14.2: Estimating and Discounting Cash Flows

Question 1

What is the initial after-tax cash flow (CF₀) in capital expenditure analysis?

Answer 1

The initial after-tax cash flow (CF₀) refers to the total cash outlay required to initiate an investment project.

This includes the initial capital cost, changes in net working capital (NWC), and associated opportunity costs.

Question 2

How is the initial after-tax cash flow (CF₀) generally estimated?

Answer 2

The initial after-tax cash flow (CF₀) is estimated using the formula:

CF₀ = C₀ + ΔNWC₀ + OC

C₀: The initial capital cost of the asset.

ΔNWC₀: The change in net working capital requirements.

OC: The opportunity costs associated with the project.

Question 3

What do the terms in the CF₀ formula reProvide an example of calculating the initial after-tax cash flow (CF₀).present?

Answer 3

Brennan Co. is evaluating the acquisition of a new milling machine with a base price of $625,000, modification costs of $25,000, and additional inventory of $100,000.

Capital cost (C₀) = $625,000 + $25,000 = $650,000

Initial cash outlay (CF₀) = C₀ + ΔNWC₀ + OC = $650,000 + $100,000 + $0 = $750,000

Question 4

What is the difference between the capital cost (C₀) and the initial after-tax cash flow (CF₀)?

Answer 4

The capital cost (C₀) includes all costs to make an investment operational and can be depreciated for tax purposes,

while the initial after-tax cash flow (CF₀) includes the total cash outlay required, including changes in net working capital and associated opportunity costs.

Question 5

What are the components included in the capital cost (C₀)?

Answer 5

The capital cost (C₀) includes:

machinery installation expenses

land-clearing costs,

and other expenses necessary to make an investment operational.

Question 6

What are expected annual after-tax cash flows (CFₜ)?

Answer 6

Expected annual after-tax cash flows (CFₜ) are those estimated to occur as a result of the investment decision.

They comprise the expected incremental increase in after-tax operating income and any incremental tax savings (or additional taxes paid) resulting from the initial investment outlay.

Question 7

How is the capital cost allowance (CCA) related to cash flow estimates?

Answer 7

The CCA is related to cash flow estimates by providing tax savings for the firm.

The amount of depreciation charged for tax purposes is called the CCA, which is established by the Canada Revenue Agency (CRA) according to the asset class associated with the cash outlay.

Question 8

What are the two ways to determine cash flows after the Capital Cost Allowance (CCA)?

Answer 8

Before-tax operating income
- CCA
= Taxable income
- Taxes payable
= After-tax income
+ CCA (non-cash expense)
= Net cash flow

OR

Before-tax operating income
- Taxes payable on operating income
= After-tax operating income
+ CCA tax savings
= Net cash flow

Question 9

Calculate the CCA for a new milling machine with a base price of $625,000 and modification costs of $25,000 in the first year, using a CCA rate of 30%.

Answer 9

C₀ = $625,000 + $25,000 = $650,000

CCA (year 1) = C₀ × CCA rate × 1/2 = $650,000 × 0.3 × 1/2 = $97,500

Question 10

What is the formula for estimating the annual operating cash flows using CCA tax savings?

Answer 10

The formula for estimating annual operating cash flows using CCA tax savings is:

CFₜ = CFBTₜ (1 - T) + CCAₜ (T)

CFBTₜ: Cash flow before taxes (incremental pre-tax operating income)

CCAₜ: CCA expense for year t
T: Firm’s marginal (effective) tax rate

Question 11

How do you calculate the present value (PV) of future CFₜ?

Answer 11

The present value (PV) of future CFₜ is calculated using the formula:

PV (Future CFₜ) = Σ [CFₜ / (1 + r)ⁿ]

where:

CFₜ: Expected annual after-tax cash flow for year t
r: Firm’s cost of capital
n: Year

PV (Future CFₜ) = Σ [$112,625 / (1.12)¹ + $143,338 / (1.12)² + $120,961 / (1.12)³ + $105,298 / (1.12)⁴ + $94,333 / (1.12)⁵] = $421,370

Question 12

What is the ending (or terminal) after-tax cash flow (ECFₙ)?

Answer 12

The ending (or terminal) after-tax cash flow (ECFₙ) is the total cash flow expected to be generated in the terminal year of a project, aside from that year’s expected after-tax cash flow.

It comprises the estimated selling or salvage value (SVₙ) of the asset.

Question 13

How do you estimate the ending (or terminal) after-tax cash flow (ECFₙ) with tax implications?

Answer 13

ECFₙ (with tax implications) = SVₙ + ΔNWCₙ - 0.5 [(SVₙ - C₀) × T] - [(SVₙ - UCCₙ) × T]

SVₙ: Estimated salvage value in year n

ΔNWCₙ: Net working capital released upon termination of the project

C₀: Original capital cost of the asset
UCCₙ: Ending undepreciated capital cost balance

T: Firm’s effective tax rate

Question 14

What is the simplified formula for estimating ECFₙ when capital gains, terminal losses, and CCA recapture do not occur?

Answer 14

The simplified formula for ECFₙ when such tax implications do not occur is:

ECFₙ = SVₙ + ΔNWCₙ

Question 15

Provide an example of calculating ECFₙ.

Answer 15

Assume the salvage value (SVₙ) of the milling machine is $132,655 and $100,000 of net working capital (ΔNWCₙ) is released.

ECFₙ = SVₙ + ΔNWCₙ = $132,655 + $100,000 = $232,655

Question 16

How do you calculate the present value (PV) of the ending cash flow (ECFₙ)?

