1. Multiple regression for analysts

The general specification is Y = β₀ + β₁X₁ + β₂X₂ + … + βₖXₖ + ε. Five core assumptions of classical linear regression: (1) linearity, (2) independence of ε, (3) homoskedasticity, (4) normality of ε, (5) no perfect multicollinearity. Violations require diagnostic tests:

1.1 Adjusted R² and F-test

R²adj = 1 − [(n − 1) / (n − k − 1)] × (1 − R²). The F-test of joint significance: F = [R²/k] / [(1 − R²)/(n − k − 1)], compared to F_{k, n−k−1, α}. Reject H₀ (all β = 0) if F > critical value.

2. International Economics — parity conditions

2.1 Covered Interest Rate Parity (CIRP)

In a no-arbitrage world, F / S = (1 + rd) / (1 + rf) where F = forward FX rate, S = spot, r_d = domestic rate, r_f = foreign rate. CIRP is enforced by arbitrageurs; deviations beyond transaction costs imply riskless profit.

2.2 Uncovered IRP and PPP

Uncovered IRP: E(St+1) / St ≈ (1 + rd) / (1 + rf). Empirically, UIP fails over short horizons (forward premium puzzle). Purchasing Power Parity (PPP) states that exchange rates adjust to equalise the price of a basket of goods: ΔS = π_d − π_f (relative PPP).

"Empirical research consistently rejects strict PPP in the short run but supports relative PPP over horizons of 5-10 years."
Source: International Monetary Fund Working Paper WP/19/27 — imf.org/en/Publications/WP

3. Consolidation accounting (IFRS 10/11/12)

3.1 Three accounting methods by ownership

3.2 Acquisition method and goodwill

When a parent acquires a subsidiary, all identifiable assets and liabilities are remeasured at fair value on the acquisition date. Goodwill = consideration transferred + non-controlling interest (NCI) + fair value of previously held interest − net identifiable assets. Under IFRS, goodwill may be measured at full goodwill (NCI at FV) or partial goodwill (NCI at proportionate share). US GAAP requires full goodwill.

4. Pension accounting (IAS 19)

For a defined benefit plan, the employer reports a net pension liability (asset) = Present Value of Defined Benefit Obligation (DBO) − Fair Value of Plan Assets. Annual changes include service cost, net interest cost (DBO − assets × discount rate), actuarial gains/losses (OCI under IAS 19R, recycled vs not), past service cost, and contributions/benefits paid. Discount rate must be a high-quality corporate bond yield matching the duration of the obligation.

5. Key takeaways

• Regression diagnostics: BP (hetero), DW (serial), VIF (multi)
• CIRP enforces arbitrage-free FX forwards
• IFRS 10 control → full consolidation; IAS 28 significant influence → equity method
• Goodwill = consideration + NCI − net identifiable FV assets
• Pension: DBO − plan assets = net liability/asset
• Actuarial gains/losses under IAS 19R go to OCI (not recycled)

For further reading

• IFRS 10 — Consolidated Financial Statements
• IAS 19 — Employee Benefits
• IMF Data — FX historical
• Pinto/Henry Equity Asset Valuation (CFA Institute Investment Series)

5. Multivariate regression — extensions

5.1 Dummy variables

Categorical regressors are encoded as dummy variables (0/1). For k categories, use k-1 dummies to avoid the dummy variable trap (perfect multicollinearity). Example: sector dummies = (Tech, Health, Energy) → 2 dummies (Tech=1 if tech 0 otherwise; Health=1 if health 0 otherwise; Energy is the baseline omitted category).

5.2 Interaction effects

Multiplying two regressors creates an interaction term. Example: returns regressed on (Beta) + (Quality) + (Beta × Quality) tests whether the effect of beta depends on quality. Coefficient significance on the interaction term confirms or rejects the hypothesis.

5.3 Logistic regression

When the dependent variable is binary (default vs no-default, M&A success vs failure), use logistic regression: P(Y=1) = 1 / (1 + e−(β₀+β₁X)). Output is a probability, not a continuous prediction.

