ACTL40011 Actuarial Studies Project 2

Portfolio Selection

Analysis

Kaushik Srikanth 762151

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Contents

1. Background ……………………………………………………………………………………………………….. 2

2. Deriving the efficient frontier under the single and three factor models (Tasks 1-2) 3

2.1 Single Factor Case: ……………………………………………………………………………………….. 3

2.1.1 Assumptions: …………………………………………………………………………………………… 3

2.1.2 Methodology: ………………………………………………………………………………………….. 3

2.1.3 Differences to account for when short selling is restricted: …………………………… 4

2.2 Three Factor Case: ………………………………………………………………………………………… 4

2.2.1 Choice of Factors ……………………………………………………………………………………. 4

2.2.2 Methodology …………………………………………………………………………………………… 4

3. Comparisons of the three-factor model and the single factor model (Task 1) ……….. 6

4. Points to take into account for the short selling and no short selling cases (Task 2) . 7

5. Stability Testing (Task 3) ……………………………………………………………………………………. 8

6. Using Blume’s technique (Task 3) ………………………………………………………………………. 9

7. Using option-market data to conduct mean-variance analysis (Task 4) ……………… 10

7.1 Methodology: ………………………………………………………………………………………………. 10

7.2 Results: ……………………………………………………………………………………………………….. 11

8. Derivation of volatility smiles for the stocks (Task 5) …………………………………………. 12

9. References ……………………………………………………………………………………………………….. 13

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1. Background

Mean variance analysis 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. 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. In both models, there is an error term that is uncorrelated for

different assets. 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 only stems from the market index.

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). 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).

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).

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2. Deriving the efficient frontier under the single and three factor models

(Tasks 1-2)

2.1 Single Factor Case:

2.1.1 Assumptions:

In working with the single factor model, we need to determine several key pieces of

information. These include:

1. 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.

2. We obtain monthly data for each stock and the index for the period 1/7/98-1/7/18

using Yahoo Finance.

3. For each month, the return for that stock is calculated as:

;#3627408453;;#3627408466;;#3627408481;;#3627408482;;#3627408479;;#3627408475;= ;#3627408438;;#3627408473;;#3627408476;;#3627408480;;#3627408470;;#3627408475;;#55349;;#56404; ;#3627408451;;#3627408479;;#3627408470;;#3627408464;;#3627408466;?;#3627408450;;#3627408477;;#3627408466;;#3627408475;;#3627408470;;#3627408475;;#55349;;#56404; ;#3627408451;;#3627408479;;#3627408470;;#3627408464;;#3627408466;

;#3627408450;;#3627408477;;#3627408466;;#3627408475;;#3627408470;;#3627408475;;#55349;;#56404; ;#3627408451;;#3627408479;;#3627408470;;#3627408464;;#3627408466;

The same methodology is also applied to the monthly returns of the index

4. The mean of the index is calculated as a geometric mean of the monthly returns. This

is calculated as:

;#3627408479;;#3627408468;=(?(1+;#3627408479;;#3627408474;;#3627408476;;#3627408475;;#3627408481;? ;#3627408470;)

240

;#3627408470;=1)

1240

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2.1.2 Methodology:

1. The return of our stock is given as ;#3627408453;;#3627408454;= ;#3627409148;+ ;#3627409149;;#3627408453;;#55349;;#56384;+ ;#55349;;#57088;, where:

;#55349;;#56376;;#3627408453;;#3627408454;= ;#3627409148;+ ;#3627409149;;#55349;;#56376;;#3627408453;;#55349;;#56384;,;#55349;;#56393;;#3627408453;;#3627408454;= ;#3627409149;2;#55349;;#56393;;#3627408453;;#55349;;#56384;+;#55349;;#56376;;#55349;;#57088;2,;#55349;;#57088;~;#3627408449;(0,;#55349;;#57102;2)

2. In order to determine ? and ? for each stock, we can run a least-squares regression on

EXCEL. This will look to minimise the quantity;#55349;;#56376;(;#3627408453;;#3627408454;?(;#3627409148;+ ;#3627409149;;#3627408453;;#55349;;#56384;))2.

3. We then use our values of ? and ? for each stock to calculate the respective means and

variances.

4. Note that;#55349;;#56376;;#55349;;#57088;2= ;#3627408455;;#3627408476;;#3627408481;;#55349;;#56398;;#3627408473; ;#3627408453;;#3627408466;;#3627408480;;#3627408470;;#3627408465;;#3627408482;;#55349;;#56398;;#3627408473; ;#3627408454;;#3627408482;;#3627408474; ;#3627408476;;#3627408467; ;#3627408454;;#3627408478;;#3627408482;;#55349;;#56398;;#3627408479;;#3627408466;;#3627408480;

;#3627408465;;#3627408467;?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.

