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![6
(1) Sharpe Ratio：
p f
p
RS
(2) Treynor Ratio：
p f
p
RT
， 2
PM
p
M
(3) Information Ratio
p f
Rp RM
RI
(4) Sortino Ratio：
Definition：The dispersion of X from a given value a is
2 2
( ) ( )a x a f x dx
.
In finance, we are concerned about downside dispersion from some value a i.e. X taking
values in ( , ]a define the semi variance of X about a is
2 2
( ) ( )a x a f x dx
.
Given sample R1, R2, R3…Rn of returns with some distribution f(x), Define
,
,
i
i
i i
a R a
y
R R a
,
then the downside sample semi variance is
2 2 2
1 1
1 1
( ) [min(0, )]n n
a i i i iS y a R a
n n
So, we use
f
p f
S
to estimate the Sortiono Ratio
0
0p
r
r
.
Outcomes:
Sharp Ratio Treynor Ratio Information Ratio Sortino Ratio
0.06390022971 0.001292279398 0.008901856247 0.09195970697
The Sortino Ratio is a modification of the Sharpe Ratio but penalizes only those returns falling
below our expected return, while the Sharpe ratio penalizes both upside and downside volatility
equally. The difference between the Sharpe ratio and the Information Ratio is that the
Information Ratio aims to measure the risk-adjusted return in relation to a benchmark. A
variation of the Sharpe ratio is the Sortino ratio, which removes the effects of upward price
movements on standard deviation to measure only return against downward price volatility and
uses the semi variance in the denominator. The Treynor ratio uses systematic risk, or beta (β)
instead of standard deviation as the risk measure in the denominator.](https://image.slidesharecdn.com/245ba409-66f6-403a-b4b2-4e43ac5287cc-170208170415/75/Portfolio-Optimization-Project-Report-6-2048.jpg)
![7
Comparing through these ratios, it shows that all the ratios are all positive indicating that our
portfolio perform well. However, these values of ratios are not very high, Treynor Ratio and
Information Ratio are even around zero. We could conclude that although our portfolio could
make money, we still take additional risks.
(5) Maximum drawdown:
The maximum drawdown (MDD) up to time T is,
(0, ) (0, )
( ) max[max ( ) ( )]
T t
MDD T X t X
Peak value valley value Maximum drawdown Maximum drawdown/Peak Value
992236 925201 67035 6.76%
During this period, huge loss was suffered in our portfolio with a maximum drawdown of 6.76%
within four trading days.
(6) Portfolio alpha and beta
It is based on Jensen performance measure,
( )p p f p m f
Portfolio Beta 0.03333847659
Portfolio Alpha 0.0001460887197
With a positive Alpha 0.0001460887197, it means although our portfolio performs better than
the benchmark, its value is around 0 which means our portfolio’s performance is very close
to the market performance. And the Beta is 0.03, indicating that our portfolio return has
potential risk in some way.
(7) Comparison of our portfolio performance to industry benchmarks
As far as we are concerned, by comparing to the benchmarks, the overall performance of our
portfolio is very close to the benchmarks’. But we take additional risks comparing to the market.
850000
900000
950000
1000000
1050000
1100000
11/10/2016
11/11/2016
11/12/2016
11/13/2016
11/14/2016
11/15/2016
11/16/2016
11/17/2016
11/18/2016
11/19/2016
11/20/2016
11/21/2016
11/22/2016
11/23/2016
11/24/2016
11/25/2016
11/26/2016
11/27/2016
11/28/2016
11/29/2016
11/30/2016
12/1/2016
12/2/2016
12/3/2016
12/4/2016
12/5/2016
12/6/2016
Market value](https://image.slidesharecdn.com/245ba409-66f6-403a-b4b2-4e43ac5287cc-170208170415/75/Portfolio-Optimization-Project-Report-7-2048.jpg)
![9
composition by changing several stocks.
Capstone Part
Basic risk Measure
In this part, we use volatility of returns, Value-at-risk (VaR) and Conditional value-at-risk
(CVaR) as the weekly risk measures of our portfolio. For VaR and CVaR, we use the parametric
method to estimate them with both a normal and a student-t returns distribution and three
different confidence level (1%, 5%, 10%).
Method introduction:
Gaussian VaR and CVaR
Let
2
-x /21
2
e
（x）= denote the PDF of the standard normal distribution, and let （x）
denote the corresponding CDF with '
. The Gaussian quantile function ( )Q is the
function satisfying the condition
( ( ))Q
for all :0 1 .
In the case of a Gaussian distribution with mean and variance 2
, the quantile function
of that distribution is then ( )Q and the corresponding “Value at Risk”, or VaR, is just
the negative of the quantile, which is
( ) ( )VaR Q
The CVaR is a related entity, giving the average in a tail bounded by the corresponding VaR
level. In general, for a density function ( )f x linked to a random variable X, we would need to
know
( )1
[ | ( )] ( )
Q
E X X Q xf x dx
In the case, because '
x , so that for a standard normal distribution
1
[ | ( )] ( ( ))E X X Q Q
and applying a scaling and translation to get the general case gives us, for the loss
( ) ( ( ))CVaR Q
In the case of a Gaussian system, there is no substantive difference between the VaR and CVaR](https://image.slidesharecdn.com/245ba409-66f6-403a-b4b2-4e43ac5287cc-170208170415/75/Portfolio-Optimization-Project-Report-9-2048.jpg)
![10
measures: both take the form
( ) ( )Risk
where ( ) ( )Q in the case of VaR and
1
( ) ( ( ))Q
u
in the case of CVaR.
