Capital asset price model(CAPM) is a model used to determine a theoretically appropriate required rate of return of an asset, to make decisions about adding assets to a well-diversified portfolio. The fomula showed as follows:
\(E(r_i)-r_f=\beta_{im}(E(r_m)-r_f)\)
\(E(r_i)\):the expected rate of return on asset i
\(r_f\):risk free rate
\(E(r_i)-r_f\): market premium
\(\beta_{im}=cov_{i,m}/var_m\):systematic risk
\(E(r_m)\):expected return of the market
\(E(r_m)-r_f\):risk premium
\(\beta_{im}\)is the systematic risk which can not be avoid through diversification. It is calculated using regression analysis, and we can think of beta as the tendency of a security’s returns to respond to swings in the market. A beta of 1 indicates that the security’s price will move with the market. A beta of less than 1 means that the security will be less volatile than the market. A beta of greater than 1 indicates that the security’s price will be more volatile than the market. For example, if a stock’s beta is 1.2, it’s theoretically 20% more volatile than the market.
The secuirty market line(SML)essentially graphs the results from the capital asset pricing model (CAPM) formula
For individual securities, we make use of the security market line (SML) and its relation to expected return and systematic risk (beta) to show how the market must price individual securities in relation to their security risk class. Once the expected/required rate of return E(R_{i}) is calculated using CAPM, we can compare this required rate of return to the asset’s estimated rate of return over a specific investment horizon to determine whether it would be an appropriate investment. To make this comparison, you need an independent estimate of the return outlook for the security based on either fundamental or technical analysis techniques, including P/E, M/B.
Though the CAPM model won the Nobel Prize in economic,it showed untestable in realistic because the formula has many assumptions showed below.
1.assume investors will not affect prices,fair.
2.all investors have the same one-period horizon,fair.
3.They all have access to the same securities and can borrow and invest at the same risk-free rate,a little strict.
4.No taxes and transaction costs,strict when we do high frequency trading.
5.Have the same information and expectations aboout securities returns and risk,strict.If so, everybody will look at same mean-variance diagram and located the same mean variance effient(MVE) portfolio.
The last assuption make it untestable in realistic.
For stock:they use excess return on 25 portfolios,formed on size and B/M equity.
Fomula:\(r_{it}-r_{ft}=a_i+b_i(r_{mt}-r_{ft})+s_iSMB_t+h_iHML_t+e_{it}\)
\(r_{it}\):expected return on stock i at time t.
\(r_{ft}\);risk free rate at time t
\(r_{mt}\):return on market portfolio of stock.
\(SMB_t\):return on small firms minus that of large firms. For year t, ME is measured at the end of June. They use NYSE breakpoints for ME to allocate NYSE, Amex and NASDAQ stock to five size quintiles.
\(hml_t\):return on high B/M firms minus that of low B/M firms.For year t, ME is measured at the end of December of t-1,BE is book common equity of the fical year year ending in calendar year t-1.They use NYSE breakpoints for BE/ME to allocate NYSE,Amex and NASDAQ stock to five book to marktet quintiles.
They conduct 25 portfolios from intersection of the size and BE/ME quintiles and calculate value-weighted monthly return on portforlio from July of t to June of t+1.Excess return choosed as dependend variables on these 25 portforlios for July 1963 to December 1991. They also have tried to use portfolio formed on E/P and D/P which which are also informative about avearge return.
The reason why they conduct 25 portolios from intersection of the size and BE/ME quintiles is that the sepeartion can reduce the cross-correlations between variables which will affect estimated beta.
For government and corperate bonds:
Formula:\(r_{it}-r_{ft}=a_i+mTEAM(t)+dDEF(t)+e_t\)
TEAM is LTG-RF,where LTG is the monthly percent long-term government bond return and RF is the one-month T-bill rate.Observed at the beginning of the month.
DEF is CB-LTG, where CB is the return on a proxy for the market portfolio of corporate bonds.
\(r_{it}-r-{ft}\):excess return on government and corporate bonds
The seven bond portfolios used as depent variables are 1- to 5- year and 6-to 10-year government and corporate bonds rated Aaa, Aa, A,Baa and below Baa(LG) by Moody’s.
Five factor model they finally come out can provide a good description of the cross-section of average return,but they do not require when we have identify the true factors.And five factors can be used to guide portfolio selection.