6. Time series

L2 introduces time series analysis: AR(p), MA(q), ARMA(p,q), ARIMA(p,d,q) and ARCH/GARCH models for volatility. Key concepts:

• Stationarity: constant mean, variance and autocovariance over time
• Unit root test (Dickey-Fuller): tests for non-stationarity
• Co-integration: two non-stationary series with a stationary linear combination
• Mean reversion: |b| < 1 in AR(1) implies series returns to long-term mean

7. International parity — full framework

8. Consolidation accounting — examples

8.1 Step acquisition (50% then 30% more)

Under IFRS 3 (revised), a step acquisition triggers remeasurement of the previously held interest to fair value with the gain/loss recognised in profit or loss. Example: Parent owned 50% of Sub at carrying value €200M. Acquires additional 30% for €180M cash. FV of 50% original = €260M. Step-up gain = 260 - 200 = €60M recognised in P&L. New goodwill computed on the full 80%.

8.2 Loss of control

When a parent loses control (sale of stake reducing to ≤50%), the remaining investment is remeasured to fair value and the difference flows through P&L. The retained stake is then accounted under equity method or fair value method depending on remaining influence.

9. Pension accounting — case study

Acme Pension Plan, FY2025 (€M):

• Defined Benefit Obligation (DBO) opening: 850
• Plan assets opening: 700
• Net pension liability opening: -150 (under-funded)
• Service cost: 45
• Interest cost at 4%: 850 * 0.04 = 34
• Expected return on plan assets at 5.5%: 700 * 0.055 = 38.5
• Actuarial losses on DBO: 22 (in OCI)
• Return on assets above expected: +12 (in OCI)
• Employer contributions: 30
• Benefits paid: 28

Calculations:

• DBO closing = 850 + 45 + 34 + 22 - 28 = 923
• Plan assets closing = 700 + 38.5 + 12 + 30 - 28 = 752.5
• Net pension liability closing = -170.5
• P&L impact = Service 45 + Net interest (DBO 34 - PA expected 38.5) = 40.5
• OCI impact = Actuarial loss 22 + Excess return on assets -12 = +10 net loss

10. Self-test quiz (5 MCQ)

11. Time series analysis — full ARMA workflow

11.1 Identification step

Before fitting an ARMA(p,q), test for stationarity using the Augmented Dickey-Fuller (ADF) test: H₀ = unit root (non-stationary), H₁ = stationary. Reject H₀ if the test statistic is more negative than the critical value. For non-stationary series, take first differences (d=1) until stationary, giving an ARIMA(p,d,q) model.

Once stationary, examine the autocorrelation function (ACF) and partial autocorrelation function (PACF):

11.2 Information criteria for model selection

Two competing models can be ranked by:

• AIC (Akaike Information Criterion): AIC = -2ln(L) + 2k. Lower = better. Tends to overfit.
• BIC / SBC (Bayesian Information Criterion): BIC = -2ln(L) + k×ln(n). Lower = better. Penalises complexity more harshly than AIC.

11.3 GARCH for volatility forecasting

Financial returns exhibit volatility clustering (large changes followed by large changes). The GARCH(1,1) model captures this:

σ²t = ω + α × ε²t-1 + β × σ²t-1

where α captures reaction to recent shocks and β captures persistence. Long-run variance = ω/(1−α−β). Typical equity GARCH: α ≈ 0.08, β ≈ 0.90, so α+β ≈ 0.98 (high persistence). Used in option pricing, VaR, and risk management.

12. International Economics — full case study

13. FRA — full IFRS-GAAP comparison

14. Full pension reconciliation example (IAS 19)

Net pension liability closing = 923 − 752.5 = €170.5M. P&L impact: total expense €51M. OCI impact: net actuarial movement +€0.5M (loss of 22 + gain of 22.5).

15. Self-test (additional 3 MCQ)

Three additional questions follow the in-leçon quiz to reinforce learning.

16. Glossary — Quant + FRA advanced

17. Bibliography

• Greene, W. (2023). Econometric Analysis, 8th ed. Pearson.
• Wooldridge, J. (2023). Introductory Econometrics, 7th ed. Cengage.
• IASB. IAS 19 Employee Benefits. London: IFRS Foundation.
• FASB. ASC 715 Compensation — Retirement Benefits. Norwalk.
• Mishkin, F. (2024). The Economics of Money, Banking and Financial Markets. Pearson.