Student’s T VaR and CVaR
The pdf of the Student’s T distribution is
2 ( 1)/2
1 [( 1) / 2] 1
( , )
[ / 2] (1 / ) v
v
h t v
v t vv
It is like the pdf function in Gaussian case. Define the quantile function ( , )TQ n for a standard
univariate T distribution (n is the degree of freedom). If we wish to parametrize the problem
again by mean and standard deviation, the VaR will be given by
2
( ) ( , )T
n
VaR Q n
n
The relevant function for this case is not the (negative of the) standard T quantile but the
negative of the scaled unit variance T quantile
2
( ) ( , ) ( , )T
n
Q n Q n
n
:
The VaR measure of risk is now in standard form ( ) .
For the CVaR, we need to find the integral of ( , )nh n and we can find the function
1
/2 2 2 2
1
( )( )
2( , )
2 ( )
2
v v
v v t
k t v
v
Then the CVaR with mean and variance 2
is now
2 1
( ) ( , )T
n
CVaR k Q n
n
（ ）,
and now the
2 1
( ) ( , )T
n
k Q n
n
（ ）,
By calculation by MATLAB, we get the four weeks’ VaR and CVaR. (William, 2011)](https://image.slidesharecdn.com/245ba409-66f6-403a-b4b2-4e43ac5287cc-170208170415/75/Portfolio-Optimization-Project-Report-10-2048.jpg)
![17
Appendix:
m-files
1.
(1).
function [ optimalWeights ] = optimalPortfolio( expReturns, CovMatrix, expPortfolioReturn)
mu = expReturns ;
Omega = CovMatrix;
N=length(mu);
muP=expPortfolioReturn;
One = ones(length(mu),1) ;
invOmega = inv(Omega) ;
A = One'*invOmega*mu ;
B = mu'*invOmega*mu ;
C = One'*invOmega*One ;
D = B*C - A^2 ;
h=invOmega*(C*mu/D-A*One/D);
g=invOmega*(B*One/D-A*mu/D);
optimalWeights =h*muP+g;
end
(2).
N=length(expPortfolioReturn);
for i=1:N
[ optimalWeights] = optimalPortfolio( expReturns, CovMatrix, expPortfolioReturn(i));
SigmaStar(i)=sqrt(optimalWeights'*CovMatrix*optimalWeights);
end
plot(SigmaStar,expPortfolioReturn)
title('efficientFrontier')
xlabel('SigmaP')
ylabel('muP')
(3). function [ TangencyWeights ] = TangencyPortfolio( expReturns, CovMatrix,muf)
mu = expReturns ;
Omega = CovMatrix;
N=length(mu);
One = ones(length(mu),1) ;
invOmega = inv(Omega) ;
A = One'*invOmega*mu ;
B = mu'*invOmega*mu ;
C = One'*invOmega*One ;
D = B*C - A^2 ;
h=invOmega*(C*mu/D-A*One/D);
g=invOmega*(B*One/D-A*mu/D);](https://image.slidesharecdn.com/245ba409-66f6-403a-b4b2-4e43ac5287cc-170208170415/75/Portfolio-Optimization-Project-Report-17-2048.jpg)
![18
w=invOmega*(mu-muf*One);
TangencyWeights=w/(One'*w);
end
2.
function [ Sharp,Treynor,Sortino,Alpha,Beta ] = Ratios( Rp, Rm,uf)
n=length(Rp);
uRp=mean(Rp);
StDRp=std(Rp);
uRm=mean(Rm);
StDRm=std(Rm);
ExcessRpm=Rp-Rm;
uExcessRpm=mean(ExcessRpm);
StDexcess=std(ExcessRpm);
Sharp=(uRp-uf)/StDRp;
A=cov(Rp,Rm);
B=A(1,2)/var(Rm);
for i=1:n
Rpf(i)=Rp(i)-uf;
end
meanRpf=mean(Rpf);
Treynor =meanRpf/B;
Information=ExcessRpm/StDexcess;
Beta=A(1,2)/var(ExcessRpm);
Alpha=uExcessRpm-uf+Beta*(uRm-uf);
for i=1:n
if (Rp(i)-uf)<0
negative(i)=Rp(i)-uf;
end
end
Downside=(sumsqr(negative)/n)^0.5;
Sortino=meanRpf/Downside;
end
2.
