What is the difference between RADR and Certainty Equivalent?

RADR adjusts the discount rate in the denominator to reflect project risk, whereas Certainty Equivalent adjusts the expected cash flows in the numerator to their risk-free equivalents.

How do you evaluate replacement decisions of differing lifespans?

Use Equivalent Annual Cost (EAC) or Annualized Equivalent NPV by dividing NPV by the Annuity Factor of the respective project lifespan.

Chapter 3: Advanced Capital Budgeting Decisions - Quick Revision Notes & Formulas Cheat Sheet | CA Final AFM

🎯 Learning Outcomes (ICAI Syllabus Focus)

According to the official ICAI study guidelines, candidates must achieve complete mastery in:

📈 Inflationary Appraisal

Differentiating nominal vs. real cash flows, Fisher equation $(1+r_n)=(1+r_r)(1+i)$, and accounting for historical tax depreciation lag.

⚖️ Risk & Uncertainty Models

Applying Probability, Variance, Standard Deviation, and Coefficient of Variation ($CV = \sigma / EV$) for risk ranking.

🎯 RADR vs. Certainty Equivalent

Discounting risky cash flows at Risk-Adjusted Discount Rates vs converting uncertain flows to certainty equivalents ($\alpha_t$) discounted at $R_f$.

🔍 Sensitivity & Scenario

Testing sensitivity margins of safety for Selling Price, Volume, and Costs; computing multi-variable Best/Base/Worst scenario expected NPVs.

🌳 Decision Tree Rollback

Constructing multi-stage decision trees across decision nodes ($\square$) and chance nodes ($\bigcirc$) using backward induction rollback.

🔄 Replacement & APV

Calculating incremental replacement cash flows (Sec 50 tax effects), Equivalent Annual Cost (EAC), and Adjusted Present Value ($\text{Base NPV} + \text{Tax Shield}$).

1 Current Trends in Capital Budgeting & Inflation Analysis

Traditional capital budgeting assumed price stability. In dynamic global markets, capital projects must explicitly adjust for Inflation, Technological Obsolescence, and Government Policy Shifts.

1.1 Impact of Inflation on Capital Investment Appraisal

⚠️

Core Exam Principle: Inflation increases both project revenues and operating expenses. However, tax depreciation does not keep pace with inflation because under Indian Income Tax law, depreciation is strictly computed on historical cost!

Annual operating cash flow after tax is given by: $$\text{CFAT} = (R - C - D)(1 - T) + D = (R - C)(1 - T) + DT$$ Where: $R$ = Revenue, $C$ = Operating Costs, $D$ = Depreciation, $T$ = Tax Rate.
While $(R - C)(1 - T)$ scales upward with inflation, the depreciation tax shield $DT$ remains frozen in nominal terms. Therefore, inflation erodes the real value of depreciation tax shields, lowering real cash flows!

The Fisher Equation (Nominal vs. Real Rate of Return)

Fisher Effect (Nominal vs. Real Rate of Return)

$$R_N = R_R + P \quad \text{[Approximation Equation]}$$

$$(1 + R_N) = (1 + R_R) \times (1 + P) \quad \text{[Exact Compounding Equation]}$$

$R_N$: Required rate of return in nominal terms (Money discount rate)

$R_R$: Required rate of return in real terms

$P$: Anticipated inflation rate

💡

Golden Consistency Rule for Discounting:
1. Nominal (Money) Cash Flows (which include inflation) must be discounted at the Nominal Discount Rate ($r_n$).
2. Real Cash Flows (expressed in constant purchasing power) must be discounted at the Real Discount Rate ($r_r$).
Mixing real cash flows with a nominal discount rate or vice-versa is a fatal examination error!

1.2 Impact of Technological Change

Rapid technological advances shorten economic product lifecycles. Companies mitigate tech risk by: (a) incorporating flexible manufacturing; (b) evaluating abandonment options if technology shifts unfavorably; and (c) adjusting discount rates upward to reflect technology obsolescence.

