This article shares key insights from a Fireside Chat on the quantitative finance industry with industry experts.

Introduction to Financial Econometrics

Definition & Objective

Financial econometrics applies statistical tools to analyze financial data.
The goal is to understand uncertainty and hidden patterns in financial markets to aid decision-making in investment, risk management, and policy formulation.

Time Series Representation

A time series is a sequence of observations collected over time.

• A time series of length T is expressed as:
  X₁, X₂, …, Xₜ
• In a more compact form:
  {Xₜ} for t = 1 to T

Multivariate Time Series

When more than one time series is observed at the same time points, they can be jointly expressed.
Example: Bivariate time series (e.g., GDP data for the UK and Singapore):

(X₁, Y₁), (X₂, Y₂), …, (Xₜ, Yₜ)

Basic Concepts for Time Series

(Assuming stationarity – discussed later)

• Expected Value (Mean):
  \(E(X_t) = \mu\)
• Variance:
  \(Var(X_t) = \sigma^2\)
• Autocovariance:
  \(Cov(X_t, X_{t-r}) = \rho_r\)
• Skewness:
  \(E\left[\left(\frac{X_t - \mu}{\sigma}\right)^3\right]\) (Measure of symmetry)
• Kurtosis:
  \(E\left[\left(\frac{X_t - \mu}{\sigma}\right)^4\right]\) (Measure of tailedness)

These are key parameters that define time series behavior.
They help in understanding the underlying uncertainty and data distribution.

Goal: Statistical Inference

The goal is to estimate and interpret these parameters to make informed financial decisions.

Stationarity

• Stationarity means that the statistical properties of a time series remain constant over time.
• In simpler terms:
  • The likelihood of certain values occurring in the past remains the same in the future.

Example of Stationary Data

On Day 1, the probability of Xₜ being 10 or 4 is 50% each.
This remains the same on Day 2, Day 3, and so on.

Types of Stationarity

1. Weak Stationarity (Second-order Stationarity)

• Mean (E(Xₜ)) and Variance (Var(Xₜ)) remain constant over time.
• Autocovariance (Cov(Xₜ, Xₜ₋ᵣ)) depends only on the time difference r, not on the absolute time t.

2. Strict Stationarity

• The joint distribution of (X₁, X₂, …, Xₙ) is the same as (X₁₊ₖ, X₂₊ₖ, …, Xₙ₊ₖ) for all n, k.
• In simpler terms, the entire distribution of the time series remains unchanged over time.

Key Relationships

• Strict stationarity → Weak stationarity (if the second moment exists).
• The reverse is only true for normally distributed time series.

Stylized Facts of Financial Time Series

1. Leptokurtosis

• Asset returns often exhibit “fat tails”, meaning extreme events (crashes/spikes) are more common than in a normal distribution.
• Implication: Risk models need to account for higher probabilities of large losses or gains.

2. Time-Varying Volatility

• Volatility (risk/uncertainty) changes over time.
• Example:
  • During market crashes, volatility spikes.
  • In stable periods, volatility declines.

3. Uncorrelatedness of Returns

• Past returns do not predict future returns.
• Implication:
  • Markets are efficient; patterns disappear once exploited.
  • No easy linear predictability for future returns.

4. Correlated Absolute/Squared Returns

• While raw returns are uncorrelated, their absolute values or squared values (volatility) are highly correlated.
• Implication:
  • Volatility clusters → If the market was volatile yesterday, it is likely to be volatile today.

5. Leverage Effect

• When stock prices fall, volatility increases.
• When stock prices rise, volatility decreases.
• Explanation: Negative shocks (losses) create uncertainty, causing risk to rise.

3. Equity Securities

Common Stocks

• Represent ownership in a company with residual claims on profits.
• Shareholders have the right to:
  • Vote on corporate decisions.
  • Receive dividends (if distributed).
  • Elect the board of directors (who oversee management).
• Executives manage the firm, but the board ensures accountability.

