Analysis Of QMJ Strategy With Black-Scholes-Merton And CAPM Models

Evaluation of Strategy Values

The QMJ (quality minus junk) strategy monthly data has been provided in the file “QMJ and FF3 factors, 1980-217”.

In the Black-Scholes-Merton model, for allhere is expected rate of return and volatility rate is denoted byare the two parameters to be estimated. is the monthly value at time t (Mamdouh Medhat, lecture 2, February 8, 2018).

From the end of June 1980 to the end of December 2017, strategy values (in $ millions)of an investment have been provided with .

Assumptions: Percentage change in the strategy values within a time period of, where, is normally distributed. The mean and variance of this distribution is directly proportional to (proportionality constants are considered to be constant) (Rachev, 2011). Sampling frequency =1 month, sampling time T=450 months.

Estimation procedure: BSM model implies that the logarithm of the strategy value at time  is normally distributed, . The corresponding one-period continuously compounded returns are given by (column C of data file) are independent and identically distributed.

Maximum Likelihood estimation procedure is used to evaluate mean and standard deviation as (for mean) and.

The MLE for and are as follows:

and.

From given QMJ strategy values,, so the estimated are in units of per month (per )(Mamdouh Medhat, lecture 2, February 8, 2018).

The ML estimates are calculated as (detailed calculation in attached excel file).

From the above results MLE for and are. Correspondingper annum values are.

The expected rate of return of 6.2713% per annum indicates that the strategy of QMJ will on average increase the portfolio value. The value of 8.8094% indicates the market volatility for the current strategy with respect to the broad sample. This low rate of volatility will help in steady growth of the portfolio and hedging will generate much more rate of return for the strategy. (79/100)

BSM parameter values are per annum.

For given initial strategy value, the expected value at time T follows log-normal distribution and. For 31^st December 1980, T=0.5 year, for 31^st December 1999, T= 19.5 years and for 31^st December 2017, T=37.5 years. Here =1 |(in $ millions)

Calculated expected values are 1.031853, 3.397028and 10.50375 |(in $ millions) for T=0.5, 19.5, 37.5 (years) correspondingly.

The detailed expected value calculation of the strategy at each month end in the sample period conditional on initial strategy value is included in excel file (attached).

The log-normal distribution of the BSM model provides closely matched expected values to the actual strategy values. The expected values provide an approximate exponential trend-line (Berger, 2012). The short term fluctuations of the actual strategy values have been smoothed out in the model. The value in the figure stands for strategy value for QMJ strategy. The expected values of the strategy significantly differ on a broad time period especially after 30^th June 2000. As observed from the figure forecast line seems to be consistent with the concept that higher prices will be obtained by quality stocks. (96/100)

From BSM model standing at 31^st December 2017, mean and s.d values can be extrapolated taking 12 months as T=1 year (Hyndman, 2008). Using BSM parameter values  per annum it is obtained that mean=(approx) and S.D= 

ML Estimates

The continuously compounded annual return between time 0 and time T is(Mamdouh Medhat, lecture 2, February 8, 2018).

 Now for T=1/12 year and T= 1 year and correspondingly.

Both the compounded annual returns have similar Means. For the time period of one year, value of variance is very less compared to the value of variance for one month (Mangano, 2012). This reflects the fact that scattering of return for longer time period smoothes out for predicted volatility.

For compounded annual return over next 12 months or 1 year to be less than 1%, which can be written as , which implies that. Hence there is 99% probability that is greater than -14.611%.

Here and it is known that.

Hence  (in$ millions), therefore there is 1% probability that the price for 1 year is below $7.84865 (in millions) (Mamdouh Medhat, lecture 2, February 8, 2018).

Here T=1 and 95% confidence interval is as  (Mamdouh Medhat, lecture 2, February 8, 2018) whereand. For continuously compounded annual return. Hence 95% confidence interval is

For the strategy value the 95% confidence interval is (Mamdouh Medhat, lecture 2, February 8, 2018).

 as the mean is 0.06271 and standard deviation is 0.088.