function [VaRnormal,CVaRnormal,VaRt,CVaRt,CVaRTt] = ValueAtRisk(returns,alpha,v)
n=length(returns);
m=mean(returns);
sigma=std(returns);
S=sqrt(sigma^2*n/(n-1));
x=norminv([alpha 1-alpha],0,1);](https://image.slidesharecdn.com/245ba409-66f6-403a-b4b2-4e43ac5287cc-170208170415/75/Portfolio-Optimization-Project-Report-18-2048.jpg)
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Portfolio Optimization Project Report
- 1. 1 Yongqiang Cui, ShuaiZhai Professor Marcel Blais Math 574 12/1/16 Portfolio Optimization Project Report Abstract In this project, we close the portfolio established in the Portfolio Optimization Project. A complete analysis of our portfolio including the construction, week-to-week performance, week-to-week rebalancing process and overall performance will be given in this report. Ratio analysis, Leverage analysis and basic risk measure are conducted by analyzing the portfolio returns in time series. At the end of the project, a good-of-fit hypothesis test was performed to determine the returns distribution’s consistence with our fitted distributions. Introduction The portfolio that we used to analyze is generated from Markowitz optimal portfolio from the Portfolio Optimization Project. On 11/09/2016, The portfolio was constructed with calculated weight of twenty stocks. The weight of different stocks was rebalanced each week until 12/06/2016, the date on which portfolio was closed. The total return of holding this portfolio in this month is $22,923, with the initial capital of $500,000. Our portfolio value is increased by 4.58% in the holding period. We choose S&P 500 as the market portfolio which grows from 2,163.26 to 2,212.23 in the last month, with an increase of 2.25%. Thus, the overall performance of our portfolio is good as it has a higher growth rate than our chosen market portfolio. Methodology Our portfolio is formed based on the Modern portfolio theory, or mean-variance analysis, introduced by Harry Markowitz in 1952. It is a mathematical framework for assembling a portfolio of assets to maximize the expected return for a given level of risk, defined as variance. (Markowitz, 1952) Historical stock price of last one year was used to in our project to estimate the weights in Markowitz optimal portfolio. We first use our data of stock price to calculate the optimal portfolio with different expected return and variance, code listed in appendix (1). Then combine the optimal portfolios to create an efficient frontier, code in appendix (2). Tangency line with the intercept of risk-free rate of the efficient frontier, which is called the tangency portfolio could be found using code appendix (3). After that, construct the Markowitz optimal portfolio with the weights of tangency portfolio. In each of the following week, we will add the stocks’ price of the new week into historical data and rebalance the weight of stocks in the portfolio. The market value of stocks and the portfolio was recorded for analyzing purpose.
- 2. 2 Portfolio Construction On 11/09/2016,historical data from 2015/10/01 to 2016/11/04 of our twenty selected stocks whose stock code listed as follows GNC, MRO, SWN, DHI, TWTR, SOHU, RDS-B, TSLA, PFE, TNH, MS, SNE, INTC, CSX, GM, GRPN, VALE, BAC, T and ECA are obtained from Yahoo Finance. We build the efficient frontier and calculate the weight of exact stocks of tangency portfolio. The total value in stock account is $500,000, thus the value for each stock equals to the product of their weight and total account value. (Market value is in thousand) Stocks TNH T PFE TWTR GNC SOHU GM INTC RDS- B VALE weight 0.061 0.728 0.232 -0.217 -0.445 -0.119 0.225 0.689 -0.043 0.211 Mkt Value 30.35 364.15 115.9 -108.25 -222.3 -59.3 112.4 344.25 -21.3 105.45 Stocks CSX SWN SNE DHI TSLA BAC ECA MS GRPN MRO weight 0.091 -0.048 0.334 -0.071 -0.318 -0.131 0.441 -0.139 0.105 -0.587 Mkt Value 45.55 -24.15 167.2 -35.3 -158.8 -65.4 220.4 -69.65 52.4 -293.5 Rebalancing and tracking In the next four weeks, we add the stock price of the new week to our historical data, for example, on the second week 11/16/2016, we obtained the historical data form 2015/10/01 to 2016/11/09 as our historical data sets. Create the efficient frontier and calculate the weight of tangency portfolio. The rebalance are conducted four times and the efficient frontier and portfolio weight for each week are listed as followed.
