ACTL40011 Actuarial Studies Project 2
Portfolio Selection Analysis
Kaushik Srikanth 762151
Contents TOC o “1-3” h z u Background PAGEREF _Toc522378671 h 2Comparisons of the single and three factor models PAGEREF _Toc522378672 h 3Deriving the efficient frontier under the single-factor model PAGEREF _Toc522378673 h 3Deriving the efficient frontier for the three factor model: PAGEREF _Toc522378674 h 4Comparisons of the three-factor model and the single factor model PAGEREF _Toc522378675 h 6Points to take into account for the short selling and no short selling cases. PAGEREF _Toc522378676 h 7Stability Testing PAGEREF _Toc522378677 h 8Using Blume’s technique PAGEREF _Toc522378678 h 8Using option-market data to conduct mean-variance analysis. PAGEREF _Toc522378679 h 10Derivation of volatility smiles for the stocks PAGEREF _Toc522378680 h 12References PAGEREF _Toc522378681 h 13
BackgroundMean variance analysis, or Modern Portfolio Theory, focuses on investors looking to maximise the risk-return trade-off i.e. given a certain level of risk (i.e. standard deviation of returns), investors will look to maximise their return for that level of risk. Similarly, we can view this in terms of investors looking to minimise their level of risk given a certain return level. An investor who believes in mean-variance analysis will only choose to invest in portfolios that lie on the efficient frontier (Joshi ; Paterson, 2013), a subset of the total opportunity set of portfolios available to the investor. Portfolios on this frontier are said to be efficient in the sense that: for any given return level, no other portfolio has a lower level of risk and for a given level of risk, no other portfolio has a greater return.
Conducting mean-variance analysis for a large number of securities can present a problem in the sense that we will be required to calculate a vast number of points in order to perform our analysis. While under traditional mean-variance analysis, asset returns and variances are known (Joshi ; Paterson, 2013), this is not true in practice. In order to reduce the number of calculations we need to make, we use factor models, with one or more factors, each of which representing a certain market index, macroeconomic factors or various sectors.
A single factor model states that the return of a security is determined by one single factor representing the market, whereas a multi-factor model will state that the return is now dependent on multiple indices. The simplicity in such models arises from the fact that all the securities in a given portfolio will owe their correlation to the factors that they all have in common (Joshi ; Paterson, 2013).
While it has been mentioned that the use of such models will lead to a reduction in the number of data points we need to calculate, these models are not without fault. For instance, the single factor model assumes that residuals are uncorrelated with each other, and that if this does not hold, then the model is a poor fit to the data (Joshi ; Paterson, 2013). It is also worth pointing out that the single factor model can be viewed as being too simplistic, as it assumes that correlation between stocks is only the result of the market. This deficiency is addressed through the use of multi-factor models, where the factors used will be uncorrelated, such as the market index and various other industries (Joshi ; Paterson, 2013).
On the other hand, while the multi-factor model can certainly provide a greater number of factors for us to work with, it is not as strong as the single factor model in its ability to predict future correlation matrices (Joshi ; Paterson, 2013). As a result, one would assume that the single factor model would be more suited to conduct mean-variance analysis.
A volatility smile is described as being the ‘discrepancy’ that arises from the fact that ‘implied volatilities vary among the different strike prices’ (Nowak ; Sibetz, 2012, p.3). In a world where the assumptions underpinning the Black-Scholes model hold true, options that ‘expire on the same date’ should have ‘the same implied volatility regardless of the strikes’ however, the existence of volatility smiles disproves this theory (Nowak ; Sibetz, 2012, p.3). It is said that options that are either ‘in- or out-of-the-money’ have a higher volatility than options that are at-the-money (Nowak ; Sibetz, 2012, p.3). These smiles are said to exist as a result of inefficiency in pricing options in the derivatives market (Narayanamurthy ; Rao, 2007, p.535).
Comparisons of the single and three factor modelsDeriving the efficient frontier under the single-factor modelIn working with the single factor model, we need to determine several key pieces of information. These include:
The index that is to be used. For this task, we have chosen the S;P 500 Index as all of the stocks are listed here.