5. To calculate the covariance between stocks:

;#3627408438;;#3627408476;;#3627408483; (;#3627408453;;#3627408454;1,;#3627408453;;#3627408454;2)=;#3627408438;;#3627408476;;#3627408483; (;#3627409148;1+ ;#3627409149;1;#3627408453;;#55349;;#56384;+ ;#55349;;#57088;1,;#3627409148;2+ ;#3627409149;2;#3627408453;;#55349;;#56384;+ ;#55349;;#57088;2)= ;#3627409149;1;#3627409149;2;#3627408438;;#3627408476;;#3627408483;(;#3627408453;;#55349;;#56384;,;#3627408453;;#55349;;#56384;)

= ;#3627409149;1;#3627409149;2;#55349;;#57102;;#55349;;#56384;2

Where ;#55349;;#57102;;#55349;;#56384;2 is the variance of the index returns.

6. To calculate the efficient frontier, we note that our portfolio takes the form

;#3627408453;;#3627408477;= ?;#3627408484;;#3627408470;;#3627408453;;#3627408470;

10

;#3627408470;=1,?;#3627408484;;#3627408470;

10

;#3627408470;=1=1

Where ;#3627408484;;#3627408470; represents the weight of the portfolio invested in the ith stock.

7. The variance of our portfolio is calculated as:

;#55349;;#56393;;#55349;;#56398;;#3627408479;(;#3627408453;;#55349;;#56387;)=;#3627408485;;#3627408455;;#3627408438;;#3627408485;

Where x represents our vector of weights and C is our covariance matrix.

8. The minimum variance portfolio (MVP) is calculated using the Solver function on

EXCEL by finding the weights that give us the lowest variance.

9. In using the Solver function, we have to impose the condition that the individual asset

weights must sum up to 1.

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10. We then take increments of 0.0002 from the return of our MVP to get the returns of

100 different portfolios.

11. To find the corresponding standard deviations, we proceed with Solver as in step 8

and make use of a macro to do this. This ensures the portfolios are efficient.

12. We plot the means and standard deviations of each portfolio, giving us our efficient

frontier.

The optimal portfolio is calculated by using the SOLVER function to maximise the Sharpe

Ratio (“Sharpe Ratio”, 2018) (;#55349;;#56376;;#3627408453;;#55349;;#56387;?;#3627408453;;#3627408467;

;#55349;;#57102;;#55349;;#56387;), where the risk-free rate ;#3627408453;;#3627408467; is the 3 month US

Treasury bill rate of 2.05% p.a. (as of 10 August 2018).

2.1.3 Differences to account for when short selling is restricted:

To incorporate the no short-selling rule in our calculations, we note down that the weights

must all be positive in the Solver function. Similarly, when the weights are between 0% and

80%, this must be stated.

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 choose returns for

different portfolios between these two portfolios and proceed in a similar manner as above to

derive the efficient frontier.

2.2 Three Factor Case:

2.2.1 Choice of Factors

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).

2.2.2 Methodology

For the three-factor model, we repeat the above process, albeit with slightly different

variations:

1. The return of the stock is given as ;#3627408453;;#3627408454;= ;#3627409148;+;#3627409149;1;#3627408453;;#55349;;#56384;1+;#3627409149;2;#3627408453;;#55349;;#56384;2+;#3627409149;3;#3627408453;;#55349;;#56384;3+

;#55349;;#57088;,;#55349;;#57088;~;#3627408449;(0,;#55349;;#57102;2), where M1 refers to GDP, M2 refers to the CPI and M3 refers to the

unemployment rate.

2. The means of the indices are calculated as follows:

a) GDP: Note that we only have quarterly data available. So a monthly

approximation for monthly growth in each quarter is

;#55349;;#56404;;#3627408474;;#3627408476;;#3627408475;;#3627408481;?;#3627408473;;#55349;;#56422;?(1+;#55349;;#56404;;#3627408478;;#3627408482;;#55349;;#56398;;#3627408479;;#3627408481;;#3627408466;;#3627408479;)

13?1.

We apply this for all 20 years of data. Our approximation to the average monthly

growth rate is since determined as

;#55349;;#56376;;#3627408448;1=(?1+;#55349;;#56404;;#3627408474;;#3627408476;;#3627408475;;#3627408481;? ;#3627408470;240;#3627408470;=1)

1240?1

b) CPI: The monthly growth in the CPI is calculated as ;#3627408470;=;#55349;;#56374;;#55349;;#56387;;#55349;;#56380;;#55349;;#56400;;#3627408482;;#3627408479;;#3627408479;;#3627408466;;#3627408475;;#3627408481; ;#3627408474;;#3627408476;;#3627408475;;#3627408481;?

