JRE 300

Section 0

Exam Logistics

Section I

Finance is the study of managing cash flows across time, space, and risk. This course focuses on time and risk. The interest rate is the exchange rate across time — it lets us convert $100 today into $110 next year (at r=10%) or vice versa. The opportunity cost of using money today is forgoing the return you could have earned by investing it.

Core formulas

Shortcut formulas for regular cash flow patterns

NPV decision rule

• NPV > 0: investment beats your opportunity cost → accept
• NPV = 0: investment matches baseline → indifferent
• NPV < 0: underperforms alternative → reject

IRR (Internal Rate of Return)

IRR is the discount rate that makes $\text{NPV} = 0$. It's a direct generalization of the one-period return to multiple cash flows. For a one-time outflow followed by inflows:

IRR decision rule: Accept if $\text{IRR} > $ opportunity cost; reject if $\text{IRR} < $ opportunity cost. For conventional projects (one sign change in cash flows), NPV and IRR rules always agree.

Section II

Stock valuation: Dividend Discount Model

A stock's price today equals the present value of all future dividends:

Case 1 — Constant dividends (perpetuity): $P_0 = D/r$ — used for mature utility-like companies.

Case 2 — Gordon Growth Model (constant growth $g < r$):

Case 3 — Multi-stage growth: Break into high-growth phase (growing annuity years $1..T$) plus stable-growth phase (growing perpetuity from $T+1$ onward, discounted back to today):

Bond valuation

A bond's cash flows: coupon payments $C$ each period (annuity) plus face value $F$ at maturity (lump sum). Price is the sum of their PVs:

Bond jargon cheat sheet

Discount, par, and premium bonds

Logic: When YTM > coupon rate, the bond pays too little relative to market rates, so its price drops below face to compensate. When YTM < coupon rate, the bond pays more than market rates, so investors bid up the price above face.

Worked example: 22-year bond, $F=\$1{,}000$, coupon rate 7.35%, yield 5%

Consols (perpetuity bonds)

British consols pay a fixed coupon forever. $\text{Value} = C/r$. If the consol pays \$40 every 6 months and yield is 5% (semi-annual APR), semi-annual rate $= 2.5\%$: $P = 40/0.025 = \$1{,}600$.

Section III

APR vs EAR

APR (Annual Percentage Rate) $=$ rate per period $\times$ periods per year. APR is just a quoting convention — it does not capture the effect of compounding frequency. Two 18% APRs can mean different actual costs depending on whether they compound monthly, daily, etc.

where $m$ = compounding periods per year. EAR puts all rates on a common annual-compounded benchmark so they can be compared directly.

Continuous compounding: As $m \to \infty$, $\text{FV} = \text{PV}\cdot e^{rt}$. Used in theoretical finance and Black-Scholes.

Rate conversion rules

• Given an APR: $r_{\text{period}} = \text{APR}/m$ (divide)
• Given an annual return (not APR): $r_{\text{monthly}} = (1 + r_{\text{annual}})^{1/12} - 1$ (recompound)
• Canadian mortgage twist: rate is a semi-annual APR but payments are monthly. Convert: semi-annual APR $\to$ EAR $\to$ monthly rate via 12th root.

Q1 worked example (practice final): Honda lease vs buy

Canonical personal finance applications

• Car loan: annuity with $\text{PV} = $ loan amount; solve for $C$ given period rate and $T$.
• Canadian mortgage: semi-annual APR $\to$ EAR $\to$ monthly rate $\to$ annuity solve.
• Retirement savings: work backwards. $\text{PV of \$1M in 40 years} = \dfrac{\$1{,}000{,}000}{(1+r_m)^{480}}$. Then annuity formula gives required monthly $C$.

Section IV

Risk, Return, and Diversification

Measuring return and risk

Portfolio return and risk

For a portfolio with weights $\mathbf{w}$:

Diversification & two types of risk

Only systematic risk earns a risk premium — rational investors won't demand extra return for bearing diversifiable risk, because it's free to eliminate. This motivates CAPM.

Sharpe ratio & tangency portfolio

Measures excess return per unit of total risk. The market portfolio maximizes the Sharpe ratio; combining it with the risk-free asset gives the Capital Allocation Line.

Section V

The CAPM equation

Only systematic risk (measured by $\beta$) is compensated.

Beta

• $\beta = 1$: moves with market (same systematic risk)
• $\beta > 1$: amplifies market movements (aggressive; cyclical)
• $0 < \beta < 1$: dampens market movements (defensive; utilities)
• $\beta = 0$: uncorrelated with market (e.g., risk-free asset)
• $\beta < 0$: moves opposite to market (rare; hedge)

Estimating beta: regress $R_{i,t} = \alpha_i + \beta_i R_{m,t} + \varepsilon_{i,t}$. The slope is $\beta$; the intercept $\alpha$ measures mispricing (positive alpha $=$ underpriced, negative alpha $=$ overpriced).