• From the problem: For

 and for ,

Conditional expectation for continuously compounded annual return for next 12 months i.e. T=1 year at the end of 31^st December 2017, given it is below -14.61% per annum is evaluated as:

Conditional expectation for the strategy value for next 12 months i.e. T=1 year at the end of 31^st December 2017, given it is below $7846891 is calculated as below:

Here and, Here

Hence

Linear regression parameters for the FF3 model are (obtained from multiple linear regression calculation table in excel file (attached)). The R-square value is 0.453296667. The t-statistic value for the estimated intercept is 8.412451.

The regression table gives a high positive correlation of 0.673 of the three factors on continuously compounded return per month (Lando, 2009). The accuracy of the linear regression fit can be explained from the significant F value which is almost zero. The p-values (two tailed) for the MKT, SMB, HML factors are almost zero which implies that the result is highly significant under 5% level of significance. The line fit plot (from excel file (attached)) of all the three variables indicates the perfect fit for the linear regression model. (89/100)

From CAPM model the continuously compounded monthly return () can be described by linear regression model. The CAPM model gives the linear regression model for continuously compounded monthly return for hedged strategy as. From both the models it can be concluded that eliminating the MKT, SMB and HML terms.

Expected Values of Strategy with Hedge Values

In the obtained linear regression model the intercept  is risk-adjusted return and is diversifiable risk. The standard error of for is the variability of idiosyncratic risk. The intercept is the return of QMJ with its factor exposures hedged out (Nichol, 2014). The QMJ factors after hedging performs better even in volatile market condition and gives higher returns. The regression model gives that the price of quality for QMJ is not affected by any other factor’s predictability. The model also indicates the fact that continuously compounded monthly return is not just driven by noise factors which are uncorrelated to the strategy’s systemic risks. For a perfect CAPM model the model becomes where the risk adjusted return vanishes. A positive signifies outperformance relative to volatile market. From a time series of the regression model systematic risk of the strategy can be evaluated with respective to the market. For given set of andthe hedged returns can be forecasted (Quality Minus Junk, Clifford S. Asness, Andrea Frazzini, and Lasse Heje Pedersen, June 5, 2017).(171/200)

Calculation for continuously compounded monthly return per month for hedge strategy is attached in Excel file (attached).

1. Value at 31^stDecember 1980 is : -0.00142 (in $ million)
2. Value at 31^stDecember 1999 is : 000926 (in $ million)

• Value at 31^stDecember 2017 is: -0.02083 (in $ million)

Table of calculation for value of the hedge strategy per month is attached in Excel file (attached).

1. Value at 31^stDecember 1980 is : 133496 (in $ million)
2. Value at 31^stDecember 1999 is : 276191 (in $ million)

• Value at 31^stDecember 2017 is:   31.28765  (in $ million)

Strategy value with expected monthly value with hedge value plot is given as a function of date:

The Strategy values given in question 1 and Hedge values calculated are very different. Hedge values are numerically greater than the QMJ strategy values (Simpson, 2008). The growth ratio of hedged values of the strategy is almost 1.0026 times that of un-hedged strategy values on a compounded basis. The expected value trend lines are a perfect fit for both the models with exponential smoothness. It can be concluded that hedging improves the strategy value for the current portfolio. Due to low volatility rate in both the strategies lesser market risk improves return values of the strategy for both the models.(100/100)

Here where is strategy  diversifiable risk, is the residual standard error. The linear regression model of the hedged strategy’s (continuously compounded) return per month from question 6(b) can be obtained as. Hence the distribution of will also follow normal distribution as, which implies that. Therefore mean of the distribution is and variance is.

Using BSM model with the similar set of assumption of question 1(a) the results for hedged strategy are as follows:

Maximum Likelihood estimation procedure is used to evaluate mean and standard deviation as (for mean) and.

The MLE for and are as follows: (per time T=1)

and.