Formula:\(r_it-r_ft=a_i+b(r_{mt}-r_{ft})+sSMB(t)+hHML(t)+mTEAM(t)+dDEF(t)+e_t\)
Fomula:\(r_{it}-r_{ft}=a_{i}+b(r_{mt}-r_{ft})+s_{it}SMB(t)+h_{it}HML(t)+p_{it}UMD_t+e_{it}\) Other variables are same as Fama-French three factors model
\(UMD_t\):Carhart sorted portfolio of mutual funds by rate of return in past 12 months and divided them into 10 levels.$UMD_t is the monthly rate of return of the first level minus that of the last level.
To evaulate the model, Carhart also sorted mutual fund by their two to five year return to prove the longer-term persistence of the model.
formula:\(r_t-r_ft=a_t+b_t(r_{mt}-r_{ft})+s_tSMB(t)+h_tHML(t)+r_tRMW(t)+c_tCMA(t)+e(t)\)
They add two new explainary factors into the three factors model representing the operating profitability and investment style(expected growth of book equity).The first three variables are same as previous model.
RWM:rate of return of robost operation minus that of weak operation.
CMA:rate of return of conservative investment minus that of aggressive investment.
In the final regression, in order to reduce the collinearity, they divided the stock into two size groups(small and big) using NTSE median as the market cap breakpoint.Then small and big stocks are allocated independently to four B/M groups(low to high) and four Investment style groups(low to high). SMB,HML,RWM and CMA constructed using 222*2 sorts on size, B/M,OP and Inv.The detailed components of factors can be see at table3 in the paper of five factor model.
Finally, the intersections of the three sorts produce 32 size-B/M-Inv portfolio.
fomula:\(E(r_i)-r_f=\beta_{MKT}E[MKT]+\beta_{ME}E[r_{ME}]+\beta_{I/A}E[r_{I/A}]+\beta_{ROE}E[r_{ROE}]\)
\(r_{I/A}\):The difference between the return on a portfolio of low investment stocks and the return on a portfolio of high investment stocks.
\(r_{ROE}\):The difference between the return on a portfolio of high profitability (return on equity, ROE) stocks and the return on a portfolio of low profitability stocks
The model develped based on the three factor model，Carhart four factor model and Q theory.In q factor model reference, Kewei,chen,Zhang take 2-period stochastic general equilibrium model for instance to further illistarted the economics fundation of the q factor model.Through the first-order condition of the equilibrum:
\(E_0[r_{i1}]=\frac{E_0[\prod_{i1}]}{1+a(\frac{I_{i0}}{A_{i0}})}\)
\(E_0[r_{i1}]\):stock return
\(\prod_{i1}\):marginal product of assets(profitability)
\(A_0\):date 0 asset
a:a > 0 is a constant parameter
\(I_{i0}\):investment on date 1 Through that equation,we can know that the expected return of the stock is assciate with its marginal profitiability which can be recognized as return on equity(ROE) or investment-to-assets(I/A).Therefore,\(r_{I/A}\) chosen as profitability factor.
Then, they used systematic regression to test the performance of q factor and compared with the three factor model and four factor model.The empirical results showed that q-factor model can represent more anomalies than three-factor model.For example:
1.q-factor has less significant alphas after making factor regressions for individual Earings momentum(SUE-1)and price momentum(R6-6) deciles.
2.q-factor model fails to fit the operating accruals(OA) anomaly but perform better for percent operating accruals(POA) anomaly.
3.When testing size and B/M quintile portfolios, q-factor appeared similar to the three-factor and four-factor model.
The main difference between q-factor and three-factor model is that the Investment factor and profitability factor in q-factor are not explained as risk factors.On the other hand, q-factor model connects the expected return of firms with characteristics of firms through econmics fundation.Therefore, the model will not be directly influenced by mispricing.
formula:\(r_t=\alpha+\beta_{MKT}MKT_t+\beta_{SMB}SMB_t+\beta_{MGMT}MGMT_t+\beta_{PERF}PERF+e_t\)
\(r_t\):the return in month t on the anomaly’s long-leg,short-leg,or long-short spread
\(MKT_t\):the excess market return
\(SMB_t\):size factor.
\(MGMT_t\):mispricing factors
\(PERF_t\):mispricing factors
The mispricing factors were based on averages of stocks’ anomaly rankings.They construct mispricing factors by averaging rankings within the set of 11 prominent anomalies.For each anomaly i they compute the spread \(R_{i,t}\), between the value weighted returns in month t on stocks in the first and tenth NYSE deciles of the ranking variable in a sort at the end of month t-1 of all NYSE/AMEX/NASDAQ stocks with share prices greater than \(5. the regression equation:\)R_{i,t}=i+b_iMKT_t+c_iSMB_t+u{i,t}$
MKT_t and SMB_t constructed same as Fama French three factor model.
