Modules 6–10: Investment Decision Rules, Capital Budgeting, Risk & Return, Cost of Capital, Financial Options
This guide covers the final five modules of the unit. Each module section gives the key concepts, the formulas (with every variable defined), and at least one worked example with the arithmetic checked. Keep these principles front of mind:
Every investment rule tries to answer one question: does this project make the firm better off? Net Present Value (NPV) answers it directly — it measures, in today's dollars, how much value a project adds. It is preferred over every other rule because it (1) uses all the project's cash flows, (2) accounts for the time value of money, (3) discounts at a rate that reflects the project's risk, and (4) is expressed in dollars of value created, so it always points to the value-maximising decision.
The other rules (IRR, payback, profitability index) are useful shortcuts or screens, but each can mislead in particular situations. The exam loves to test exactly those situations.
NPV is the present value of all cash flows, discounted at the project's cost of capital r.
NPV = Σ CFₜ / (1 + r)ᵗ
= CF₀ + CF₁/(1+r) + CF₂/(1+r)² + ... + CFₙ/(1+r)ⁿ
Where CFₜ = cash flow in period t (CF₀ is usually a negative initial outlay), r = discount rate / cost of capital, and N = project life.
A project costs $50,000 today and returns $20,000, $25,000, and $30,000 at the ends of years 1–3. Cost of capital = 10%.
PV₁ = 20,000 / 1.10 = 18,182 PV₂ = 25,000 / 1.10² = 20,661 PV₃ = 30,000 / 1.10³ = 22,539 NPV = -50,000 + 18,182 + 20,661 + 22,539 = +$11,382
NPV > 0, so accept.
The IRR is the discount rate that makes NPV = 0 — the project's break-even rate of return.
0 = Σ CFₜ / (1 + IRR)ᵗ
The IRR rule: accept if IRR > cost of capital. For a "normal" project (one sign change — an outflow followed by inflows), the IRR rule agrees with NPV.
| Pitfall | What goes wrong |
|---|---|
| Multiple / no IRR | If cash flows change sign more than once, there can be several IRRs (or none) and the rule is meaningless. Use NPV. |
| Lending vs borrowing | If positive cash flows come first (you receive money now, pay later), the rule reverses: accept when IRR < r. |
| Ranking mutually exclusive projects | IRR ignores scale and the timing of cash flows. A small project can show a higher IRR but add less value. Always rank by NPV. |
Project S: -$100 now, +$150 in 1 yr → IRR = 50%, NPV@10% = +$36.4 Project L: -$1,000 now, +$1,300 in 1 yr → IRR = 30%, NPV@10% = +$181.8
IRR prefers S, but L creates far more value. Choose L (higher NPV). This is the single most common IRR trap.
The number of years to recover the initial investment. Rule: accept if payback is shorter than a chosen cut-off.
Payback = number of years until cumulative cash flow = 0
Weaknesses: it ignores the time value of money, ignores all cash flows after the cut-off, and the cut-off itself is arbitrary. It is only a quick liquidity screen, never a substitute for NPV. (The discounted payback variant fixes the time-value problem but still ignores later cash flows.)
PI = PV(future cash flows) / Initial investment (accept if PI > 1)
PI measures value created per dollar invested, which makes it the right tool for capital rationing — when funds are limited, rank projects by PI to squeeze the most value out of a fixed budget.
Using the Module 6.2 project: PV of inflows = $61,382 on a $50,000 outlay.
PI = 61,382 / 50,000 = 1.23 → > 1, accept (Note: NPV $11,382 = PV $61,382 - outlay $50,000, so PI > 1 ⇔ NPV > 0.)
| Rule | Accept if | Time value? | Best used for |
|---|---|---|---|
| NPV | NPV > 0 | Yes | The default — always gives the correct decision |
| IRR | IRR > r | Yes | Single, "normal" projects |
| Payback | < cut-off | No | Quick liquidity / risk screen |
| Profitability Index | PI > 1 | Yes | Capital rationing (limited funds) |
Module 6 told us to discount cash flows. Module 7 tells us which cash flows. The rule: discount the project's incremental free cash flows — the extra cash the firm receives because it takes the project — not accounting earnings. Accounting earnings include non-cash items (depreciation) and exclude real cash items (capital expenditure, working-capital changes), so they must be converted to cash.
