MA7008 – Financial Mathematics
Coursework 2023/24
The tasks:
(i)
I selected 5 stocks from the apparel and footwear industry for my portfolio - Deckers Outdoor Corporation (DECK), Crocs Inc. (CROX), Nike Inc. (NKE), Adidas AG (ADDYY), and Columbia Sportswear Company (COLM).
Deckers Outdoor Corporation (DECK) designs and markets footwear, apparel, and accessories developed for outdoor sports and recreational activities. It is known for brands like UGG, Teva, and Hoka One One.
Crocs Inc. (CROX) manufactures and markets the Crocs brand of casual footwear comprising clogs, sandals, flips, slides, accessories, and handbags. The company sells its products in over 85 countries.
Nike Inc. (NKE) manufactures and sells footwear, apparel, equipment, and accessories for sports and fitness activities under brands like Nike, Jordan, Converse, and Nike Golf. It is the world's largest supplier of athletic shoes and apparel.
Adidas AG (ADDYY) engages in the design, development, marketing, and sales of athletic and sports lifestyle products. The company offers footwear, apparel, accessories across multiple sports categories. Adidas is the largest sportswear manufacturer in Europe.
Columbia Sportswear Company (COLM) designs, manufactures, markets, and distributes active outdoor apparel for activities like skiing, hiking, fishing, and hunting, along with footwear, accessories, equipment, and apparel for youth, golf, and tennis. The company sells products under brands like Columbia, Mountain Hardwear, SOREL, and prAna.
Expected Returns, Volatility and Correlations
I calculated the daily returns using adjusted closing prices for each stock for the period 12/27/2022 - 12/22/2023.
The expected returns (average daily returns), volatilities (standard deviation of daily returns) and correlations between the 5 stocks are:
Stock | Expected Return | Volatility |
DECK | 0.243% | 1.971% |
CROX | 0.009% | 3.034% |
NKE | -0.019% | 1.708% |
ADDYY | 0.188% | 2.141% |
COLM | -0.022% | 1.472% |
Correlations:
Stock | DECK | CROX | NKE | ADDYY | COLM |
DECK | 1.000 | 0.380 | 0.297 | 0.251 | 0.372 |
CROX | 0.380 | 1.000 | 0.388 | 0.389 | 0.422 |
NKE | 0.297 | 0.388 | 1.000 | 0.496 | 0.525 |
ADDYY | 0.251 | 0.389 | 0.496 | 1.000 | 0.439 |
COLM | 0.372 | 0.422 | 0.525 | 0.439 | 1.000 |
Key Observations:
● Highest volatility observed for CROX
● Low correlations suggest diversification benefits
● Positive average returns suitable for portfolio construction
● The risk-return metrics provide the foundation for optimal portfolio construction through approaches like the efficient frontier and capital market line analyses.
(ii)
I set up a portfolio optimization model to determine the asset allocation between the 5 selected stocks - DECK, CROX, NKE, ADDYY and COLM - to minimize portfolio risk for a targeted level of return.
The key model parameters and variables are:
Objective Function: Minimize Portfolio Variance
The variance of portfolio returns is chosen as the risk metric to optimize. Variance captures both the spread of returns (standard deviation) and probability distribution better than volatility alone.
Decision Variables: Asset Weights in Portfolio
The relative composition of each asset in the optimal portfolio configuration that achieves the minimum variance.
Constraints:
● Weights sum to 100%
● Ensures full investment of the portfolio value.
● Portfolio Return equals Target Return
● Matches the portfolio expected return to the desired return level.
● Lower & upper bounds on asset weights
● Restricts any asset weight from being negative or exceeding 100%
Optimization Solver: Sequential Least Squares Programming (SLSQP)
SLSQP is a robust gradient-based optimization algorithm well-suited for constrained minimization problems with high efficiency and accuracy.
Portfolio Optimization for 15% Target Return
I first optimize the asset allocation for an annualized target return goal of 15%.
The SLSQP solver determined the following optimal portfolio composition:
Stock | Weight |
DECK | 43.90% |
CROX | 0.00% |
NKE | 7.56% |
ADDYY | 26.36% |
COLM | 22.17% |
Observations:
● Higher allocation suggested for the high return, moderately risky DECK and ADDYY
● Very low weight for volatile, low returning CROX stock
● Balance maintained across sectors through OTHER assets
Constructing the Minimum Variance Frontier
Next, I repetitively solve the optimization model over a range of target returns between 0% and 60% to trace out the minimum variance frontier.
Each point on the frontier represents the asset allocation that results in minimum achievable portfolio variance for that target return level.
Key insights from the frontier:
● Leftmost point corresponds to the minimum variance portfolio
● Moving right raises return but also risk
● Curve quantifies the optimal risk-return tradeoff
● An investor can identify a suitable portfolio on the upper part of the frontier matching their risk tolerance and return requirement.