Then, they computed the correlation matrix of the estimated reisiduals of regression and divided the 11 anomalies into two clusters.The first cluster of anomalies includes net stock issues, composite equity issues, accruals, net operating assets, asset growth, and investment to assets, which represent quantities that firm’s management.The second cluster of anomalies includes distress, O-score, momentum,gross profitability, and return on assets, which related to performance.The first clusters construct the \(MGMT_t\) factor and second construct the \(PERF_t\) factor.
When constucting size factor in this model,they depart more significantly from the approach in studies before.The stocks they use to form size factors in a given month are the stocks not used in forming either of the mispricing factors.And value of SMB in a given month is the return on the small-cap leg minus large-cap return.
To test the sentiment effect,they seperated the parameter estimates from mispricing pricing model into three panels.First is estimates for long-short spreads, second is for long legs and third is for short legs.
The main differece between mispricing model and other models is that the mispricing factors are combining the information in multiple anomalies,as opposed to being single-anomaly factors.
For example,we can first set the Fama French three factor model’s empritical results as bench mark.Then adding a new factor into the model like the profitability factor,RMM.Of crouse, we need to reduce the collinearity between the new variables and original variables using sort skills.Next, we should look as intercepts of two models produced by the factors after individual linear regression.If after adding the new factor, the alpha reduces(or t statistic of alpha reduce,alpha becomes insignificant) and the t statistic of new factors is in the significant range(large t value). The new factor is efficient in the model.
1.p quant and q quant are two seperate branches of the quantitive finance.Q quant focus more on modeling and theory.The quants who operate in the Q world of derivatives pricing are specialists with deep knowledge of the specific products they model. Securities are priced individually, and thus the problems in the Q world are low-dimensional in nature.P quant focus more on the data analysis.They use advanced statistics and econometric techniques to build model and use emprical result to price asset.The goal of q quant is somewhat like model the future.
2.the core knowledge of two branches
p quant:stochastic calculus, finanicial mathematics.
q quant:econometrics,machine learning, time-series data analysis.
3.From my point of view,p quant has a brighter future in the Chinese market. With the explosion of the information and data in the recent world, using the real world data and doing emprical research are obligatory. Focusing on the classic model is unrealistic, we need more advanced technique to handle the big data and create new model to accommodate the future.By the way,the models in the reference paper can be all seperated to the P quant range and Q quant is generate approaching p quant.
我理解的量化投资就是用高级的金融模型寻找投资策略，机器学习分析数据，编写交易策略程序实现高频交易等。量化投资会是一个非常辛苦，压力很大的工作，但是一个充满奇迹的行业，会有非常多的机会，努力也能得到应当回报的职业。
想做的原因主要是在本科大二的时候选择了金融辅修之后对金融觉得非常感兴趣，本身对数学，大数据分析以及建模都有与生俱来的天赋和兴趣。然后慢慢了解到了quant这个职业，既然这是一个我喜欢的方向并且有着不错的回报，自然成为了我人生的追求。既然能找到自己喜欢干的事情之后也就努力出国向着这个方向发展，最后选择来到香港这个金融大圈子发展。虽然这个职业的要求真的非常高，但是我还是向着这个方向踏实地慢慢努力，想着如果真的不行就去读一个phd（所以在master选择了跟着导师做项目以备以后能向phd发展），然后再往这方面试试或者最好是在硕士毕业后能找到这方面的实习或者工作是最好的。
能胜任这个工作的点：能吃苦耐劳，学习能力强，自律能力强。编程基础不错，有金融的基础概念，工商数据分析结合的交叉背景。
来香港之后觉得压力过大，用眼过多之后或者工作太久后就会选择去打篮球或者健身，打了5年多的篮球也参加过大学院队比赛还是觉得比单纯的跑步或者去健身房更加有意思点，因为打球能和朋友更多地互动交流，而且进球的感觉一直都会让人有快感，而且小时候非常爱看nba,更是崇拜篮球明星的那些投篮扣篮的动作。最初的健身就是为了能让自己的身体素质更加好，同时也为了参加比赛做的日常训练，现在越发觉得健身的重要性，身体是革命的本钱，日常健身更能提高自我学习的效率和生活的状态。