1.3 Impact of Government Policy Shifts

Changes in corporate tax rates, import tariffs, environmental carbon taxes, or export subsidies drastically alter domestic and cross-border project viability.

2 Dealing with Risk and Uncertainty in Investment Decisions

Parameter	Risk	Uncertainty
Definition	A situation where the future outcome of an investment is not known, but the probability distribution of outcomes can be objectively or subjectively quantified.	A situation where future outcomes are completely unpredictable and no probabilities can be assigned.
Quantifiability	Can be mathematically modeled using standard deviation, variance, and expected values.	Cannot be measured mathematically; relies entirely on managerial judgment or game-theoretic rules (Maximin, Minimax regret).
Insurability	Generally insurable or hedgeable using derivative contracts.	Uninsurable due to lack of historical actuarial baseline.

Reasons for Adjusting for Risk in Capital Budgeting

⚖️ Opportunity Cost Alignment

Investors demand higher risk premia for riskier projects. The cost of capital must reflect the specific risk profile of the venture.

📉 Real Value of Inflows

Uncertain future cash flows have lower present utility than certain cash flows. Discounting penalizes distant, uncertain streams.

3 Internal and External Factors Affecting Capital Budgeting

🏢 Internal Factors (Micro / Firm-Level)

1. Project-Specific Risk: Errors in estimating sales volume, technological delays, construction overruns, or operational glitches unique to this specific project.
2. Company-Specific Risk: Corporate governance quality, industrial labor relations, internal capital structure leverage, and management competence.

🌍 External Factors (Macro / Market-Level)

1. Industry-Specific Risk: Sectoral regulatory bans, raw material cartels, or cyclical demand downturns.
2. Market Risk (Systematic): Macroeconomic interest rates, inflation, and general stock market crashes.
3. Competition Risk: Aggressive price-war discounting by new market entrants.
4. International Risk: Foreign exchange volatility, geopolitical sanctions, and sovereign host-government expropriation.

4.1 Statistical Techniques for Risk Incorporation

When probability distributions of cash flows can be estimated, finance managers evaluate expected cash flows ($\text{ENCF} = \sum P_i \times \text{NCF}_i$), dispersion around the mean (Variance and Standard Deviation), and relative risk:

⚠️

Decision Rule for Coefficient of Variation ($CV = \frac{\sigma}{\bar{X}}$):
- When two mutually exclusive projects have the same Expected NPV, choose the project with the lower Standard Deviation ($\sigma$).
- When projects have different Expected NPVs and different outlays, Standard Deviation is misleading! Always select the project with the lower Coefficient of Variation (CV).

4.2 Conventional Techniques (RADR & Certainty Equivalent)

4.2.1 Risk-Adjusted Discount Rate (RADR)

The use of risk adjusted discount rate (RADR) is based on the concept that investors demand higher returns from risky projects. Under this method, NPV is calculated as follows:

Risk-Adjusted Discount Rate (RADR)

$$\text{Risk adjusted discount rate (RADR)} = \text{Risk free rate } (R_f) + \text{Risk premium}$$

$$\text{NPV} = \sum_{t=1}^n \frac{\text{NCF}_t}{(1 + k)^t} - I$$

Under CAPM Model: $\text{RADR } (k) = R_f + \beta \times (R_m - R_f)$

$\text{NCF}_t$: Net cash flow for period $t$

$k$: Risk adjusted discount rate (RADR)

$I$: Initial Investment

$t$: Period ($t = 1$ to $n$)

4.2.2 Certainty Equivalent (CE) Approach

Certainty Equivalent expresses risky future cash flows in terms of the certain cash flow which the decision maker considers equivalent. Under this method, NPV is calculated as follows:

Certainty Equivalent (CE) Method

$$\text{Certainty Equivalent Coefficient } (\alpha_t) = \frac{\text{Certain cash flow}_t}{\text{Risky or expected cash flow}_t}$$

$$\text{NPV} = \sum_{t=1}^n \frac{\alpha_t \times \text{NCF}_t}{(1 + R_f)^t} - I$$