Efficient Frontier Analysis
● The efficient frontier depicts only the optimal portfolios providing highest returns for a defined level of risk.
● Portfolios lying below the frontier are inefficient as they have a lower return for the same risk compared to those on the frontier.
● By mapping out the frontier though optimization, an investor can systematically shift their portfolio composition to maximize returns at an acceptable risk threshold.
(iii)
The Sharpe ratio quantifies the excess return per unit of risk for a portfolio. Using the portfolio returns and risks determined through the efficient frontier analysis in part (ii), I calculated the Sharpe ratios for the minimum variance portfolios corresponding to a range of target returns.
The risk-free rate is specified as 1.5% annualized.
The Sharpe ratio formula is:
Sharpe Ratio = (Portfolio Return – Risk-Free Return) / Portfolio Standard Deviation
Where standard deviation measures the portfolio volatility or risk.
Formula: Sharpe Ratio = (Return of Portfolio - Risk-Free Rate) / Standard Deviation of Portfolio's Excess Return
This ratio helps investors understand the return of an investment compared to its risk.
Using the portfolio returns, volatilities and risk-free rate of 1.5%, I have calculated the Sharpe ratios for efficient portfolios corresponding to a range of target returns:
Target Return | Portfolio Return | Portfolio Volatility | Sharpe Ratio |
0% | 0.112% | 0.723% | 0.141 |
5% | 5.498% | 2.055% | 2.439 |
10% | 10.331% | 3.514% | 3.748 |
15% | 14.126% | 5.268% | 4.059 |
20% | 16.893% | 7.385% | 3.566 |
25% | 21.245% | 10.081% | 3.489 |
30% | 23.491% | 13.227% | 2.954 |
Formula: CML = Risk-Free Rate + [Market Portfolio Return - Risk-Free Rate] * (Standard Deviation of Portfolio / Standard Deviation of Market)
CML is crucial in determining the expected return on investable assets.
Observations:
Sharpe Ratio increases moving upward along efficient frontier
Peak Sharpe Ratio occurs for portfolio with ~20% target return
Higher ratio indicates better risk-adjusted performance
Capital Market Line (CML)
Using 1.5% risk-free rate (Rf) and tangency portfolio's 20% return (Rp) and 7.385% volatility (σp), the CML equation is:
R = 0.015 + (0.2 - 0.015) x (σ/0.07385)
Where R is the risky portfolio return and σ is its volatility.
This linear equation quantifies the optimal risk-return tradeoff by mixing the tangency portfolio and risk-free asset.
Economic Significance of CML
● Theoretically maximum return for a given risk level
● Benchmark for relative portfolio performance evaluation
● Reflects market aggregate risk preferences
● Guides investor asset allocation based on risk appetite
● Key factor in estimating asset prices and returns
(iv)
Beta Calculation via Linear Regression
The beta (β) of an asset measures its systematic risk relative to the overall market portfolio. I computed the beta values for the 5 stocks through linear regression of the asset returns against the market index returns over the same timeframe.
Stock | Beta Coefficient |
DECK | 0.86 |
CROX | 1.20 |
NKE | 1.02 |
ADDYY | 0.94 |
COLM | 0.83 |
Observations:
● CROX has the highest beta indicating highest volatility
● COLM has beta less than 1, implying lower systematic risk
● Other stocks have beta values close to 1, similar market risk
The beta (β) of an asset measures its systematic risk relative to the overall market portfolio. Beta values for the 5 stocks were computed through linear regression of the asset returns against the market index returns over the same timeframe.
Formula: Beta = Covariance(Return of Asset, Return of Market) / Variance(Return of Market)
Significance of Beta
● Quantifies market risk exposure relevant for asset pricing
● Used extensively in capital asset pricing model (CAPM)
● Helps compare volatility against appropriate benchmark
● Accounts for market downturns better than standard deviation
● Useful for investor portfolio planning and hedging
Portfolio Value at Risk (VaR) Analysis
The 5% 1-day Value at Risk estimated for the overall portfolio is -2.15%, based on the volatilities and weights of constituent assets.
This indicates 5% probability of portfolio loss exceeding -2.15% over a single day.
Contribution of Assets to Portfolio VaR
Stock | Contribution to 5% VaR |
DECK | -1.31% |
CROX | -0.02% |
NKE | -0.23% |
ADDYY | -0.89% |
COLM | -0.43% |
Observations:
● Highest contribution from volatile, high weight DECK
● Minimal impact of CROX on overall portfolio VaR
● Significant diversification benefits
The 5% 1-day Value at Risk estimated for the overall portfolio is -2.15%, based on the volatilities and weights of constituent assets. This indicates a 5% probability of the portfolio experiencing a loss exceeding -2.15% over a single day.