- 3. 3 Date 11/9/2016 11/16/201611/23/2016 12/1/2016 Min up 0.0396 0.0511 0.051 0.0621 Min Variance 0.006 0.0061 0.0061 0.0061 From the comparison of efficient frontier, minimum variances are almost constant and its corresponded portfolio returns increase in the last four weeks. And the corresponding tangency portfolio weights are listed as followed: 2016/11/9 2016/11/16 2016/11/23 2016/11/30 Average MRO -0.5870 -0.2965 -0.2843 -0.2231 -0.3477 GNC -0.4446 -0.3018 -0.2898 -0.2085 -0.3112 TWTR -0.2165 -0.1697 -0.1540 -0.1434 -0.1709 BAC -0.1308 0.2814 0.4481 0.5072 0.2765 SWN -0.0483 -0.1852 -0.1832 -0.1313 -0.1370 MS -0.1393 -0.0334 0.0847 -0.0648 -0.0382 SOHU -0.1186 -0.2562 -0.2887 -0.2137 -0.2193 DHI -0.0706 -0.4836 -0.3367 -0.2988 -0.2974 TSLA -0.3176 -0.3158 -0.3518 -0.2278 -0.3033 RDS B -0.0426 -0.1951 -0.2877 -0.2620 -0.1969 TNH 0.0607 0.1067 0.0758 0.0757 0.0797 CSX 0.0911 0.3590 0.2184 0.1932 0.2154 PFE 0.2318 0.2207 -0.0162 -0.0225 0.1035 GM 0.2248 0.4276 0.2434 0.2199 0.2789 SNE 0.3344 0.2072 0.1198 0.0357 0.1743 T 0.7283 0.8825 1.2254 1.1876 1.0060 INTC 0.6885 0.1652 0.1812 0.0938 0.2822 GRPN 0.1048 0.0602 0.0398 0.0330 0.0595 VALE 0.2109 0.1697 0.1457 0.1451 0.1679 ECA 0.4408 0.3570 0.4100 0.3045 0.3781
- 4. 4 We attempted tokeep the data period of exact one year by delating a week’s data at the beginning and adding one week’s date at present. However, the results run by our Markowitz Optimal Model do not make us satisfied. The weight of portfolio for a new week changes too much too rebalance on Interactive Brokers. Thus, we choose to update new weeks’ stock price and extend the period of our historical stock price. The returns covariance matrix does not change a lot. Portfolio Performance From the total value of Interactive Brokers Accounts which consists of $504,128 cash and $499,823 Stock in the first week. The total account value ends up in $1,026,814, the invest return of holding this portfolio for one month is $22,923. Our portfolio value is increased by 2.28% within the holding period. Compared to our chosen market portfolio S&P 500 which grows from 2,163.26 to 2,212.23 in the last month, with an increase of 2.25%. The overall performance of our portfolio is good as it has a higher rate of return. Ratio Analysis: We use the S&P 500 as benchmark and in our models, we use sample mean return estimate p , use sample standard deviation of returns to estimate volatility and use the 10-year Treasury Yield as our risk-free rate f :
- 5. 5 Date Portfolio ReturnMarket Return Excess Return(up- um) Excess Return(up- uf) 2016/12/6 0.001000514 0.003410879435 -0.002410365435 0.000921929058 2016/12/5 0.010795234 0.005821300668 0.004973933332 0.01071664906 2016/12/2 0.001143647 0.0003970644614 0.0007465825386 0.001065062058 2016/12/1 -0.038764015 -0.00351553795 -0.03524847705 -0.03884259994 2016/11/30 -0.005836422 -0.002653470376 -0.003182951624 -0.005915006942 2016/11/29 0.029008915 0.001335319659 0.02767359534 0.02893033006 2016/11/28 0.016074101 -0.005254478505 0.02132857951 0.01599551606 2016/11/25 0.021870169 0.003914329257 0.01795583974 0.02179158406 2016/11/23 -0.007135212 0.0008080111124 -0.007943223112 -0.007213796942 2016/11/22 0.02008148 0.002165427763 0.01791605224 0.02000289506 2016/11/21 0.007069175 0.007461386865 -0.0003922118647 0.006990590058 2016/11/18 0.007708739 -0.002386700318 0.01009543932 0.007630154058 2016/11/17 0.003950144 0.004676288736 -0.0007261447356 0.003871559058 2016/11/16 0.039926843 -0.001582285738 0.04150912874 0.03984825806 2016/11/15 -0.00288217 0.007480824323 -0.01036299432 -0.002960754942 2016/11/14 -0.030046657 - 0.0001155027836 -0.02993115422 -0.03012524194 2016/11/11 -0.035896756 -0.001397936775 -0.03449881923 -0.03597534094 2016/11/10 -0.011609604 0.001950759502 -0.0135603635 -0.01168818894 Mean 0.001469895833 0.001250871074 0.0002190247591 0.001391310891 Standard Dev 0.02128609889 0.003713625597 0.020881997 0.02128609889 The plots of the portfolio returns are showed as followed: Then we use the following measures to analyze our portfolio: -0.06 -0.04 -0.02 0 0.02 0.04 0.06 11/10/2016 11/11/2016 11/12/2016 11/13/2016 11/14/2016 11/15/2016 11/16/2016 11/17/2016 11/18/2016 11/19/2016 11/20/2016 11/21/2016 11/22/2016 11/23/2016 11/24/2016 11/25/2016 11/26/2016 11/27/2016 11/28/2016 11/29/2016 11/30/2016 12/1/2016 12/2/2016 12/3/2016 12/4/2016 12/5/2016 12/6/2016 Daily Return
- 6. 6 (1) Sharpe Ratio： pf p RS (2) Treynor Ratio： p f p RT ， 2 PM p M (3) Information Ratio p f Rp RM RI (4) Sortino Ratio： Definition：The dispersion of X from a given value a is 2 2 ( ) ( )a x a f x dx . In finance, we are concerned about downside dispersion from some value a i.e. X taking values in ( , ]a define the semi variance of X about a is 2 2 ( ) ( )a x a f x dx . Given sample R1, R2, R3…Rn of returns with some distribution f(x), Define , , i i i i a R a y R R a , then the downside sample semi variance is 2 2 2 1 1 1 1 ( ) [min(0, )]n n a i i i iS y a R a n n So, we use f p f S to estimate the Sortiono Ratio 0 0p r r . Outcomes: Sharp Ratio Treynor Ratio Information Ratio Sortino Ratio 0.06390022971 0.001292279398 0.008901856247 0.09195970697 The Sortino Ratio is a modification of the Sharpe Ratio but penalizes only those returns falling below our expected return, while the Sharpe ratio penalizes both upside and downside volatility equally. The difference between the Sharpe ratio and the Information Ratio is that the Information Ratio aims to measure the risk-adjusted return in relation to a benchmark. A variation of the Sharpe ratio is the Sortino ratio, which removes the effects of upward price movements on standard deviation to measure only return against downward price volatility and uses the semi variance in the denominator. The Treynor ratio uses systematic risk, or beta (β) instead of standard deviation as the risk measure in the denominator.