Using Yahoo Finance, we obtain monthly data for each stock for the period 1/7/98-1/7/18.
For each month, the return for that stock is calculated as:
Return= Closing Price-Opening Price Opening PriceThe same methodology is also applied to the monthly returns of the index
The mean of the index is calculated as a geometric mean of the monthly return. This is calculated as:
rg=i=12401+ri1240-1The methodology for the single factor model is as follows:
The return of our stock is given as RS= ?+ ?RM+ ?, where:
ERS= ?+ ?ERM, VRS= ?2VRM+E?2, ?~N0, ?2 In order to determine ? and ? for each stock, we can run a least-squares regression on EXCEL. This will look to minimise the quantityE(RS-?+ ?RM)2.
We then use our values of ? and ? for each stock to calculate the respective means and variances.
Note thatE?2= Total Residual Sum of Squaresdf-1, where df is the total degrees of freedom (number of data points, which is 240). The total residual sum of squares can be inferred from the regression output in the ANOVA table.
To calculate the covariance between stocks:
Cov RS1, RS2=Cov ?1+ ?1RM+ ?1, ?2+ ?2RM+ ?2= ?1?2CovRM, RM= ?1?2?M2Where the mean of the index is rg (from above) and the variance can be calculated using EXCEL.
To calculate the efficient frontier, we note that our portfolio takes the form
Rp= i=110wiRi, i=110wi=1Where wi represents the weight of the portfolio invested in the ith stock. We look to calculate the minimum variance portfolio first.
The variance of our portfolio is calculated as:
VarRP=xTCxWhere x represents our vector of weights and C is our covariance matrix.
The minimum variance portfolio is calculated using the Solver function on EXCEL by finding the weights that give us the lowest variance.
We then take increments of 0.0002 from the return of our minimum variance portfolio to get the returns of 100 different portfolios.
To find the corresponding standard deviation, we proceed with Solver as in step 7, only this time, we make use of a macro to speed up the calculation process.
We plot the calculated portfolios in the mean/standard deviation space and note down the shape of the efficient frontier.
To incorporate the no short-selling rule in our calculations, we note down that the weights must all be positive in addition to summing up to 1 when using the Solver function. Similarly, when the weights are between 0% and 80%, we state this constraint in the Solver function.
In addition, we also calculate the portfolio with the maximum return in these two cases (by finding the weights that allow this to take place). As there are clear upper bounds on how much of any one asset we can hold, this can be calculated. We then take the difference in returns for this portfolio and our minimum variance portfolio and divide this by 50 to give us our increment (as in the previous case) which we use to calculate the returns of 50 different portfolios on the efficient frontier.
Furthermore, the tangent portfolio is calculated by using the SOLVER function to maximise the Sharpe Ratio (“Sharpe Ratio”, 2018) (ERP-Rf?P), where the risk-free rate Rf is the 3 month US Treasury bill rate of 2.05% p.a. (as of 10 August 2018).
Deriving the efficient frontier for the three factor model:
For the three-factor model, we have chosen a macroeconomic model with factors being the unemployment rate, the growth in real gross domestic product (GDP) and the growth in the consumer price index (CPI), which is a proxy for the rate of inflation. The choice of a macroeconomic model is attributed to the fact that macroeconomic factors can ‘affect the performance of a business’ and ‘as they are beyond the control of an organisation’, it is imperative for businesses to be able to predict their ‘heterogeneous effect’ on ‘future corporate performance’ (Issah ; Antwi, 2017, p.2).
For the three-factor model, we repeat the above process, albeit with slightly different variations:
The return of the stock is given asRS= ?+?1RM1+?2RM2+?3RM3+ ?, ?~N(0, ?2), where M1 refers to GDP, M2 refers to the CPI and M3 refers to the unemployment rate.
The means of the indices are calculated as follows:
GDP: Note that we only have quarterly data available. To determine a monthly approximation here,gmonthly?1+gquarter13-1, applying this for all 20 years of data. Our approximation to the average monthly growth rate is since determined as EM1=i=12401+gmonth i1240-1
CPI: The monthly growth in the CPI is calculated as i=CPIcurrent monthCPIpast month-1The mean approximation to the average change in the CPI is:
EM2=i=12401+imonth i1240-1 Unemployment rate: We calculate how much unemployment increases on a monthly basis by subtracting the past month’s rate from the current’s month rate. The expectation of the index here is an arithmetic average of the historic monthly changes.