Preferred Stocks

• Priority in dividends: Paid before common stockholders.
• No voting rights: Cannot influence corporate decisions.
• Hybrid nature: Shares features of both equity and debt.

4. Overview of Financial Markets

1. Financial Assets vs. Real Assets

• Real Assets: Tangible assets producing goods/services (e.g., factories, equipment).
• Financial Assets: Claims on income generated by real assets (e.g., stocks, bonds).

2. Money Market

• Short-term debt instruments (maturity < 1 year).
• Examples: T-bills, Commercial Paper, Certificates of Deposit (CDs).
• Low risk, traded in large denominations.

3. Bond Market (Capital Market)

• Long-term debt securities (maturity > 1 year).
• Examples: Treasury Bonds, Corporate Bonds, Mortgage-Backed Securities.
• Provide fixed or formula-based income.

4. Equity Market

• Common Stock: Ownership, voting rights, residual claim on income.
• Preferred Stock: Priority in dividends, no voting rights.

5. Stock Market Indexes

• Track overall market performance.
• Examples: S&P 500, NASDAQ, Dow Jones, FTSE 100.
• Weighting Methods:
  • Price-weighted (e.g., Dow Jones).
  • Market-value weighted (e.g., S&P 500).
  • Equally weighted.

6. Market Classifications

• Primary Market: New securities issued (e.g., IPOs).
• Secondary Market: Trading of existing securities (e.g., NYSE, LSE).
• Auction Markets: Centralized order matching.
• Over-the-Counter (OTC) Markets: Decentralized dealer trading.

7. Market Participants

• Brokers: Match buyers and sellers, charge commissions.
• Dealers: Trade on own account, profit from bid-ask spread.
• Investment Banks: Handle underwriting, M&A, financial structuring.
• Financial Intermediaries: Banks, mutual funds, pension funds, hedge funds.

Financial Econometrics - Problem Set

Question 1

Denote by Zₜ the closing stock price of Apple Inc. (NASDAQ:AAPL) today.
Supposing that {Zₜ} is identically distributed over time, discuss, using probability notation for Zₜ, under what circumstances you can fully understand what will happen tomorrow for Apple’s stock price.

Answer 1

Since {Zₜ} is identically distributed over time, its probability distribution remains the same for all time periods.

To have a complete understanding of Zₜ₊₁ (Apple’s stock price tomorrow), we must know:

P(a ≤ Zₜ ≤ b), for all values of a and b.

This means:

• We know the probability distribution of Zₜ completely.
• Since it is identically distributed, the probability distribution of Zₜ₊₁ is the same as that of Zₜ.
• Therefore, knowing P(a ≤ Zₜ ≤ b) for all values a and b is equivalent to knowing the full behavior of Zₜ₊₁.

This implies that we fully understand what will happen tomorrow for Apple’s stock price in a probabilistic sense. However, it does not mean we can predict the exact stock price due to inherent randomness.

Question 2

Let C and D be two uncorrelated Gaussian random variables with mean zero and variance σ².

Define the time series:
Xₜ = C cos(2πft) + D sin(2πft), where t ∈ Z and f is a constant.

Discuss whether {Xₜ} is weakly stationary, and identify the distribution of Xₜ for any t ∈ Z.

Answer 2

Step 1: Compute the Mean of Xₜ

By taking expectations:

E(Xₜ) = E(C) cos(2πft) + E(D) sin(2πft) = 0.

Since C and D have mean zero, the expected value of Xₜ is 0, which is constant over time.

Step 2: Compute the Autocovariance

The autocovariance function is given by:

E(Xₜ Xₜ₊τ) = E([C cos(2πft) + D sin(2πft)][C cos(2πf(t+τ)) + D sin(2πf(t+τ))]).

Expanding this expression and using the properties of expectation:

= E(C²) cos(2πft) cos(2πf(t+τ)) + E(D²) sin(2πft) sin(2πf(t+τ))
= σ² cos(2πft) cos(2πf(t+τ)) + σ² sin(2πft) sin(2πf(t+τ))
= σ² cos(2πf(t - (t+τ)))
= σ² cos(2πfτ).

