$ = Number of periods

Project A:

$NPV_A = -\frac{1150000}{(1+0.13)^0} + \frac{300000}{(1+0.13)^1} + \frac{300000}{(1+0.13)^2} + \frac{300000}{(1+0.13)^3} + \frac{300000}{(1+0.13)^4} + \frac{300000}{(1+0.13)^5}$

$NPV_A = -1150000 + \frac{300000}{1.13} + \frac{300000}{1.2769} + \frac{300000}{1.44289} + \frac{300000}{1.63046} + \frac{300000}{1.84232}$

$NPV_A = -1150000 + 265486.73 + 234943.23 + 207927.32 + 183998.77 + 162838.61$

$NPV_A = -1150000 + 1055194.66 = -94805.34$

Project B:

$NPV_B = -\frac{1470000}{(1+0.13)^0} + \frac{418000}{(1+0.13)^1} + \frac{418000}{(1+0.13)^2} + \frac{418000}{(1+0.13)^3} + \frac{418000}{(1+0.13)^4} + \frac{418000}{(1+0.13)^5}$

$NPV_B = -1470000 + \frac{418000}{1.13} + \frac{418000}{1.2769} + \frac{418000}{1.44289} + \frac{418000}{1.63046} + \frac{418000}{1.84232}$

$NPV_B = -1470000 + 369911.50 + 327355.33 + 289683.56 + 256361.05 + 226887.71$

$NPV_B = -1470000 + 1470200 = 208299.15$

(c) Profitability Index (PI)

The formula for PI is:

$PI = \frac{PV \text{ of future cash flows}}{Initial Investment}$

Project A:

$PV \text{ of future cash flows} = \frac{300000}{1.13} + \frac{300000}{1.2769} + \frac{300000}{1.44289} + \frac{300000}{1.63046} + \frac{300000}{1.84232} = 1055194.66$

$PI_A = \frac{1055194.66}{1150000} = 0.9176$

Project B:

$PV \text{ of future cash flows} = \frac{418000}{1.13} + \frac{418000}{1.2769} + \frac{418000}{1.44289} + \frac{418000}{1.63046} + \frac{418000}{1.84232} = 1470200$

$PI_B = \frac{1470200}{1470000} = 1.000136$

3. Final Answer

Payback period of project A: 3.8333 years

Payback period of project B: 3.5167 years

Net present value of project A: -$94805.34

Net present value of project B: $200

Profitability index of project A: 0.9176

Profitability index of project B: 1.000136