Answer 16

PV (ECFₙ) = ECFₙ / (1 + r)ⁿ

Example:
PV (ECFₙ) = $232,655 / (1.12)⁵ = $132,015

Question 17

What is the formula for calculating the Net Present Value (NPV) of an investment?

Answer 17

The formula for NPV is:

NPV = Σ [CFₜ / (1 + k)ⁿ] - CF₀ = PV(Future CFs) - CF₀

where:

CFₜ: Cash flow in year t
k: Discount rate
CF₀: Initial cash outlay

Question 18

How can the NPV be broken down into components?

Answer 18

The NPV can be expressed as:

NPV = PV(Annual CFs) + PV(ECFₙ) - CF₀

Question 19

Provide an example of calculating the NPV for Brennan Co.’s project.

Answer 19

NPV = PV(Operating CFs) + PV(ECFₙ) - CF₀
NPV = $421,370 + $132,015 - $750,000 = -$196,615

Therefore, Brennan Co. should reject the project.

Question 20

How is the present value (PV) of operating cash flows calculated when they are assumed to be the same every year?

Answer 20

PV(Operating Cash Flows) = CFBT(1 - T) × [ (1 - (1 + k)⁻ⁿ) / k ]

Question 21

What is the formula for the present value (PV) of the CCA tax shield?

Answer 21

PV(CCA Tax Shield) = [(C₀)(d)(T) / (d + k)] × [ (1 + 0.5k) / (1 + k) ] - [(UCCₙ)(d)(T) / (d + k)] × [ (1 / (1 + k)ⁿ) ] - [(SVₙ - UCCₙ)(T) / (1 + k)ⁿ]

where:

d: CCA rate
k: Discount rate
T: Effective tax rate
C₀: Initial cost
UCCₙ: Ending UCC balance
SVₙ: Salvage value

Question 22

Provide an example calculation of the PV of the CCA tax shield.

Answer 22

PV(CCA Tax Shield) = [($650,000)(0.30)(0.45) / (0.30 + 0.12)] × [ (1 + 0.5×0.12) / (1 + 0.12) ] - [($132,655)(0.30)(0.45) / (0.30 + 0.12)] × [ (1 / (1 + 0.12)⁵ ] = $173,541

Question 23

How is the PV of capital gains taxes paid calculated if a capital gain occurs?

Answer 23

PV(Capital Gains Taxes Paid) = [(SVₙ - C₀) × T] / (1 + k)ⁿ

SVₙ: Salvage value
C₀: Original cost

Question 24

What is the accelerated depreciation in the context of the Accelerated Investment Incentive?

Answer 24

Accelerated depreciation is a method that creates a higher depreciation expense in the earlier years and lower charges in the later periods.

It was introduced as part of new tax measures to incentivize capital investments by allowing depreciation expenses to be claimed at a rate three times higher than the usual rate for eligible assets acquired between November 21, 2018, and the end of 2023.

Question 25

How does the Accelerated Investment Incentive impact the calculation of CCA tax shield?

Answer 25

The calculation of the CCA tax shield is adjusted to account for the tripling of the CCA in the year of acquisition.

The equations for PV(CCA Tax Shield) are modified to:

PV(CCA Tax Shield) = [(C₀)(d)(T) / (d + k)] × [ (1 + 1.5k) / (1 + k) ] - [(UCCₙ)(d)(T) / (d + k)] × [ (1 / (1 + k)ⁿ )]

PV(CCA Tax Shield) = [(C₀)(d)(T) / (d + k)] × [ (1 + 1.5k) / (1 + k) ] - [(SVₙ - UCCₙ)(T) / (1 + k)ⁿ]

Question 26

Provide an example of calculating the NPV using the valuation by components approach for an asset purchased after November 20, 2018, and eligible for accelerated depreciation.

Answer 26

CF₀ = $750,000

PV(Operating Cash Flows) = $247,828

a. ECFₙ = $150,000 and PV(ECFₙ) = $85,114
PV(CCA Tax Shield) = $213,616

NPV = $247,828 + $213,616 + $85,114 - $750,000 = -$203,442

b. ECFₙ = $300,000 and PV(ECFₙ) = $170,228
PV(CCA Tax Shield) = $175,315

NPV = $247,828 + $175,315 + $170,228 - $750,000 = -$156,629

Even with accelerated depreciation and higher salvage value, the project is still financially unacceptable due to the large initial cash outlay.

Question 27

How do changes in the discount rate (k) affect the NPV calculation?

Answer 27

Changes in the discount rate (k) affect the present value of future cash flows and tax shields.

Lowering the discount rate increases the present value of future cash flows, potentially making the NPV less negative.

However, in the given example, even with a discount rate of 10%, the NPV remains negative, indicating the project should still be rejected.

Question 28

Provide an example of recalculating the NPV with a lower discount rate of 10%.

Answer 28

CF₀ = $750,000
PV(Operating Cash Flows) = $260,617
PV(CCA Tax Shield) = $181,604
PV(ECFₙ) = $144,460

NPV = $260,617 + $181,604 + $144,460 - $750,000 = -$163,319

The project is still rejected, but the NPV is less negative compared to using a higher discount rate.

Question 29

Provide an example of recalculating the NPV when operating cash flows are 10% higher than original estimates.

Answer 29

CF₀ = $750,000
CFBT_new = CFBT_old × 1.1 = $137,500
PV(Operating Cash Flows) = $286,678
PV(CCA Tax Shield) = $181,604
PV(ECFₙ) = $144,460

NPV = $286,678 + $181,604 + $144,460 - $750,000 = -$137,258

The project is still rejected, although the NPV is higher (less negative) due to the increased estimated operating cash flows.