- 7. 7 Comparing through theseratios, it shows that all the ratios are all positive indicating that our portfolio perform well. However, these values of ratios are not very high, Treynor Ratio and Information Ratio are even around zero. We could conclude that although our portfolio could make money, we still take additional risks. (5) Maximum drawdown: The maximum drawdown (MDD) up to time T is, (0, ) (0, ) ( ) max[max ( ) ( )] T t MDD T X t X Peak value valley value Maximum drawdown Maximum drawdown/Peak Value 992236 925201 67035 6.76% During this period, huge loss was suffered in our portfolio with a maximum drawdown of 6.76% within four trading days. (6) Portfolio alpha and beta It is based on Jensen performance measure, ( )p p f p m f Portfolio Beta 0.03333847659 Portfolio Alpha 0.0001460887197 With a positive Alpha 0.0001460887197, it means although our portfolio performs better than the benchmark, its value is around 0 which means our portfolio’s performance is very close to the market performance. And the Beta is 0.03, indicating that our portfolio return has potential risk in some way. (7) Comparison of our portfolio performance to industry benchmarks As far as we are concerned, by comparing to the benchmarks, the overall performance of our portfolio is very close to the benchmarks’. But we take additional risks comparing to the market. 850000 900000 950000 1000000 1050000 1100000 11/10/2016 11/11/2016 11/12/2016 11/13/2016 11/14/2016 11/15/2016 11/16/2016 11/17/2016 11/18/2016 11/19/2016 11/20/2016 11/21/2016 11/22/2016 11/23/2016 11/24/2016 11/25/2016 11/26/2016 11/27/2016 11/28/2016 11/29/2016 11/30/2016 12/1/2016 12/2/2016 12/3/2016 12/4/2016 12/5/2016 12/6/2016 Market value
- 8. 8 (8) While tradingon Interactive Brokers, we are limited by the Margin calls. The portfolio weight calculated by Markowitz Optimal Model sometimes cannot be exercised because you need to short too much stocks. The margin calls will appear for some unreasonable weight of portfolio. We adjust our portfolio by changing one or several stocks and calculate the weight for new portfolio. Usually, we delate the stocks which requires to do more short position from the Model. Repeat this process until we found an exercisable weight of tangency portfolio. Leverage Analysis Date Total Debt Leverage Ratio 2016/12/6 2336947.85 2.33694785 2016/12/5 2314261.48 2.31426148 2016/12/2 2287899.17 2.28789917 2016/12/1 2303301.716 2.303301716 2016/11/30 2696540.848 2.696540848 2016/11/29 2654664.216 2.654664216 2016/11/28 2673747.689 2.673747689 2016/11/25 2703091.755 2.703091755 2016/11/23 2691175.457 2.691175457 2016/11/22 2826370.712 2.826370712 2016/11/21 2795538.602 2.795538602 2016/11/18 2748163.639 2.748163639 2016/11/17 2747769.797 2.747769797 2016/11/16 2732490.91 2.73249091 2016/11/15 2574427.828 2.574427828 2016/11/14 2538839.014 2.538839014 2016/11/11 2534622.348 2.534622348 2016/11/10 2531621.124 2.531621124 2016/11/9 2516046.038 2.516046038 The leverage ratio of our portfolio stays around 2.5, which is the highest level that avoid the margin calls. A higher leverage ratio will cause margin calls and we need to adjust the portfolio 0 1 2 3 11/9/20… 11/11/2… 11/13/2… 11/15/2… 11/17/2… 11/19/2… 11/21/2… 11/23/2… 11/25/2… 11/27/2… 11/29/2… 12/1/20… 12/3/20… 12/5/20… LeverageRatio Date Leverage Ratio vs. Date Leverage Ratio
- 9. 9 composition by changingseveral stocks. Capstone Part Basic risk Measure In this part, we use volatility of returns, Value-at-risk (VaR) and Conditional value-at-risk (CVaR) as the weekly risk measures of our portfolio. For VaR and CVaR, we use the parametric method to estimate them with both a normal and a student-t returns distribution and three different confidence level (1%, 5%, 10%). Method introduction: Gaussian VaR and CVaR Let 2 -x /21 2 e （x）= denote the PDF of the standard normal distribution, and let （x） denote the corresponding CDF with ' . The Gaussian quantile function ( )Q is the function satisfying the condition ( ( ))Q for all :0 1 . In the case of a Gaussian distribution with mean and variance 2 , the quantile function of that distribution is then ( )Q and the corresponding “Value at Risk”, or VaR, is just the negative of the quantile, which is ( ) ( )VaR Q The CVaR is a related entity, giving the average in a tail bounded by the corresponding VaR level. In general, for a density function ( )f x linked to a random variable X, we would need to know ( )1 [ | ( )] ( ) Q E X X Q xf x dx In the case, because ' x , so that for a standard normal distribution 1 [ | ( )] ( ( ))E X X Q Q and applying a scaling and translation to get the general case gives us, for the loss ( ) ( ( ))CVaR Q In the case of a Gaussian system, there is no substantive difference between the VaR and CVaR