$\alpha_t$: Certainty Equivalent coefficient for period $t$ $(0 \le \alpha_t \le 1)$

$\text{NCF}_t$: Expected risky cash flow for period $t$

$R_f$: Risk-free rate of interest (discount rate)

$I$: Initial Investment

Comparison: RADR vs. Certainty Equivalent (CE)

Criteria	Risk-Adjusted Discount Rate (RADR)	Certainty Equivalent (CE)
Adjustment Mechanism	Adjusts denominator (discount rate).	Adjusts numerator (cash inflows).
Compounding of Risk	Implicitly assumes risk increases compounding exponentially over time as $(1 + r^*)^t$ grows, which may not be realistic.	Allows separate, independent risk evaluation for each specific year ($\alpha_1, \alpha_2, \dots, \alpha_n$).
Applicable Discount Rate	Discounts using elevated rate ($R_f + \text{Risk Premium}$).	Strictly discounts using Risk-Free Rate ($R_f$).
Practical Popularity	Extremely popular in corporate practice due to simplicity.	Theoretically superior, but subjective to estimate $\alpha_t$.

4.3 Sensitivity Analysis (What-If Modeling)

🔍

Definition: Sensitivity Analysis is a modeling technique that examines how the project's NPV (or IRR) changes when one input variable is altered at a time, keeping all other variables strictly constant (ceteris paribus).

Steps in Sensitivity Analysis

1

Identify Critical Variables

Identify variables influencing project cash flows: Initial Outlay, Annual Sales Volume, Selling Price per unit, Variable Cost per unit, Fixed Operating Costs, and Project Life.

2

Establish Mathematical Relationships

Formulate annual net cash inflow equations: $\text{NCF} = [(\text{Sales Units} \times (\text{Price} - \text{VC})) - \text{Fixed Costs}](1 - T) + \text{Depr}(T)$.

3

Apply Adverse Variations & Compute Margin of Safety

Test an adverse shift (e.g., 2.5%, 5%, 10%) in each variable, or calculate the maximum percentage adverse change each variable can tolerate before $\text{NPV} = 0$.

Sensitivity Analysis Margin of Safety

$$\text{Sensitivity } (\%) = \frac{\text{Base Case NPV}}{\text{Present Value of Variable under consideration}} \times 100$$

Decision Rule: The variable with the lowest Sensitivity Percentage is the most sensitive / critical variable, because the smallest adverse deviation will wipe out the project's entire NPV!

📌

Universal ICAI Exam Pattern: Across almost all ICAI problems (e.g. Illustration 9), Selling Price per Unit consistently exhibits the highest sensitivity because changes in selling price directly alter total revenue without being dampened by costs.

4.4 Scenario Analysis & Monte Carlo Simulation

4.4.1 Scenario Analysis (Multi-Variable Modeling)

Unlike sensitivity analysis which isolates one variable, Scenario Analysis evaluates the impact of changing multiple mutually correlated variables simultaneously. A company evaluates three classic macroeconomic states:

Scenario	Underlying Assumptions	Typical Cash Flow Profile
1. Pessimistic (Worst Case)	Severe recession, high inflation, intense competition, low unit sales, low price, cost overruns.	Depressed or negative cash flows; negative NPV.
2. Most Likely (Base Case)	Normal market conditions, planned sales targets achieved, standard operating cost inflation.	Expected normal cash flows; baseline positive NPV.
3. Optimistic (Best Case)	Booming economic expansion, high market share, strong pricing power, operational cost efficiencies.	Surge in operating cash inflows; exceptionally high NPV.

📊

$$\text{Expected NPV} = \sum_{s=1}^3 P_s \times \text{NPV}_s = [P_{\text{pess}} \times \text{NPV}_{\text{pess}}] + [P_{\text{base}} \times \text{NPV}_{\text{base}}] + [P_{\text{opt}} \times \text{NPV}_{\text{opt}}]$$

4.4.2 Monte Carlo Simulation Analysis

When real-world project variables have continuous probability distributions with infinite combinations, finance managers use Monte Carlo Simulation.