Start with incremental earnings, then convert to cash.
Incremental Earnings = (Revenues - Costs - Depreciation) × (1 - τℂ)
Unlevered Net Income = EBIT × (1 - τℂ)
Free Cash Flow (FCF) = (Revenues - Costs - Depreciation) × (1 - τℂ)
+ Depreciation
- Capital Expenditure (CapEx)
- Δ Net Working Capital
Where τℂ = corporate tax rate. Depreciation is added back because it is a non-cash expense — it was only subtracted to capture its tax benefit. ΔNWC is the change in net working capital (current assets − current liabilities, e.g. inventory + receivables − payables): an increase ties up cash and is subtracted.
FCF = (Rev - Costs) × (1 - τℂ) + τℂ × Depreciation - CapEx - ΔNWC
The term τℂ × Depreciation is the depreciation tax shield — the cash the firm saves on tax because depreciation is deductible. Both FCF formulas give the same answer; this one isolates the tax effect.
| Item | Include? | Why |
|---|---|---|
| Sunk costs | No | Already spent and unaffected by the decision (e.g. past R&D, a completed market study). |
| Opportunity costs | Yes | The value of the best forgone alternative (e.g. the market value of land you already own and would otherwise sell or rent). |
| Externalities / side effects | Yes | Effects on the rest of the firm — most importantly cannibalization, where a new product steals sales from an existing one. |
| Financing costs / interest | No | Captured in the discount rate (WACC), not in the cash flows. Including them double-counts. |
| Unchanged overhead | No | Only incremental overhead caused by the project counts. |
When an asset is sold at the end of the project, the cash received is the after-tax salvage value:
After-tax salvage = Sale price - τℂ × (Sale price - Book value)
If you sell above book value you pay tax on the gain; if below, you receive a tax credit. Also recover the net working capital invested in the project at the end — it returns as a cash inflow.
Revenue $100,000; cash operating costs $40,000; depreciation $20,000; no CapEx this year; NWC rises by $5,000; tax rate 30%.
EBIT = 100,000 - 40,000 - 20,000 = 40,000
Unlevered net income = 40,000 × (1 - 0.30) = 28,000
+ Depreciation = 20,000
- CapEx = 0
- ΔNWC = 5,000
FCF = 28,000 + 20,000 - 5,000 = $43,000
Check (shortcut): (100,000-40,000)(0.70) + 0.30×20,000 - 5,000
= 42,000 + 6,000 - 5,000 = $43,000 ✓
Once you have the FCF for every year, discount them at the project's cost of capital (Modules 8–9) and apply the NPV rule from Module 6.
This module measures risk, shows how diversification removes part of it, and explains how the market prices the risk that cannot be diversified away. The payoff is the CAPM, which gives the expected return (and therefore the cost of equity used in Module 9).
Expected return: E[R] = Σ pᵢ × Rᵢ (probability-weighted average) Variance: Var(R) = Σ pᵢ × (Rᵢ - E[R])² Standard deviation: SD(R) = √Var(R) (the "volatility")
Where pᵢ = probability of state i and Rᵢ = return in that state. From historical data, the average return is (1/T)ΣRₜ and the variance is [1/(T-1)]Σ(Rₜ - R̄)². Standard deviation measures total risk.
Boom (p = 0.5): R = +30% Bust (p = 0.5): R = -10% E[R] = 0.5(0.30) + 0.5(-0.10) = 10% Var = 0.5(0.30-0.10)² + 0.5(-0.10-0.10)² = 0.5(0.04)+0.5(0.04) = 0.04 SD = √0.04 = 20%
Portfolio return: E[Rₚ] = w₁E[R₁] + w₂E[R₂] + ... Two-asset variance: Var(Rₚ) = w₁²σ₁² + w₂²σ₂² + 2 w₁ w₂ ρ₁₂ σ₁ σ₂ Covariance: Cov(R₁,R₂) = ρ₁₂ σ₁ σ₂ Correlation: ρ₁₂ = Cov / (σ₁σ₂), with -1 ≤ ρ ≤ +1
Where w = portfolio weights (they sum to 1), σ = standard deviation, and ρ = correlation. The key insight: the lower the correlation between assets, the greater the risk reduction. The cross term 2w₁w₂ρσ₁σ₂ is what makes a portfolio's risk fall below the weighted average of the individual risks — that is diversification at work.