Since the autocovariance depends only on τ (the time lag) and not on t, the process {Xₜ} is weakly stationary.

Step 3: Identify the Distribution of Xₜ

Since C ~ N(0, σ²) and D ~ N(0, σ²), their linear combination in the given form implies that:

Xₜ ~ N(0, σ²).

Thus, Xₜ follows a normal distribution with mean 0 and variance σ².

Question 3

A white noise process {εₜ} is defined as a sequence of uncorrelated random variables such that:

• E(εₜ) = με (constant mean).
• Cov(εₜ, εₜ₊τ) = σ²ε if τ = 0, and Cov(εₜ, εₜ₊τ) = 0 if τ ≠ 0.

Is a white noise process weakly stationary or strictly stationary?

Answer 3

A white noise process is weakly stationary because:

• E(εₜ) = με, which is constant over time.
• Variance is σ²ε, which is also constant.
• Autocovariance depends only on τ (zero for τ ≠ 0).

However, we cannot conclude whether it is strictly stationary unless we know the full distribution.

• If εₜ follows a Gaussian distribution, then it is strictly stationary because Gaussian distributions are fully determined by their mean and variance.
• Otherwise, more information is needed to determine strict stationarity.

Question 4

The time series {Yₜ} is defined as:

Yₜ = β₀ + β₁t + Xₜ,

where Xₜ = εₜ + 0.6εₜ₋₁ and εₜ is a white noise sequence with mean zero and variance σ²ε.

1. Show that Yₜ is not stationary.
2. Show that the variance of Yₜ is 1.36 and its lag-1 autocovariance is 0.6.

Answer 4

(i) Show that Yₜ is not stationary

Taking expectations:

E(Yₜ) = β₀ + β₁t,

which depends on t. Since the mean changes over time, Yₜ is not weakly stationary, and therefore not strictly stationary.

(ii) Show that Var(Yₜ) = 1.36 and Cov(Yₜ, Yₜ₋₁) = 0.6

From the definition:

Cov(Yₜ, Yₜ₋₁) = E([Yₜ - E(Yₜ)][Yₜ₋₁ - E(Yₜ₋₁)])
= E(Xₜ Xₜ₋₁).

Expanding Xₜ:

E((εₜ + 0.6εₜ₋₁)(εₜ₋₁ + 0.6εₜ₋₂)) = 0.6.

Thus, Cov(Yₜ, Yₜ₋₁) = 0.6.

Question 5

Company A has issued both common and preferred stocks. Common shares trade at S$50.
If an investor buys 1000 common shares, answer:

1. What is she entitled to?
2. What is the potential gain?
3. What is the potential loss?
4. How does this change for preferred stock?

Answer 5

(i) Investor’s Entitlements

• Receives dividends (if issued).
• Has voting rights (worth 1000 votes).

(ii) Potential Gain

• Unlimited potential gains, as stock prices have no upper bound.

(iii) Potential Loss

• Maximum loss = Investment amount → 1000 × S$50 = S$50,000.

(iv) Preferred Stock Differences

• Dividends must be paid before common shares.
• No voting rights.
• Lower risk but limited growth potential.

5. Risk & Return in Financial Markets

Returns Computation

Holding Period Return (HPR)

The Holding Period Return (HPR) measures total return over a holding period:

HPR = (End Price - Start Price + Dividend) / Start Price

Where:

• End Price = Final asset price
• Start Price = Initial asset price
• Dividend = Cash distributed

Log Return

Logarithmic return is another common return measure:

Log Return = ln(End Price / Start Price)

Risk Premium & Excess Returns

Risk Premium

The Risk Premium is the additional return expected for taking on risk:

Risk Premium = Expected Return - Risk-Free Rate

Excess Return

The Excess Return is the difference between actual return and the risk-free rate:

Excess Return = Actual Return - Risk-Free Rate

Investment Analysis

Market Conditions and Expected Returns

Consider an investment in a financial asset. The asset’s price movement depends on market conditions in 1 year. The risk-free rate is 4%.