1

Identify Model Variables

Define endogenous variables (NPV, IRR) and stochastic exogenous variables (market size, market share, unit cost, project life).

2

Specify Probability Distributions

Assign cumulative probability distributions to each uncertain exogenous variable.

3

Generate Random Numbers

Sample random numbers (e.g. from 00 to 99) to select specific values from cumulative distribution curves.

4

Compute Simulated Trial NPV

Determine the NPV for the selected set of randomly generated variables.

5

Iterate Thousands of Times

Repeat steps 3 and 4 a massive number of times (e.g. 10,000 iterations) using computer software.

6

Plot Output Distribution

Construct the frequency distribution of all simulated NPVs, calculating mean expected NPV and standard deviation to assess downside loss probability.

4.5 Decision Tree Analysis & Rollback Induction

🌳

Core Concept: Capital budgeting often involves sequential investment decisions over time (e.g., spending ₹2 Lakhs on an R&D pilot test today, and deciding whether to commit ₹15 Lakhs for commercial rollout next year only if the pilot succeeds). A Decision Tree maps this visual progression.

Nodes in a Decision Tree Diagram

⏹️ 1. Decision Node (Square $\square$)

Represents an active choice available to the finance manager (e.g., Invest vs. Abandon; Commercialize vs. Drop). The manager has complete control over which branch to choose. At a decision node, always choose the branch with the Highest Expected Net Present Value.

⏺️ 2. Chance / Event Node (Circle $\bigcirc$)

Represents an uncertain future market event beyond managerial control (e.g., High Demand vs. Low Demand; Success vs. Failure). Each branch stemming from a chance node carries an assigned probability (sum of probabilities = 1.0). At a chance node, compute the Expected Monetary Value (EMV).

The Rollback (Backward Induction) Procedure

⬅️

Work from Right to Left (Backward Induction)

Start at the terminal branches on the extreme right of the tree, discounting final cash flows back to the intermediate nodes.

✖️

Evaluate Chance Nodes First

Calculate Expected Value at each chance node: $\text{EV} = \sum (\text{Probability} \times \text{Branch PV})$.

✂️

Prune / Truncate Suboptimal Branches at Decision Nodes

Compare options at each decision node. Select the highest net payoff branch and prune (truncate with double slash //) inferior alternatives. Continue rolling backward until the initial time-zero decision node is reached.

5 Replacement Decisions & Optimum Replacement Cycle

Evaluating whether to replace an existing functional asset with a modern asset is an incremental cash flow analysis.

5.1 Step-by-Step Replacement Cash Flow Architecture

I

Initial Net Cash Outflow (Time 0)

$$\text{Net Initial Outlay} = \text{Cost of New Asset} - \text{Salvage of Old Asset} \pm \text{Tax Consequences on Sale}$$ Tax Rule (Sec 50 Income Tax Act): Under the block-of-assets system, sale proceeds are deducted from the written-down value (WDV) of the block. If a standalone single-asset block ceases to exist: $$\text{Tax Payable / Saved} = (\text{Book Value of Old Equipment} - \text{Sale Price}) \times \text{Tax Rate}$$

II

Incremental Annual Operating Cash Inflow (CFAT)

$$\Delta \text{CFAT} = [(\Delta \text{Revenues} - \Delta \text{Operating Costs}) - \Delta \text{Depreciation}](1 - T) + \Delta \text{Depreciation}$$ Where $\Delta \text{Depreciation} = \text{Depreciation on New Asset} - \text{Depreciation on Old Asset}$.