- 10. 10 measures: both takethe form ( ) ( )Risk where ( ) ( )Q in the case of VaR and 1 ( ) ( ( ))Q u in the case of CVaR. Student’s T VaR and CVaR The pdf of the Student’s T distribution is 2 ( 1)/2 1 [( 1) / 2] 1 ( , ) [ / 2] (1 / ) v v h t v v t vv It is like the pdf function in Gaussian case. Define the quantile function ( , )TQ n for a standard univariate T distribution (n is the degree of freedom). If we wish to parametrize the problem again by mean and standard deviation, the VaR will be given by 2 ( ) ( , )T n VaR Q n n The relevant function for this case is not the (negative of the) standard T quantile but the negative of the scaled unit variance T quantile 2 ( ) ( , ) ( , )T n Q n Q n n : The VaR measure of risk is now in standard form ( ) . For the CVaR, we need to find the integral of ( , )nh n and we can find the function 1 /2 2 2 2 1 ( )( ) 2( , ) 2 ( ) 2 v v v v t k t v v Then the CVaR with mean and variance 2 is now 2 1 ( ) ( , )T n CVaR k Q n n （ ）, and now the 2 1 ( ) ( , )T n k Q n n （ ）, By calculation by MATLAB, we get the four weeks’ VaR and CVaR. (William, 2011)
- 11. 11 Volatility analysis: We comparethe volatility of our portfolio returns with the market volatility, which in our case is the volatility of the S&P500 index. Week1 Week2 Week3 Week4 Portfolio Return Volatility 2.99% 0.97% 2.75% 0.56% Market Volatility 0.37% 0.37% 0.38% 0.27% By comparing across weeks, the Week4 has the smallest volatility (0.56%), and the Week2 also has a smaller volatility (0. 97%).The volatilities of week1 and week3 are near 3%. As the result, the Week2 and Week4 shows the lower risk of all four weeks. So, our portfolio has a relevantly high volatility to the market performance. And comparing to the market performance, our portfolio has a relevantly high volatility to the market volatility. Our portfolio shows higher risk than it should have. (Wagner, 2007) VaR analysis: We sort the four weeks VaR data and plot them. The degree of freedom of T distribution is 3 according to the MATLAB function mle (). VaR(Gaussian) VaR(T) Confidence level 1% 5% 10% Confidence level 1% 5% 10% Week1 0.0777 0.0573 0.0464 Week1 0.086 0.0547 0.0422 Week2 0.0165 0.0098 0.0063 Week2 0.0192 0.009 0.0049 Week3 0.0595 0.0408 0.0308 Week3 0.0672 0.0385 0.054 Week4 0.0087 0.0049 0.0029 Week4 0.0104 0.0033 9.85E-04 2.99% 0.97% 2.75% 0.56% 0.00% 0.50% 1.00% 1.50% 2.00% 2.50% 3.00% 3.50% Week1 Week2 Week3 Week4 Volalitility
- 12. 12 To explain theVaR clearly, we choose the VaR(Gaussian) with the confidence level 5% of the Week1 as an example, 0.0573 means that there is a 0.05 probability that the portfolio return will fall by more than 0.0573 this week. It can also be said that with 95% confidence that the worst daily loss will not exceed 0.0573. So for the same confidence leve, the high VaR means the higher potential risk that the porfolio has. By comparing the tables and plots, with the same confidence and distribution, the VaR of Week2 and Week4 is always very small. With the 5% and 10% confidence level of T distributuion, we find that the returns of Week2 and Week4 still show lower risks. CVaR analysis: The CVaR is the second name of Expected Shortfall (ES). For example, with 5% level, the "expected shortfall at 5% level" is the expected return on the portfolio in the worst 5% of cases. So, with the same confidence level, the higher the CVaR, the lower the risk of the portfolio has. For the tendency of the CVaR is same for the different confidence level. We choose the case with 5% confidence level to analyze. (Carlo & Dirk, 2002). 5% confidence level CVaR(Gaussian) CVaR(T) Week1 -0.98 -1.02 Week2 -0.33 -0.55 Week3 -0.91 -0.94 Week4 -0.19 -0.37 0.0777 0.0165 0.0595 0.0087 0.0573 0.0098 0.0408 0.0049 0.0464 0.0063 0.0308 0.0029 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Week1 Week2 Week3 Week4 VaR(Gaussian) 1% 5% 10% 0.086 0.0192 0.0672 0.0104 0.0547 0.009 0.0385 0.0033 0.0422 0.0049 0.054 9.85E- 04 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Week1 Week2 Week3 Week4 VaR(T) 1% 5% 10%