Market Scenarios and Returns

Questions

1. What would be the excess returns if the market turns out to be ‘excellent’?
2. What is the expected return of the investment?
3. What is the risk premium of the investment?
4. What is the risk of the investment measured by variance?

Answers

1. Excess Return in the ‘Excellent’ Market

Excess Return = Rₜ - Risk-Free Rate
= 0.4 - 0.04 = 0.36 (i.e., 36%)

2. Expected Return of the Investment

Formula:
E(Rₜ) = Σ [P(Rₜ) × Rₜ]

Substituting values:

E(Rₜ) = (0.2 × 0.4) + (0.7 × 0.1) + (0.1 × -0.2)
E(Rₜ) = 0.08 + 0.07 - 0.02 = 0.13 (i.e., 13%)

3. Risk Premium of the Investment

Risk Premium = Expected Return - Risk-Free Rate

= 0.13 - 0.04 = 0.09 (i.e., 9%)

4. Variance of the Investment

Formula:
Var(Rₜ) = Σ [(Rₜ - E(Rₜ))² × P(Rₜ)]

Substituting values:

Var(Rₜ) = (0.4 - 0.13)² × 0.2 + (0.1 - 0.13)² × 0.7 + (-0.2 - 0.13)² × 0.1

= (0.27)² × 0.2 + (-0.03)² × 0.7 + (-0.33)² × 0.1
= 0.0729 × 0.2 + 0.0009 × 0.7 + 0.1089 × 0.1
= 0.01458 + 0.00063 + 0.01089
= 0.0261 (i.e., 2.61%)
—

6. Portfolio Theory & Diversification

Portfolio Return & Risk

Portfolio Return

The return of a portfolio is the weighted sum of individual asset returns:

Portfolio Return = w₁R₁ + w₂R₂ + … + wₙRₙ

Where:

• w₁, w₂, …, wₙ = Weights of assets in the portfolio
• R₁, R₂, …, Rₙ = Returns of each asset

Portfolio Variance

The total risk (variance) of a portfolio is given by:

Portfolio Variance = Σ Σ wᵢ wⱼ Cov(Rᵢ, Rⱼ)

Where Cov(Rᵢ, Rⱼ) is the covariance between asset returns.

Correlation & Diversification

Key Correlation Cases:

• Perfect Correlation (ρ = 1) → No diversification benefit.
• Perfect Negative Correlation (ρ = -1) → Perfect hedge (zero risk possible).

Diversification Effect

Diversification reduces risk through low or negative correlation:

Portfolio Risk = sqrt( w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂ )

Where:

• ρ = Correlation coefficient between asset returns.

Capital Allocation & Sharpe Ratio

Capital Allocation Line (CAL)

Represents the tradeoff between risk-free assets and risky portfolios:

Expected Return of Portfolio = Risk-Free Rate + (Sharpe Ratio × Portfolio Standard Deviation)

Sharpe Ratio (Risk-Return Tradeoff)

Measures return per unit of risk:

Sharpe Ratio = (Portfolio Return - Risk-Free Rate) / Portfolio Standard Deviation

A higher Sharpe Ratio indicates a better risk-adjusted return.

7. Markowitz Portfolio Theory

Efficient Frontier & Optimal Portfolio

• Efficient Frontier → Portfolios that provide the highest return for a given level of risk.
• Minimum Variance Portfolio → The portfolio with the lowest possible variance.
• Optimal Portfolio → The portfolio that maximizes the Sharpe Ratio.

Diversification with Multiple Assets

For a portfolio with N assets, the portfolio variance is:

Portfolio Variance = Σ Σ wᵢ wⱼ Cov(Rᵢ, Rⱼ)

As N → ∞, the total risk reduces to:

Portfolio Risk ≈ Average Asset Covariance

This demonstrates risk reduction through diversification.