III

Terminal Incremental Cash Flow (Year n)

$$\Delta \text{Terminal Cash Flow} = \text{Salvage Value of New Asset} - \text{Salvage Value of Old Asset (at Year n)}$$

5.2 Optimum Replacement Cycle (Equivalent Annual Cost - EAC)

When an asset must be repeatedly replaced over an indefinite time horizon, finance managers evaluate the optimum replacement frequency (e.g., replacing every 1 year vs. 2 years vs. 3 years) using Equivalent Annual Cost (EAC):

Equivalent Annual Cost (EAC)

$$\text{EAC} = \frac{\text{PV of All Costs} - \text{PV of Salvage Value}}{\text{PVAF}_{r, n}}$$

PV of All Costs: Initial Cost + $\sum$ PV of operating costs across $n$ years

PV of Salvage: Present value of scrap resale recovery at end of year $n$

$\text{PVAF}_{r, n}$: Cumulative Present Value Annuity Factor at discount rate $r$ for $n$ years

Optimal Decision: Select the replacement cycle that Minimizes the EAC!

6 Adjusted Present Value (APV) Approach

💡

Core Philosophy: Traditional capital budgeting uses WACC to bundle investment risk and financing tax shields into a single discount rate. Adjusted Present Value (APV) separates the investment decision from the financing decision by valuing them additively!

Adjusted Present Value (APV)

$$\text{APV} = \text{Base-Case NPV} + \text{PV of Tax Benefit on Interest}$$

1. Base-Case NPV: Project NPV assuming all-equity unlevered financing, discounted at the unlevered cost of equity ($K_e$).
2. PV of Tax Benefit on Interest: Discounted value of debt tax shields: $\sum \frac{\text{Interest} \times T}{(1 + K_d)^t}$
3. Decision Rule: Accept if $\text{APV} > 0$.

🏆

Decision Rule: Accept the project if $\text{APV} > 0$. A project that has a negative base-case NPV under pure equity financing may become economically viable and value-accretive once interest tax shield benefits are added!

7 Solved ICAI Master Illustrations & Problem Walkthroughs

Case 1: Sensitivity Analysis (ICAI Illustration 9 Walkthrough)

Given Data: Initial Capital Outlay = ₹400 Cr; Annual Sales = 5 Cr units; Selling Price = ₹100/unit; Variable Cost = ₹50/unit; Fixed Costs = ₹50 Cr/year; Project Life = 3 years; Discount Rate = 6%. (No taxes). PVAF(6%, 3 years) = 2.673.
Required: Calculate Base NPV, and compute impact on NPV considering a 2.5% adverse variance in each variable. Determine the most sensitive variable.

Variable Tested	Base Value	Adverse Value (2.5%)	Revised Annual Cash Flow	Revised NPV (₹ Cr)	Change in NPV (₹ Cr)	% Drop in NPV
Base Case	—	—	(5 × 50) - 50 = ₹200 Cr	(200 × 2.673) - 400 = + ₹134.60 Cr	—	—
1. Selling Price per unit	₹100	₹97.50 (-2.5%)	[5 × (97.5 - 50)] - 50 = ₹187.5 Cr	(187.5 × 2.673) - 400 = + ₹101.19 Cr	- ₹33.41 Cr	- 24.82% (MOST SENSITIVE!)
2. Variable Cost per unit	₹50	₹51.25 (+2.5%)	[5 × (100 - 51.25)] - 50 = ₹193.75 Cr	(193.75 × 2.673) - 400 = + ₹117.89 Cr	- ₹16.71 Cr	- 12.41%
3. Sales Volume (Units)	5 Cr	4.875 Cr (-2.5%)	(4.875 × 50) - 50 = ₹193.75 Cr	(193.75 × 2.673) - 400 = + ₹117.89 Cr	- ₹16.71 Cr	- 12.41%
4. Initial Capital Outlay	₹400 Cr	₹410 Cr (+2.5%)	₹200 Cr	(200 × 2.673) - 410 = + ₹124.60 Cr	- ₹10.00 Cr	- 7.43%
5. Fixed Operating Costs	₹50 Cr	₹51.25 Cr (+2.5%)	(5 × 50) - 51.25 = ₹198.75 Cr	(198.75 × 2.673) - 400 = + ₹131.26 Cr	- ₹3.34 Cr	- 2.48% (LEAST SENSITIVE)

🎯

ICAI Exam Conclusion: Selling Price per Unit is by far the most sensitive variable, causing a 24.82% collapse in NPV for a mere 2.5% price cut! Fixed costs are the least sensitive variable.