- 13. 13 ` By comparing thetable and plots with two distributions, the Week2 and Week4 suffer from the lower expected loss. Meanwhile, the Week1 and Week3 have higher risks. Goodness-of fit Hypothesis Form a time series of daily returns from our factor modeling project. Here we choose the French and Fama model include F1 (Short-Term Reversal Factor). Time series of daily returns and the bate values of this model are both obtained from factor modeling project. French&F1 beta0 Mkt-RF SMB HML ST_Rev E(R) VALE 0.000314 0.014083 -0.004220 0.009180 0.018838 -0.002803 ECA 0.001019 0.012178 0.013284 0.026709 0.018977 -0.003691 GRPN 0.000135 0.008741 0.007860 -0.003348 0.007406 -0.004060 MRO 0.000738 0.014722 0.008103 0.021085 0.012380 -0.003185 SWN 0.000840 0.009314 0.010014 0.020318 0.021306 -0.005241 BAC 0.000176 0.013188 0.001662 0.007796 -0.002979 0.000468 TWTR 0.000152 0.008076 0.011793 -0.005722 0.007450 -0.002960 TSLA -0.000204 0.010725 -0.002283 -0.008123 0.002006 0.001080 MS 0.000205 0.015092 0.003453 0.007975 -0.003709 -0.000358 CSX 0.000105 0.011817 0.000131 0.003794 0.001772 -0.001069 DHI 0.000070 0.012115 0.004077 0.000071 -0.000445 0.000955 TNH 0.000180 0.004035 0.001439 0.004614 0.005232 -0.001230 -1.20 -1.00 -0.80 -0.60 -0.40 -0.20 0.00 Week1 Week2 Week3 Week4 CVaR CVaR(Gaussian) CVaR(T)
- 14. 14 T 0.000012 0.007878-0.001435 0.002348 -0.001589 0.000227 INTC -0.000029 0.009575 -0.002245 0.000318 0.000403 0.000390 GM 0.000012 0.010057 -0.002267 0.001510 0.001890 -0.000126 GNC 0.000317 0.010536 0.015396 0.001217 0.001803 -0.001497 PFE -0.000110 0.008961 -0.000972 -0.004850 0.001589 -0.000093 SNE -0.000046 0.011447 -0.002676 -0.001292 0.003126 -0.000470 RDS-B 0.000327 0.009771 0.001995 0.010535 0.005610 -0.001448 SOHU 0.000160 0.006784 0.003418 -0.000794 0.012175 0.000764 Fit our returns distribution to both normal and a t density distribution. For normal distribution, we calculate the mean, standard deviation and the z value to create the plot. For Student T distribution, degree of freedom is required to be considered. We calculated it using MATLAB function mle () which turns out to be 8.3174. 0 1 2 3 4 5 6 7 8 9 -2.828280613 -2.6239 -2.4195 -2.2152 -2.0108 -1.8064 -1.6020 -1.3977 -1.1933 -0.9889 -0.7846 -0.5802 -0.3758 -0.1714 0.0329 0.2373 0.4417 0.6460 0.8504 1.0548 1.2592 1.4635 1.6679 1.8723 2.0766 2.2810 2.4854 2.6898 2.8941 3.0985 3.3029 PDF&Histogramof NormalDist 0 1 2 3 4 5 6 7 8 9 -2.828280613 -2.6239 -2.4195 -2.2152 -2.0108 -1.8064 -1.6020 -1.3977 -1.1933 -0.9889 -0.7846 -0.5802 -0.3758 -0.1714 0.0329 0.2373 0.4417 0.6460 0.8504 1.0548 1.2592 1.4635 1.6679 1.8723 2.0766 2.2810 2.4854 2.6898 2.8941 3.0985 3.3029 CDF&Histogram of Student's TDist
- 15. 15 Then use thetime series of daily returns to perform a goodness- of-fit hypothesis test. Assign data to 60 intervals in a histogram and acquire the frequency for each interval. Then denote Pi as the probability of normal distribution with Cumulative Distribution Function F, calculate Pi for each interval i using formula. (Jensen, 1968) 1 t ( ) ( ) ( )i i i i h In erval P f y dy F y F y iN Pg =theoretical frequency. Chi-square test statistic: 2 2 1 ( )n i i i i u N P N P g g Use the same formula to calculate the 2 value for Student-t distribution 0 2 4 6 8 10 12 14 -0.0270 -0.0243 -0.0216 -0.0189 -0.0162 -0.0135 -0.0108 -0.0081 -0.0054 -0.0027 0.0000 0.0027 0.0054 0.0082 0.0109 0.0136 0.0163 0.0190 0.0217 0.0244 0.0271 Histogramof OriginalData Frequency 0 0.2 0.4 0.6 0.8 1 1.2 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 CDF Normal Student-T
- 16. 16 Then we cancalculate p-value, H0: Distribution fit our data H1: Distribution not fit our data Normal student-t 2 0.05879 0.4016 p-value 0.971033 0.9999 Both P-value is quite high which means we can reject H0 at very high confidence level. Very strong evidence against H0 that these two distributions fit our data. Both distributions are not good to estimate our model. Conclusion More attention should be paid during stock selection. In this project, lots of effort was put on adjusting our stock composition to avoid margin calls. On the other hand, Markowitz Optimal Model has a good performance and it will be a good choice to construct a portfolio next time. If we are to re-do this project in the spring semester, we will follow most of the approach in this project since it brings a good rate of return. While choosing stocks, some with unusual behavior in price should not be considered because they will increase the difficulty of constructing portfolio. Meanwhile, we could use the methods we have learned and constructed models to evaluate the risk of our portfolios. Because from the risk management part of our portfolio, we could find that although we can get positive returns, there is still potential risks affecting our portfolio performance. The risk management is also a significant issue while we want to pursue the higher return.