;#55349;;#56374;;#55349;;#56387;;#55349;;#56380;;#3627408477;;#55349;;#56398;;#3627408480;;#3627408481; ;#3627408474;;#3627408476;;#3627408475;;#3627408481;??1

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The mean approximation to the average change in the CPI is:

;#55349;;#56376;;#3627408448;2=(?1+;#3627408470;;#3627408474;;#3627408476;;#3627408475;;#3627408481;? ;#3627408470;

240

;#3627408470;=1)

1240

?1

c) 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.

3. The mean return of the stock is now calculated as:

;#55349;;#56376;;#3627408453;;#3627408454;= ;#3627409148;+;#3627409149;1;#55349;;#56376;;#3627408453;;#55349;;#56384;1+;#3627409149;2;#55349;;#56376;;#3627408453;;#55349;;#56384;2+;#3627409149;3;#55349;;#56376;;#3627408453;;#55349;;#56384;3

4. The variance of the stock returns is calculated as:

;#55349;;#56393;;#3627408453;;#3627408454;= ;#3627409149;12;#55349;;#57102;;#55349;;#56384;12+;#3627409149;22;#55349;;#57102;;#55349;;#56384;22+;#3627409149;32;#55349;;#57102;;#55349;;#56384;32+ ;#55349;;#56376;;#55349;;#57088;2

5. The covariance between two stocks is now calculated as:

;#3627408438;;#3627408476;;#3627408483;(;#3627408453;;#3627408454;1,;#3627408453;;#3627408454;2)=;#3627408438;;#3627408476;;#3627408483;(;#3627409148;1+;#3627409149;11;#3627408453;;#55349;;#56384;1+;#3627409149;21;#3627408453;;#55349;;#56384;2+;#3627409149;31;#3627408453;;#55349;;#56384;3+;#55349;;#57088;1,

;#3627409148;2+;#3627409149;12;#3627408453;;#55349;;#56384;1+;#3627409149;22;#3627408453;;#55349;;#56384;2+;#3627409149;32;#3627408453;;#55349;;#56384;3+;#55349;;#57088;2)

= ;#3627409149;11;#3627409149;21;#3627408438;;#3627408476;;#3627408483;(;#3627408453;;#55349;;#56384;1,;#3627408453;;#55349;;#56384;1)+ ;#3627409149;21;#3627409149;22;#3627408438;;#3627408476;;#3627408483;(;#3627408453;;#55349;;#56384;2,;#3627408453;;#55349;;#56384;2)

+ ;#3627409149;31;#3627409149;32;#3627408438;;#3627408476;;#3627408483;(;#3627408453;;#55349;;#56384;3,;#3627408453;;#55349;;#56384;3)

= ;#3627409149;11;#3627409149;21;#55349;;#57102;;#55349;;#56384;12+ ;#3627409149;21;#3627409149;22;#55349;;#57102;;#55349;;#56384;22+ ;#3627409149;31;#3627409149;32;#55349;;#57102;;#55349;;#56384;32

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3. Comparisons of the three-factor model and the single factor model (Task

1)

Figure 1: Efficient Frontiers from the Single and Three Factor Models.

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 MVP 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. However, the optimal

portfolio in the single factor model has a higher return, and the shape of the two efficient

frontiers suggests that as we take on more risk, the single factor model offers higher returns.

Furthermore, we note that for each of the stocks, residual variance is much higher in the

single factor case than the three factor case. Although it was mentioned that multi-factor

models are meant to be superior, the factors used here have led to a model that does a poor

job of explaining variability in the returns of the stocks. 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. There is also the issue that the

parameters of each model are static i.e. parameters such as betas are ‘unstable through time’

and ‘should change’ with ‘company fundamentals’ and ‘capital structures’ (Mullins Jr.,

1982).

Figure 2: MVP portfolios of both factor models

Figure 3: Tangent portfolios of both factor models MVP (Single Factor)MVP (Three Factor)

Mean Return0.0080963950.00942038

Standard Deviation0.0376161440.026024083 Optimal (Single Factor)Optimal (Three Factor)

Mean Return0.014694390.011547272

Standard Deviation0.0535980260.029387892

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4. Points to take into account for the short selling and no short selling cases

(Task 2)

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, we have an upper bound for our efficient frontier.

No short selling means that we cannot have any negative covariance terms that reduces our

portfolio variance (Note that none of the assets have negative covariance terms with each

other). We actually see that returns increase at a decreasing rate, because of inability to fully

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.

Figure 4: Efficient Frontiers for Task 2a Figure 5: Efficient Frontiers for Task 2b

Figure 6: MVP portfolios of all 3 cases using the

single factor model

Figure 7: Optimal portfolios of all three cases

using the single factor model MVP (Short Sell)MVP (No Short Sell)MVP (