Case 2: Optimum Replacement Cycle (ICAI Illustration 13 Walkthrough)

Problem Statement: X Ltd. is a taxi operator. Initial cost of each taxi = ₹4,00,000. Useful life = 3 years. Discount rate = 10%.
Operating costs: Year 1 = ₹1,20,000; Year 2 = ₹1,40,000; Year 3 = ₹1,80,000.
Salvage Values: End of Yr 1 = ₹2,40,000; End of Yr 2 = ₹1,80,000; End of Yr 3 = ₹1,20,000.
PV factors (10%): Yr 1 = 0.909; Yr 2 = 0.826; Yr 3 = 0.751. Cumulative PVAF: 1 yr = 0.909; 2 yrs = 1.735; 3 yrs = 2.486.
Required: Recommend the optimum replacement cycle (1, 2, or 3 years).

Cycle	Present Value of Outlay & Operating Costs (₹)	Less: Present Value of Salvage Value (₹)	Total Net PV of Costs (₹)	Cumulative PVAF (10%)	Equivalent Annual Cost (EAC) (₹)
Replace Every 1 Year	4,00,000 + (1,20,000 × 0.909) = 5,09,080	2,40,000 × 0.909 = 2,18,160	5,09,080 - 2,18,160 = 2,90,920	0.909	$\frac{2,90,920}{0.909} =$ ₹ 3,20,044
Replace Every 2 Years	4,00,000 + (1,20,000 × 0.909) + (1,40,000 × 0.826) = 6,24,720	1,80,000 × 0.826 = 1,48,680	6,24,720 - 1,48,680 = 4,76,040	1.735	$\frac{4,76,040}{1.735} =$ ₹ 2,74,375 (MINIMUM!)
Replace Every 3 Years	6,24,720 + (1,80,000 × 0.751) = 7,59,900	1,20,000 × 0.751 = 90,120	7,59,900 - 90,120 = 6,69,780	2.486	$\frac{6,69,780}{2.486} =$ ₹ 2,69,421

🏆

Decision Rule Check: Replace the taxi every 3 years if EAC ₹2,69,421 is lowest, or every 2 years depending on specific tax block depreciation. In this illustration without tax, Year 3 minimizes EAC to ₹2,69,421!

8 Quick Revision Master Summary & Review Questions

Technique	Formula / Operational Decision Rule	Key Exam Nuance
Fisher Formula	$(1 + r_n) = (1 + r_r)(1 + i)$	Discount nominal flows with nominal rate; real flows with real rate. Tax shield is nominal.
Coefficient of Variation	$CV = \frac{\sigma}{\bar{X}}$	Use when projects have unequal sizes or unequal expected NPVs. Lower CV = Better.
Certainty Equivalent	$\text{NPV} = \sum \frac{\alpha_t \text{NCF}_t}{(1 + R_f)^t} - I_0$	Discount at Risk-Free Rate ($R_f$), NOT at WACC!
Sensitivity %	$\text{Sensitivity } \% = \frac{\text{Base Case NPV}}{\text{PV of Variable}} \times 100$	Lowest percentage = Most sensitive variable (typically Selling Price).
Decision Tree	$\square$ = Decision node (Max NPV); $\bigcirc$ = Chance node (EMV)	Solve from right to left using rollback induction.
Optimum Replacement	$\text{EAC} = \frac{\text{PV of Costs} - \text{PV of Salvage}}{\text{PVAF}_{r, n}}$	Select replacement frequency that minimizes EAC.
Adjusted Present Value	$\text{APV} = \text{Base-Case NPV} + \text{PV of Tax Shields}$	Base NPV is unlevered (all-equity); evaluates financing side effects additively.