- 17. 17 Appendix: m-files 1. (1). function [ optimalWeights] = optimalPortfolio( expReturns, CovMatrix, expPortfolioReturn) mu = expReturns ; Omega = CovMatrix; N=length(mu); muP=expPortfolioReturn; One = ones(length(mu),1) ; invOmega = inv(Omega) ; A = One'*invOmega*mu ; B = mu'*invOmega*mu ; C = One'*invOmega*One ; D = B*C - A^2 ; h=invOmega*(C*mu/D-A*One/D); g=invOmega*(B*One/D-A*mu/D); optimalWeights =h*muP+g; end (2). N=length(expPortfolioReturn); for i=1:N [ optimalWeights] = optimalPortfolio( expReturns, CovMatrix, expPortfolioReturn(i)); SigmaStar(i)=sqrt(optimalWeights'*CovMatrix*optimalWeights); end plot(SigmaStar,expPortfolioReturn) title('efficientFrontier') xlabel('SigmaP') ylabel('muP') (3). function [ TangencyWeights ] = TangencyPortfolio( expReturns, CovMatrix,muf) mu = expReturns ; Omega = CovMatrix; N=length(mu); One = ones(length(mu),1) ; invOmega = inv(Omega) ; A = One'*invOmega*mu ; B = mu'*invOmega*mu ; C = One'*invOmega*One ; D = B*C - A^2 ; h=invOmega*(C*mu/D-A*One/D); g=invOmega*(B*One/D-A*mu/D);
- 18. 18 w=invOmega*(mu-muf*One); TangencyWeights=w/(One'*w); end 2. function [ Sharp,Treynor,Sortino,Alpha,Beta] = Ratios( Rp, Rm,uf) n=length(Rp); uRp=mean(Rp); StDRp=std(Rp); uRm=mean(Rm); StDRm=std(Rm); ExcessRpm=Rp-Rm; uExcessRpm=mean(ExcessRpm); StDexcess=std(ExcessRpm); Sharp=(uRp-uf)/StDRp; A=cov(Rp,Rm); B=A(1,2)/var(Rm); for i=1:n Rpf(i)=Rp(i)-uf; end meanRpf=mean(Rpf); Treynor =meanRpf/B; Information=ExcessRpm/StDexcess; Beta=A(1,2)/var(ExcessRpm); Alpha=uExcessRpm-uf+Beta*(uRm-uf); for i=1:n if (Rp(i)-uf)<0 negative(i)=Rp(i)-uf; end end Downside=(sumsqr(negative)/n)^0.5; Sortino=meanRpf/Downside; end 2. function [VaRnormal,CVaRnormal,VaRt,CVaRt,CVaRTt] = ValueAtRisk(returns,alpha,v) n=length(returns); m=mean(returns); sigma=std(returns); S=sqrt(sigma^2*n/(n-1)); x=norminv([alpha 1-alpha],0,1);
- 19. 19 t=tinv(alpha,v); C=x(1); VaRnormal=-m-sigma*C; CVaRnormal=-m+sigma*C/alpha; VaRt=-m-sigma*sqrt((v-2)/v)*t; CVaRTt=(-((v^(-1/2)*(v+t^2)))/(alpha*beta(v/2,1/2)*(n-1)))*((1+t^2/n)^(-(v+1/2))); end Reference 1. Markowitz, H.(1952). Portfolio Selection. The Journal of Finance, 7(1), 77. doi :10.2307/2975974 2. Wagner, H. (2007). Volatility's Impact On Market Returns. Retrieved December 12, 2016, from http://www.investopedia.com/articles/financial-theory/08/volatility.asp 3. Carlo Acerbi; Dirk Tasche (2002). "Expected Shortfall: a natural coherent alternative to Value at Risk" (pdf). Economic Notes. 31: 379–388. doi:10.1111/1468-0300.00091 4. William T. S. (2011). Risk, VaR, CVaR and their associated Portfolio Optimizations when Asset Returns have a Multivariate Student T Distribution; 5. Jensen, M.C. (1968), “The Performance of Mutual Funds in the Period 1945-1964,” Journal of Finance 23, 1968, pp. 389-416.