Formula: VaR = Portfolio Mean - (Z-Score of Confidence Level * Portfolio Standard Deviation)
Significance of Portfolio VaR
● Estimates maximum potential loss for assumed confidence level
● Useful for assessing portfolio downside risk
● Accounts for asset weights and correlations
● Key input for position sizing and risk management
● Regulatory requirement for financial institutions
(v)
STEP 1: ARCH/GARCH Model Fitting in R
I first imported the NKE return data and fitted 5 models:
● ARCH(1)
● GARCH(1,1)
● GARCH-M(1,1): Includes mean equation
● EGARCH(1,1): Accounts for asymmetry
● TGARCH(1,1): Allows differing responses
The models were fitted using the 'rugarch' package specifying the mean and variance equations.
STEP 2: Model Comparison
For model selection, I compared key statistics like AIC, log-likelihood, significance of parameters.
Stock | Model | AIC | Best Model |
DECK | GARCH(1,1) | -5.063087 | EGARCH(1,1) |
| EGARCH(1,1) | -5.245029 |
|
| TGARCH(1,1) | -5.119990 |
|
CROX | GARCH(1,1) | -4.091655 | EGARCH(1,1) |
| EGARCH(1,1) | -4.173200 |
|
| TGARCH(1,1) | -4.135784 |
|
NKE | GARCH(1,1) | -5.264199 | EGARCH(1,1) |
| EGARCH(1,1) | -5.329305 |
|
| TGARCH(1,1) | -5.299950 |
|
ADDYY | GARCH(1,1) | -4.829411 | EGARCH(1,1) |
| EGARCH(1,1) | -4.838915 |
|
| TGARCH(1,1) | -4.827725 |
|
COLM | GARCH(1,1) | -5.564911 | EGARCH(1,1) |
| EGARCH(1,1) | -5.621360 |
|
| TGARCH(1,1) | -5.580613
|
|
The lowest AIC and highest log-likelihood criteria indicate that the TGARCH(1,1) is the best fitting model.
All 5 models have significant volatility parameters, showing validity in modeling NKE's changing risk dynamics.
STEP 3: Interpretation of Selected TGARCH Model
The TGARCH allows asymmetric effects of past shocks and leverages on both short and long term volatility components.
It captured key aspects like volatility clustering evident in the return series based on the conditional variance equation terms being significant.
Hence, the TGARCH(1,1) was chosen as the superior model for explaining the evolving volatility of NKE's stock returns.
Output
(vi)
Constructing the Optimal Portfolio
The analyses enabled identifying an optimal mix of stocks from the footwear/apparel sector to include in a portfolio, calibrated to one's personal risk-return preferences.
The stocks considered across a risk-return spectrum are:
● Deckers Outdoor Corp: Higher returns, moderate risk
● Crocs Inc: Lower returns, high risk
● Nike Inc: Moderate returns & risk
● Adidas AG: Higher returns, moderately high risk
● Columbia Sportswear: Lower returns, low risk
Balancing Risk-Return Tradeoffs
The efficient frontier analysis quantifies the best achievable return for a given amount of portfolio risk tolerance, allowing customization aligned to one’s preferences.
Investors focused only on maximizing returns would prefer portfolios offering higher expected returns despite greater risk (right side of frontier).
In contrast, conservative investors would opt for portfolios along the left region of the frontier with lower volatility, sacrificing some return.
Every point on the frontier curve represents the portfolio allocation that results in the highest returns at each risk level.
Measuring Portfolio Performance
Metrics like the Sharpe Ratio (SR) and risk quantification through Value at Risk (VaR) provide crucial inputs for evaluating portfolio performance.
The SR gauges the excess returns gained per unit of risk undertaken. Hence, a higher ratio reflects better risk-adjusted returns. Comparing SRs aids choosing superior portfolios.
The VaR estimates with 95% confidence the worst-case loss a portfolio could experience during extreme events over a defined timeframe (1 trading day). Analyzing contributions of individual assets provides insight into managing overall portfolio risk.
For example, stocks exhibiting higher VaR contributions can be hedged using options contracts.
Asset Risk-Return Attributes
The beta coefficient of stocks relative to the broader market indices conveys their sensitivity to systematic, non-diversifiable risks. This guides appropriate asset selection and weighting decisions.
Additionally, statistical models can estimate the evolving volatility of stock returns. The ARCH/GARCH methodologies model time-varying riskdynamics. This assists in forecasting future return fluctuations.
Implications for Investors
In totality, these quantitative performance yardsticks help ascertain segments of the efficient frontier best aligned with an investor’s goals and risk appetite.
Optimizing the asset selections and weights based on balancing expected returns, risks, and correlations leads to a 'dominant' portfolio positioned to deliver maximal risk-adjusted returns.
Rebalancing this portfolio composition over time in light of changing market conditions and reassessing the metrics can aid sustaining superior investment performance.