Security Market Line (SML)

Fig. 1 — The SML plots expected return against beta. Equilibrium assets lie on the line.

Every asset should lie on the SML in equilibrium. Assets above the line (α>0) are underpriced; below (α<0) are overpriced. Arbitrage forces prices back onto the line.

Project beta vs firm beta (the Q3 trap)

Fig. 2 — Using firm-wide beta leads to false negatives (low-risk projects rejected) and false positives (high-risk projects accepted).

Key insight: You must use the project's beta, not the firm's overall beta, when evaluating capital projects. A conglomerate with $\beta_{\text{firm}} = 1.3$ evaluating a low-risk utility project with $\beta_{\text{project}} = 0.6$ should use the lower hurdle rate. Using the firm's rate systematically rejects positive-NPV low-risk projects and accepts negative-NPV high-risk projects.

Using CAPM as hurdle rate

Accept the project if $\text{IRR} > \text{Hurdle}$ (equivalently, $\text{NPV} > 0$ at that discount rate).

Q3 worked logic: timber expansion (practice final)

Beyond CAPM: multifactor models

The Fama-French 3-factor model extends CAPM with size (SMB) and value (HML) factors:

Multifactor models fit the cross-section of returns better empirically, but CAPM remains the workhorse for corporate finance applications because of its simplicity. The unifying principle: only systematic risk earns a premium, however many dimensions it has.

Section VI

Cost of Capital & WACC

Three funding sources, three costs

• Equity (issuing new shares or retaining earnings): $r_E$ from CAPM using stock's $\beta$
• Debt (issuing bonds or loans): $r_D = $ YTM on corporate bond
• Retained earnings: another form of equity financing (opportunity cost $= r_E$)

After-tax cost of debt: $r_D(1 - \tau_c)$. Interest is tax-deductible, so every \$1 of interest reduces taxes by $\tau_c$ dollars.

Weighted Average Cost of Capital

Weights use market values, not book values. WACC is the hurdle rate for projects that match the firm's average systematic risk.

Example: Computing WACC

Section VII

The frictionless benchmark (Modigliani-Miller)

In a frictionless world (no taxes, no bankruptcy costs), capital structure is irrelevant to firm value. WACC is constant regardless of leverage. Why? As debt rises (cheaper), equity becomes riskier (more expensive) and the two effects cancel exactly.

Equity beta with leverage

If debt is risk-free $(\beta_D = 0)$: $\beta_E = \beta_{\text{Project}}(1 + D/E)$. More leverage $\Rightarrow$ higher equity beta.

Cost of equity rises linearly with leverage

$r_0 = $ unlevered cost of equity (return if firm had no debt). The leverage premium compensates equity holders for bearing amplified systematic risk.

With taxes: tax shield & WACC falls

Each dollar of debt reduces WACC because interest is tax-deductible. Lower WACC → higher firm value and more positive-NPV projects.

Fig. 3 — With taxes: $r_E$ rises linearly, WACC falls with leverage until bankruptcy costs dominate.

Bankruptcy costs & the trade-off theory

Direct costs: legal, accounting, court fees (3-7% of firm value). Indirect costs: lost customers, suppliers demanding cash, talented employees leaving, underinvestment in good projects, fire-sale asset disposals.

Fig. 4 — Trade-off theory: tax shield lowers WACC at low leverage, bankruptcy costs dominate at high leverage. The minimum of WACC is the optimal capital structure.

High-β firms (cyclical revenues, airlines) should use less debt. Stable-cash-flow firms (utilities) can safely take on more.

Q4 worked example (practice final): Debt levels & maximum debt

Section VIII

Central Banks & the Real Economy

Policy rate transmission chain

When a central bank (Bank of Canada, Fed, ECB) raises its policy rate:

1. Banks' opportunity cost of funds rises $\to$ banks charge higher rates on loans
2. $r_f$ and $r_D$ rise throughout the economy
3. WACC rises for all firms (via CAPM: $r_E = r_f + \beta(\mathbb{E}[R_m] - r_f)$)
4. Fewer projects have positive NPV $\to$ the "investment space" shrinks
5. Firms cut capital spending $\to$ less hiring $\to$ unemployment rises
6. Economic growth slows, putting downward pressure on inflation

This is the real channel of monetary policy: the central bank affects the real economy through its effect on the cost of capital and investment decisions.

Open market operations

Central banks hit their policy rate target by buying/selling government bonds. Buying injects reserves (lowers rate); selling drains reserves (raises rate).