| Firm-specific risk | Systematic (market) risk |
|---|---|
| Also: idiosyncratic / diversifiable / unique | Also: undiversifiable / market-wide |
| Examples: a lawsuit, a CEO resigning, a factory fire | Examples: a recession, interest-rate moves, inflation |
| Eliminated by holding many stocks | Cannot be diversified away |
| Not rewarded with extra expected return | Investors are compensated for bearing it |
As stocks are added to a portfolio, volatility falls and then flattens out — the residual is market risk. An efficient portfolio (e.g. the market portfolio / S&P 500) holds only market risk; its risk cannot be reduced further without giving up expected return.
Beta (β) measures a security's systematic risk — how sensitive its returns are to the overall market. The market portfolio has β = 1. Beta is not the same as total volatility; it measures only the market-risk component.
βᵢ = Cov(Rᵢ, Rₘₖₜ) / Var(Rₘₖₜ)
E[Rᵢ] = R + βᵢ × (E[Rₘₖₜ] - R)
Where R = risk-free rate and (E[Rₘₖₜ] − R) = the market risk premium. The whole second term is the security's risk premium. This is the equation of the Security Market Line (SML): expected return rises linearly with beta.
R = 3%, market return = 11%, stock β = 1.4 Market risk premium = 11% - 3% = 8% E[R] = 3% + 1.4 × 8% = 3% + 11.2% = 14.2%
The cost of capital is the discount rate used in NPV. A firm raises money from two main sources — equity and debt — each with its own required return. We estimate each, then blend them by their market-value weights into the Weighted Average Cost of Capital (WACC).
rₐ = R + βₐ × (E[Rₘₖₜ] - R)
Directly from Module 8. This is the most common approach because it links the cost of equity to the firm's systematic risk.
rₐ = (Div₁ / P₀) + g
Where Div₁ = next year's expected dividend, P₀ = current share price, and g = the constant dividend growth rate. The first term Div₁/P₀ is the dividend yield. Use this when the stock pays a stable, growing dividend.
CAPM: R = 4%, β = 1.2, premium = 6%
rₐ = 4% + 1.2 × 6% = 11.2%
Dividend model: Div₁ = $2, P₀ = $40, g = 4%
rₐ = 2/40 + 0.04 = 5% + 4% = 9%
The cost of debt is the yield to maturity (YTM) on the firm's existing debt — the rate the market currently demands — not the coupon rate. Because interest is tax-deductible, what matters for WACC is the after-tax cost:
After-tax cost of debt = rₜ × (1 - τℂ)
For risky debt the expected cost is a little below the promised YTM (because of default risk), but for exam purposes the YTM is normally used as rₜ.
WACC = (E/V) × rₐ + (D/V) × rₜ × (1 - τℂ)
Where E = market value of equity, D = market value of debt, and V = E + D = total firm value. Use market values for the weights, not book values.
WACC is the correct discount rate for a project only if the project has the same risk as the firm as a whole. For a project with different risk, estimate a project-specific cost of capital (e.g. take beta from comparable "pure-play" firms).
E = $60m, D = $40m → V = $100m, E/V = 0.6, D/V = 0.4
rₐ = 12%, rₜ = 6%, tax = 30%
WACC = 0.6(12%) + 0.4(6%)(1 - 0.30)
= 7.2% + 0.4 × 4.2%
= 7.2% + 1.68% = 8.88%
A financial option gives its holder the right, but not the obligation, to buy or sell an underlying asset at a fixed price by (or at) a set date. Because it is a right and not an obligation, the holder will only exercise when it is profitable to do so — this asymmetry is the source of an option's value.
| Term | Meaning |
|---|---|
| Call | The right to buy the underlying at the strike price. |
| Put | The right to sell the underlying at the strike price. |
| Strike / exercise price (K) | The fixed price at which the holder may trade. |
| Expiration | The last date the option may be exercised. |
| American option | May be exercised any time up to expiration. |
| European option | May be exercised only at expiration. |
| Long / short (writer) | The buyer who holds the right / the seller who receives the premium and bears the obligation. |
Moneyness: an option is in-the-money if exercising now gives a positive payoff, at-the-money if the stock price S = K, and out-of-the-money if exercising would lose money.