Self-Test Practice Questions (ICAI TYK)

Q1: Explain the Certainty Equivalent (CE) approach in capital budgeting. How is it superior to RADR?

Model Answer: Under the Certainty Equivalent (CE) approach, risky future cash inflows are adjusted into riskless equivalent cash flows by multiplying by certainty equivalent coefficients $\alpha_t = \frac{\text{Certain Cash Flow}}{\text{Risky Cash Flow}}$ (where $0 \le \alpha_t \le 1$). These certain flows are then discounted at the Risk-Free Rate ($R_f$).
Superiority over RADR: RADR compounds the risk premium exponentially over time $(1 + r^*)^t$, which presumes that risk increases strictly at a constant compounding rate with time. The Certainty Equivalent approach is theoretically superior because it does not assume uniform risk compounding; it permits independent risk evaluation for each separate year ($\alpha_1, \alpha_2, \dots$).

Q2: How is capital budgeting appraisal conducted under inflationary conditions? Explain the tax shield distortion.

Model Answer: Under inflationary conditions, finance managers must observe the rule of consistency: nominal cash flows must be discounted at the nominal rate of return ($r_n$), while real cash flows must be discounted at the real rate of return ($r_r$), linked via Fisher’s equation: $(1 + r_n) = (1 + r_r)(1 + i)$.
Tax Shield Distortion: Under tax laws, tax depreciation is calculated strictly on historical asset costs. While revenues and operating expenses inflate, the depreciation tax deduction remains unchanged in nominal terms. Consequently, real taxable profits rise, causing tax outflows to inflate faster than revenues. Thus, inflation diminishes the real value of the depreciation tax shield, adversely affecting project NPV.

Q3: What are the sequential steps involved in Monte Carlo Simulation Analysis for capital budgeting?

Model Answer: Monte Carlo Simulation follows six systematic steps: (1) Define the mathematical relationship connecting endogenous variables (NPV/IRR) with exogenous input variables; (2) Specify probability distributions and construct cumulative probability ranges for each random variable; (3) Select random numbers from a random number table; (4) Map random numbers to the cumulative distributions to pick stochastic input values and compute trial NPV; (5) Replicate steps (3) and (4) over thousands of iterations to generate a sample distribution of simulated NPVs; (6) Construct a probability distribution of NPVs, calculating the mean expected return, standard deviation, and downside probability of negative NPV.

Technique	Formula / Operational Decision Rule	Key Exam Nuance
Fisher Formula	\((1 + r_n) = (1 + r_r)(1 + i)\)	Discount nominal flows with nominal rate; real flows with real rate. Tax shield is nominal.
Coefficient of Variation	\(CV = \frac{\sigma}{\bar{X}}\)	Use when projects have unequal sizes or unequal expected NPVs. Lower CV = Better.
Certainty Equivalent	\(\text{NPV} = \sum \frac{\alpha_t \text{NCF}_t}{(1 + R_f)^t} - I_0\)	Discount at Risk-Free Rate (\(R_f\)), NOT at WACC!
Sensitivity %	\(\text{Sensitivity } \% = \frac{\text{Base Case NPV}}{\text{PV of Variable}} \times 100\)	Lowest percentage = Most sensitive variable (typically Selling Price).
Decision Tree	\(\square\) = Decision node (Max NPV); \(\bigcirc\) = Chance node (EMV)	Solve from right to left using rollback induction.
Optimum Replacement	\(\text{EAC} = \frac{\text{PV of Costs} - \text{PV of Salvage}}{\text{PVAF}_{r, n}}\)	Select replacement frequency that minimizes EAC.
Adjusted Present Value	\(\text{APV} = \text{Base-Case NPV} + \text{PV of Tax Shields}\)	Base NPV is unlevered (all-equity); evaluates financing side effects additively.