Quantitative Methods in Finance
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Quantitative Methods in FinanceMSc Finance 2015-2016Question 1Dissert upon the suitability of the Carhart model as a way to explain the excess returns of the stock. Pay special attention to the regression output and diagnostic tests as conducted in class and in the tutorials.The Carhart ModelIn a risk-neutral world, the return of a stock should be equal to the risk-free rate. However, in the real world, the volatility of a stock among other factors creates active returns, or α, when the returns exceed those of a benchmark with a similar level of risk. Different models can be used to determine the excess return, but when it comes to finding α, the suitability of a model greatly varies. For example, Fama and French deem the CAPM invalid as the assumptions it makes are unrealistic and lead to excess returns greater than they should be, thus, overestimating α.Designed over 20 years ago by Eugene Fama and Kenneth French, the Fama-French three-factor model is used to describe the stock returns. They based their model on the fact that small caps and stocks with a high book-to-market ratio tend to outperform the market, which is something the CAPM omitted. As a result, Fama and French designed two portfolios supporting this idea and added the newly formed parameters to the traditional CAPM.In that regards, the Carhart model is an extension of the Fama-French model, which has first been pushed forward by Carhart in 1997.Whilst the Fama-French model uses three factors, Carhart’s is a four-factor model, which includes the momentum factor. The momentum factor is an empirical observation, which links the future movements of a stock to its past performance. In that matter, if a stock sees its price rise, its momentum should push the price to rise even further. On the other hand, a stock with a declining value should see its price fall even lower. Adding this extra empirical observation should, as a result, assess more efficiently the true α.Thus, the Carhart model should explain the return of a stock by encompassing the risks and returns of four factors affecting stocks:Market-Risk factor – The Market-Risk factor represent the returns of the market minus the risk-free free rate. The Market-Risk factor captures the sensitivity of a stock to the market returns.SMB (Small-Minus-Big) factor – The Small-Minus-Big factor represents a zero-investment portfolio, which is long on small capitalization stocks and short on big capitalization stocks. The idea behind this factor is that small firms tend to outperform larges ones. Hence, the SMB aims at capturing abnormal returns based on the market cap.HML (High-Minus-Low) factor – The High-Minus-Low factor represents a zero-investment portfolio, which is long on high book-to-market ratio stocks (value stocks) and short on low book-to-market ratio stocks (growth stocks). The rationale behind the HML factor is that value stocks tend to outperform growth stocks. Hence, the HML aims at capturing abnormal returns based on the stock value derived from the book-to-market ratio.UMD (Up-Minus-Down, a.k.a. WML, Winners-Minus-Losers) factor – The Up-Minus-Down factor represents a zero-investment portfolio, which is long on previous 12-month winning stocks and short on the previous 12-months losing stocks. The rationale behind the UMD factor is that rising stocks will keep rising further while falling stocks keep falling down. Hence, the UMD factor aims at capturing abnormal returns based on a stock momentum.However, we want to understand whether the Carhart model is actually suitable to explain excess returns, so as to know whether or not we can use it further in our analysis and our strategies.To establish the suitability of this model, we have decided to test it on a stock. Amazon appeared to us as an interesting stock since its volatility is really high. Moreover, Amazon’s stock exhibits a strong positive momentum, leading the stock to more than double over the last 12 months. Thus, this positive upward trend should directly impact our results. We believe this is beneficial to our analysis in order to find Amazon’s true α.
In order to conduct an analysis of quality, we have selected a one-year period ranging from the 31st of October 2014 to the 30th of October 2015. As for the risk-free rate, the market excess return, the SMB, the HML, and the Momentum, we obtained them directly from Kenneth French’s website.Running the Carhart ModelTo evaluate this Carhart model, we have used a linear regression using four explanatory variables (the market-risk factor, the SMB factor, the HML factor, and the Momentum).We built and ran the following model: [pic 1]Thanks to this model, we obtained the following results:Regression StatisticsMultiple R0,5643R Square0,3184Adjusted R Square0,3074Standard Error0,0172Observations252 dfSSMSFSignificance FRegression40,034020,008528,84741,11025E-19Residual2470,07280,0003Total2510,1068    CoefficientsStandard Errort StatP-valueLower 95,0%Upper 95,0%Intercept0,00240,00112,18590,02980,00020,0045X Variable 1 – EXMKT0,00990,00128,27368,14E-150,00750,0122X Variable 2 – SMB-0,00620,0023-2,70810,0072-0,0107-0,0017X Variable 3 – HML-0,01280,0029-4,49701,06E-05-0,0185-0,0072X Variable 4 – UMD-0,00390,0017-2,30910,0218-0,0074-0,0006