The mean return of the stock is now calculated as:
ERS= ?+?1ERM1+?2ERM2+?3ERM3The variance of the stock returns is calculated as:
VRS= ?12?M12+?22?M22+?32?M32+ E?2The covariance between two stocks is now calculated as:
CovRS1, RS2=Cova1+?11RM1+?21RM2+?31RM3+?1, ?2+?12RM1+?22RM2+?32RM3+?2= ?11?21CovRM1, RM1+ ?21?22CovRM2, RM2+ ?31?32CovRM3, RM3 = ?11?21?M12+ ?21?22?M22+ ?31?32?M32Note that across both models, the error terms are uncorrelated for different stocks while the in the three-factor model, the returns on each of the indices are independent of each other.
Comparisons of the three-factor model and the single factor model
We calculate the respective betas of the stocks in the single factor model and find that these range considerably, from 0.33 to 1.88, indicating that the sensitivity of the individual stocks with respect to the S;P 500 is quite variable. In comparison, the three factor model is considerably more volatile with regards to the calculated values of the betas. We find that the majority of the stocks move in line with the growth in GDP whilst most of them move in the opposite direction to the change in the unemployment rate. The expected returns in the three factor model are higher, and the same is true regarding the stocks’ variances.
The minimum variance portfolio in the three factor model is much better than its counterpart in the single factor model, with a superior return as well as a smaller standard deviation. Despite this, the efficient frontier in the single factor case is more “convex” than the three-factor. As standard deviation increases, we see that the efficient frontier of the single factor model converges to the three factor model, and actually starts to overtake it for high returns (>2% per month).
We had previously stated that the single factor model was worse than the multi-factor model due to the fact that it only considers the impact of one factor on asset returns, which is viewed as being unrealistic. However, in this case, we note that for each of the stocks, the R squared term (which explains how much of the variability in the returns can be explained using the model) is much higher in the single factor case. Although it was mentioned that macroeconomic factors can considerably impact companies and their returns, the factors used here have led to a mode that does a poor job of explaining variability in the model. As such, it would be prudent to conclude that the data does not fit the model and further testing would need to use an entirely different model or different macroeconomic factors.
305752529591000Points to take into account for the short selling and no short selling cases.
When comparing the short selling and no short selling cases, we see that the slope of the short selling case is considerably greater. We can attribute this to the fact that the ability to short sell gives us a greater ability to diversify our risks and actually maximise our returns for a given level of standard deviation. To increase our returns without drastically taking on more risk (i.e. increasing standard deviation of returns), we can just choose to short sell a greater amount of our poorest performing assets and hold greater amounts of our better assets. Because of the negative covariance terms that arise from having negative weights, we will witness a fall in the portfolio variance. We also see that the short selling case is better because the tangent portfolio here provides a superior return.
On the other hand, with no short selling, it is clear that we have an upper bound on the maximum return that we can achieve. This is effectively a portfolio of one asset (which generates the largest expected return). No short selling means that we cannot have any negative covariance terms that reduce our variance. We actually see that returns increase at a decreasing rate, because of inability to full diversify our risks i.e. we must hold a non-negative amount of each asset. As a result, the marginal risks outweigh the marginal returns in this portfolio.
Comparing the two short selling cases separately, note that they are near identical, but as returns get higher and higher, the effect of the weight restriction comes into play. We see that while a restriction of 80% of asset weights reduces the maximum possible return, it allows for more diversification and as a result, carries less risk. This point is best emphasised with the maximum returns of the two efficient frontiers, where in the 80% case, the return is only slightly smaller, but the level of risk carried is significantly smaller.
This reinforces the idea that we need short selling in our assets and at the very least, we need to be able to diversify our portfolio as much as possible in order to eliminate the non-systemic risk from it. However, it is prudent to take into account of the fact that while short selling can eliminate the non-systemic risk arising from returns, we may be open to other risks (need to look up risks of excessive short-selling).