Let S = stock price at expiration and K = strike price.
Long call payoff = max(S - K, 0) Long put payoff = max(K - S, 0) Short call payoff = -max(S - K, 0) Short put payoff = -max(K - S, 0) Profit (long) = Payoff - Premium paid Profit (short) = Premium received - Payoff
Buy a call, K = $50, premium = $4. If S = $62: payoff = max(62-50,0) = $12; profit = 12 - 4 = $8 If S = $45: payoff = max(45-50,0) = $0; profit = 0 - 4 = -$4 (lose premium) Break-even: S = K + premium = 50 + 4 = $54
For European options on the same stock with the same strike and expiry, call and put prices are linked:
C + PV(K) = P + S₀ → C = P + S₀ - K/(1 + r)ᵗ
Where C = call price, P = put price, S₀ = current stock price, and PV(K) = the strike discounted at the risk-free rate. Intuitively, a "protective put" (hold the stock + a put, S₀ + P) has the same payoff as a "fiduciary call" (a call + cash equal to PV(K), C + PV(K)), so the two must cost the same.
S₀ = $50, K = $50, r = 5%, T = 1 yr, put price P = $3 PV(K) = 50 / 1.05 = $47.62 C = P + S₀ - PV(K) = 3 + 50 - 47.62 = $5.38
| If this factor increases → | Call value | Put value |
|---|---|---|
| Stock price (S) | Up | Down |
| Strike price (K) | Down | Up |
| Volatility (σ) | Up | Up |
| Time to expiration (T) | Up | Up* |
| Risk-free rate (r) | Up | Down |
*Generally true for American options; for European puts the time effect can be ambiguous. The key intuition is that higher volatility raises both call and put values, because an option caps the downside (you simply don't exercise) while preserving the upside — more uncertainty is good for the option holder.
Option value = intrinsic value + time value. Intrinsic value is the in-the-money payoff if exercised now; time value is the extra worth from the chance the option moves further into the money before expiry.
These modules form a single chain for valuing and financing an investment: decision rule → cash flows → discount rate → a separate valuation tool for options.
Cross-references the exam may probe:
r used to discount the FCFs from Module 7.(1 − tax) only to the debt term.max(S−K,0) or max(K−S,0) first, then subtract the premium for profit, and remember put-call parity links the two prices.| Topic | Formula |
|---|---|
| NPV | Σ CFₜ/(1+r)ᵗ — accept if > 0 |
| IRR | rate where NPV = 0 — accept if IRR > r |
| Profitability Index | PV(inflows) / Investment — accept if > 1 |
| Free Cash Flow | (Rev − Costs − Dep)(1 − τ) + Dep − CapEx − ΔNWC |
| Depreciation tax shield | τℂ × Depreciation |
| After-tax salvage | Sale − τℂ(Sale − Book) |
| Expected return | Σ pᵢRᵢ |
| Variance / SD | Σ pᵢ(Rᵢ − E[R])² ; SD = √Var |
| Portfolio variance (2 assets) | w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂ |
| Covariance | ρ₁₂σ₁σ₂ |
| Beta | Cov(Rᵢ, Rₘₖₜ) / Var(Rₘₖₜ) |
| CAPM | R + β(E[Rₘₖₜ] − R) |
| Cost of equity (dividend) | Div₁/P₀ + g |
| After-tax cost of debt | rₜ(1 − τℂ) |
| WACC | (E/V)rₐ + (D/V)rₜ(1 − τℂ) |
| Call / Put payoff | max(S − K, 0) / max(K − S, 0) |
| Put-call parity | C + PV(K) = P + S₀ |