Corporate tax policy & equity market stability

Higher $\tau_c$ acts as an automatic stabilizer on equity markets: the $(1 - \tau_c)$ term in $\beta_E$ dampens the leverage effect on systematic risk. Countries with higher corporate tax rates (Canada vs US) tend to have lower equity market volatility for this reason.

Section IX

Options, Hedging & Put-Call Parity

Calls and puts

A call gives the right (not obligation) to buy at strike $K$ at expiration $T$: payoff $= \max(S_T - K, 0)$. A put gives the right to sell at $K$: payoff $= \max(K - S_T, 0)$.

Moneyness

• Call in-the-money when $S_0 > K$; put ITM when $S_0 < K$
• Call at-the-money when $S_0 = K$
• Call out-of-the-money when $S_0 < K$; put OTM when $S_0 > K$

Hedging with options

Airlines use call options on oil to lock in a ceiling on fuel costs. If oil spikes above strike, airline exercises; if oil falls, airline lets option expire (losing only the premium). Asymmetric payoff = insurance against adverse moves.

Why not hedge by buying oil company stock? (i) stocks have systematic risk that moves with the market, adding unwanted beta exposure; (ii) bilateral risk — if oil falls, stock also falls, so you lose twice; (iii) oil stock returns are imperfectly correlated with oil prices.

Put-Call Parity

Consider two portfolios with the same expiration $T$ and strike $K$:

• Portfolio A: Long call + bond paying $K$ at $T$  (cost today $= C + K e^{-rT}$)
• Portfolio B: Long stock + long put  (cost today $= S + P$)

Both have identical payoffs $= \max(S_T, K)$ in every state. By law of one price:

This lets us synthesize any one security from the other three. If the relationship breaks, arbitrage is possible.

Section X

Core idea: price by replication, not expected value

We don't estimate expected payoffs and apply a risk-adjusted discount rate. Instead, we construct a portfolio of stock and bond that has identical payoffs to the option in every possible state. By law of one price, the option must cost the same as the replicating portfolio. No assumptions about probabilities, risk preferences, or expected returns are needed.

Single-period binomial

Stock at $S_0$ goes to $S_u$ (up) or $S_d$ (down) at time $T$. Call has payoffs $C_u = \max(S_u - K, 0)$ and $C_d = \max(S_d - K, 0)$.

Step 1 — Riskless portfolio (long $\Delta$ shares, short 1 call). Set payoffs equal in both states:

Step 2 — Price the riskless portfolio. Since it pays the same in every state, discount at $r_f$:

Solve for $C_0$.

Equivalent risk-neutral pricing

Under the risk-neutral measure $q$, stocks grow at the risk-free rate in expectation. $q$ is not the real probability; it's a pricing construct that makes no-arbitrage pricing work by discounting at $r_f$.

Multi-period: backward induction

1. Start at expiration: compute option payoffs at all terminal nodes
2. At each intermediate node, apply single-period pricing using next period's values as "payoffs"
3. Work backwards node by node until you reach t=0

Q6 worked example (practice final): 2-period call

Section XI

Black-Scholes & The Greeks

The Black-Scholes formula

Five inputs: $S_0$ (stock price), $K$ (strike), $T$ (time to expiration), $\sigma$ (volatility), $r$ (risk-free rate, continuous). $N(\cdot)$ is the standard normal CDF.

Assumptions: constant volatility, log-normal stock prices, no dividends, continuous trading with no transaction costs, no arbitrage, constant $r_f$, European exercise only.

Put-call parity (continuous compounding)

The Greeks

Section XII

Real options: flexibility has value

• Option to wait: delay investment until uncertainty resolves
• Option to expand: scale up successful projects
• Option to abandon: cut losses when conditions worsen
• Option to switch: flexible operations (e.g., coal/gas plant)

Traditional DCF assumes a fixed plan; real options capture the value of managerial discretion.

Real options are most valuable when uncertainty is high, there's flexibility, and investment is staged.

Employee stock options (ESOs)

Call options granted to employees as compensation. Complications vs market options:

• Non-transferable: can't sell ESOs to diversify
• Forfeiture risk: options lost if employee leaves before vesting
• Dilution: exercising creates new shares
• Early exercise: often exercised before expiry for diversification

These complications reduce the employee's value below Black-Scholes: typically $C_{\text{ESO}} = 50\text{-}70\%$ of $C_{\text{BS}}$. Standard haircut is 30–50% reduction. However, the company records the full Black-Scholes value as the accounting cost of granting the options.

Q5 worked structure (practice final): Company A vs B

Reference

Official Formula Sheet (provided on exam)

Practice Exam

Full Mock Final (6 Questions)

Matches the length and structure of the actual Winter 2026 final. Budget 150 minutes total. Show formula setup on every part. Click "Show Answer" to reveal model solutions.

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