For all stocks and factors, we have used data starting from 1/7/1992 to 30/6/2012. However, we encountered an issue with SRE because we only had data for 20 years’ on hand. As a result, it was decided that we would use a sub-section of this data on hand, focusing on the period 1/7/2001 to 30/6/2015 just for this stock.
We find that the returns generated using different data results in considerably different efficient frontiers: in both cases, the frontier gives us portfolios that offer higher returns. This is not something we would expect because in the long-run, we expect high growth assets such as the chosen stocks here to provide returns that are consistent and stable. However, we see that asset returns over the new 20 year period clearly differ to the returns offered from July 1998 to July 2018. In this case, it becomes clear that the model’s weakness is down to its inability to account for different time periods i.e. the outputs that we get from the model will be dependent on the data that we use in order to calibrate the relevant parameters.
Similarly, the same weakness is present when performing stability testing on the three factor model. Our model can only account for historical data and is static in this particular regard. From our testing, it is reasonable to come to the conclusion that we would need a model that allows us to have more robust parameters that will help provide more reasonable outputs.
Using Blume’s technique
We use Blume’s adjustment technique in order to provide an estimate of the stock betas that are more ‘forward looking’. The reasoning behind this formula is that betas ‘tend to the average market beta of 1 over time’ (Skardziukas, 2010, p.23).
The technique is given as follows (Skardziukas, 2010, p.23):
?adj=0.33+0.67?rawWhere ?raw is the beta that was estimated using the least squares regression in Task 1.
We see that that efficient frontier created with Blume’s adjustment is quite close to the efficient frontier from the single factor model. The graph actually shows us that returns under the adjustment are slightly higher at first. However, as we take on more risk, it leads to all three efficient frontiers converging towards similar values, and as we take on even more risk, the frontiers from Task 1 begin to outperform the frontier using Blume’s adjustment. This suggests that either the original models were not as accurate, providing returns that did not account for the fact that static parameters would not be realistic.
Using option-market data to conduct mean-variance analysis.
For each security, we choose a call and put option on a particular date with the same strike.
We also choose a call and put option with the same strike from the market index on the same date (here, we have chosen 17 August 2018). Note that our choice of options is determined by how close the strike price was to the stock price at the time at which calculations were performed (10 August 2018).
Using the Black-Scholes formulae for call and put options, we calculate the implied volatility for each of the securities.
Black-Scholes Call Price=SNd1-Ke-rTN(d2)Black-Scholes Put Price=Ke-rTN-d2-SN-d1, whereNx is the cumulative distribution function of N~(0,1) evaluated at x
dj= lnSK+r+-1j-1?22T?T, j=1,2K is the strike price, S is the spot price on the day, T is the time left till the option expires and r is the risk-free rate.
The risk-free rate used is the same one as above (2.05% p.a.).
According to Buss and Vilkov (2011), the volatility of the stock is given as:
IVStock= IVCall+IV(Put)2Where the options are at-the-money (p.10)
We repeat step 5 for the options on the S&P 500.
According to French, Groth and Kolari (as cited in S. Chen & J.Y. Yoon, 2012), we calculate the implied beta of the stock as:
?= ?i, M×IV(Stock)IV(Market), ?i, M= ?i?i2/?M2This figure is then substituted in the single factor model, replacing the value of beta that was calculated using linear regression.
We then proceed in the same manner as before, calculating the covariance matrix and the respective efficient frontiers.
We see that the efficient frontiers that were generated using option data sit below the efficient frontiers using the original single factor model. This is mainly due to the fact that the option data now understates the returns of the individual securities as well as the respective variances. While the frontiers are relatively the same early on, we see that there is a clear divergence between the two and as a result, the use of the option data results in a portfolio that offers lower returns as well as lower levels of risk. One of the main reasons for this is that is that the return of SWKS (which is the asset with the largest return) is now lower, and as a result, a no short selling constraint means that the portfolio cannot generate a return that is as high as expected. We find that this also holds true when we restrict the weight of individual assets to 80%.