BSM model fitted hedge strategy ML parameter values are

Hence the per annum values are  and  whereas for un-hedged QMJ model per annum ML parameter values were and. The hedged strategy improves the mean and variance. The growth ratio of hedged values of the strategy is almost 1.0026 times that of un-hedged strategy values on a compounded basis. From the excel file (attached) data it can be observed that hedged model improves the strategy values significantly with a factor of 3.225. The value of strategy at 31^st December 2017 is 31.28765 (in $ millions) compared to 9.081344 (in $ millions) of un-hedged strategy.

Linear Regression Parameters

Un-hedged strategy (QMJ) value along with expected monthly value with hedged strategy value along with expected monthly value plot is given as a function of date:

Expected value of the hedged strategy is as follows: (table in excel file (attached))

1. Value at 31^stDecember 1980 is : 048088 (in $ million)
2. Value at 31^stDecember 1999 is : 244811 (in $ million)

• Value at 31^stDecember 2017 is:   33.87231  (in $ million)

The hedging of the QMJ strategy improves the strategy values as explained in answer of 6(c). Here expected values of hedged strategy have been plotted and a perfect fit with strategy values can be observed. The expected value curve has exponential smoothness and can act as a line of fit for the strategy values. From the figure the rapid growth of the hedged values are clear after the year 1997. At the end of the broad time period of 37.5 years, the jump in the hedged curve is significant. After 30^th June 2000 the disparity of the four curves start exhibiting. (101/100)

• Here T=1 and 95% confidence interval is as whereand. For continuously compounded annual return. Hence 95% confidence interval is

For the strategy value the 95% confidence interval is as the mean is 0.0939356 and standard deviation is 0.06506.

For compounded annual return over next 12 months or 1 year to be less than 1%, which can be written as , which implies that. Hence there is 99% probability that is greater than -5.954%.

Conditional expectation for continuously compounded annual return for next 12 months i.e. T=1 year at the end of 31^st December 2017, given it is below -7.60% per annum is evaluated as:

is taken as the most recent peak value of the strategy. Peak values for the time period 30^th June 1980 to December 2017 for both models are found in excel format (attached in excel file (attached)). The peak values until a peak has been found are initially set to zero.

The drawdown values for the time period 30^th June 1980 to December 2017 for both models are found in excel format (attached in excel file (attached)). The formula (Mamdouh Medhat, lecture 2, February 8, 2018) was used to find the drawdown values. It was observed that at the downturn points the drawdown values were negative whereas it was positive for upturn values with respective to the most recent peak value. The initial drawdown values were zero until a peak value was obtained.

The maximum drawdown values for the time period 30^th June 1980 to December 2017 for both models are found in excel format (attached in excel file (attached)). The formula  (Mamdouh Medhat, lecture 2, February 8, 2018) was used for calculation of maximum drawdown values. Initial maximum drawdown was zero for zero drawdown values.

The values are given in tabular format from excel file (attached).

Type of Data	31st December 1980	31st December 1999	31st December 2017
Peak QMJ	1.078825067	3.589166171	9.269426845
Peak Hedge	1.135104123	6.27619057	31.94628602
DD QMJ	0.0	-0.0474	-0.0203
DD Hedge	-0.0014	0.0	-0.0206
MDD QMJ	0.0067	0.1637	0.2305
MDD Hedge	0.0014	0.1041	0.1041

The QMJ regression model had the three factors of CAPM model (MKT, SMG and HML) on the right hand side with negative coefficients whereas the hedged model had the same three factors on the right hand side with positive coefficients due to the negative signs in the model. This particular scenario was based on an uprising market and therefore the hedged model gave much better returns and at the end of the time period the strategy values were almost three times the QMJ strategy values. The expected values gave the trend line for both the models with exponential smoothness. Negative load of HML implies that low quality stocks increases the risk adjusted return for QMJ (Quality Minus Junk, Clifford S. Asness, Andrea Frazzini, and Lasse Heje Pedersen, June 5, 2017). For bear market scenario hedged model may not outperform the QMJ model. The income volatility may be avoided with the application of hedging. Mean of the hedged strategy was much higher compared to the un-hedged model provided much higher returns. The low standard deviation of the hedged strategy compared to the un-hedged model given a long time span indicated less scattering of strateg.y values which indicated the less volatility of the model. (202/200)

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