Derivation of volatility smiles for the stocksThe volatility smile for each stock is derived by taking all call options that expire on the same day and graphing the implied volatilities with respect to the strike price. We also do the same thing for all the put options that expire on the same day and observe whether a volatility smile exists or not.
Our data shows that the majority of our options do not have a volatility smile. While there are those options that represent the general features we expect to see in smiles i.e. higher volatilities for out-in-the-money options than at in the money options, they are far from the perfect curves that we would have expected to see. In particular, the more unusual shapes come from the stocks that have a higher number of particular options available.
The shapes that we get resemble what are known as ‘volatility smirks’ or ‘skews’, owing to the fact that they are more skewed to one side or another (“Volatility Skew”, 2018). The majority of the option data analysed here produces skews that are biased towards the right. This means that implied volatility figures are generally higher for call options that are out of the money and for put options in the money. According to Xing, Zhang and Zhao (2010), this is a trait exhibited in ‘the majority of individual stock options’ and that there is an inverse relationship between the ‘steepness’ of the curves and the corresponding equity returns (p.661).
Yahoo Finance also quotes the implied volatilities for some options as being 0%, which is nonsensical especially given that if one attempts to calculate the price of an option using the Black-Scholes formula, they would find that a volatility of 0 will not produce an answer. Therefore, we must also consider the effect of such data affecting our analysis.
ReferencesBureau of Economic Analysis. (2018, August). Retrieved from https://apps.bea.gov/iTable/iTable.cfm?isuri=1&reqid=19&step=2&0=survey.
Bureau of Labor Statistics Data. (2018, August). Retrieved from https://data.bls.gov/timeseries/CUUR0000SA0L1E?output_view=pct_12mths.
Bureau of Labor Statistics Data. (2018, August). Retrieved from https://data.bls.gov/timeseries/LNS14000000.
Buss, A., Vilkov, G. (2011). Measuring Equity Risk with Option Implied Correlations, 1-42. Retrieved from https://www.tilburguniversity.edu/upload/4f6f5ece-f17d-4172-9977-227d07e9a999_vilkov.pdf.
Chen, S., Yoon, J.Y. (2012). Option-Implied Betas: Positive Bias and Term Structure. (Master’s Thesis, Weinberg College – Northwestern University). Retrieved from http://mmss.wcas.northwestern.edu/thesis/articles/get/773/Chen;Yoon2012.pdf.
Daily Treasury Yield Curve Rates. (2018, August). Retrieved from https://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/TextView.aspx?data=yield.
Issah, M., Antwi, S. (2017). Role of macroeconomic variables on firms’ performance: Evidence from the UK. Cogent Economics & Finance, 1-18. doi: https://doi.org/10.1080/23322039.2017.1405581.
Joshi, M.S., & Paterson, J.M. (2013). Introduction to Mathematical Portfolio Theory. The Edinburgh Building, Cambridge: Cambridge University Press.
Narayanamurthy, V., Rao, KCS. (2007). Volatility Smiles in Indian Option’s Market- A Study. The Management Accountant, July 2007, 535-542. Retrieved from https://www.researchgate.net/publication/304798302/download.
Nowak, P., Sibetz, P. (2012). Volatility Smile, 1-51. Retrieved from http://www.fam.tuwien.ac.at/~sgerhold/pub_files/sem12/s_sibetz_nowak.pdf.
Sharpe Ratio | Investopedia. (2018, August). Retrieved from https://www.investopedia.com/terms/s/sharperatio.asp.
Skardziukas, D. (2010). Practical Approach to Estimating Capital, 1-75. Retrieved from https://mpra.ub.uni-muenchen.de/31325/1/Practical_Approach_to_CoC_v3.pd.
Volatility Skew. (2018, August). Retrieved from https://www.investopedia.com/terms/v/volatility-skew.asp.
Xing, Y., Zhang, X., Zhao, R. (2010). What Does the Individual Option Volatility Smirk Tell Us About Future Equity Returns? Journal of Financial and Quantitative Analysis, 45(3), June 2010, 641-662. doi: 10.1017/S0022109010000220.
Yahoo Finance – Business Finance, Stock Markets, Quotes, News. (2018, August). Retrieved from https://finance.yahoo.com/.