Project One
Construction of the Efficient Frontier and the Optimal Risky Portfolio
Xie Liwei(3035450785)
Miao Yiwei(3035449487)
Bai Wenxuan(3035448677)
FINA2320: Investments & Portfolio Analysis
Author Note
Part One: Recommended Portfolio Page 1-3
Part Two: Additional Comment & Warning Page 4-7
Part Three: Modification and Inclusion of Latest News Page 7-10
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Part One
1.1 Raw Data Process
According to the monthly return index of the nine given stocks, the following average, standard deviation, variance-covariance matrix and correlation matrix has been generated through Excel.
The notations from Re1 to Re9 represent Cathay Pacific Airways, China Mobile, CITIC, Cheung
Kong Infrastructure, Power Assets Holdings Limited, PCCW, Sun Hung Kai Properties, Sino
Land and Wharf Holdings respectively.
Average and Sample Standard Deviation data:
Variance and Covariance data:
Correlation Matrix:
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1.2 Return Curve Construction
Using data from exhibits above and the solver function in Excel, individual expected returns with an increment of 0.1% for 12 data points has been input to the calculation. The solver function then processes those information and construct a portfolio of which the variances are the smallest. The result is a table of 12 individual weights of the 9 stocks with different weights which can be later used for constructing the Minimum Variance Frontier.
Weights of individual stocks for 12 points:
Those points were then threw into a single graph representing the Minimum Variance Frontier:
MVF Graph:
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To calculate the tangency portfolio, Excel solver was used again. There were no constraints on the expected return and variance this time, but the slope of the tangent, or sharpe ratio, was set to maximum. The result is as follows:
With an expected return of 0.84868% and s.d. of 0.03807.
Using the risk free point at s.d. = 0, Re = 0.1%, the linear function of tangency (Capital
Allocation Line) is:
Re = 0.19666 * s.d. + 0.1%
At Re = 0.8, s.d. = 0.03559. The approximate percentage that should be put into risk-free asset is
6.5%.
The final weighting of the portfolio is:
With an expected return of 0.8% and a s.d. of 0.03559.
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Part Two
2.1 Comparison with the market
According to our calculation, the average expected return of Hung Seng Index during the past 5 years is around 0.8365%, with an overall standard deviation of 0.0468 representing risks.
Market Final client’s portfolio
Return 0.8365% 0.8000%
Standard deviation 0.0468 0.03559
Considering the final portfolio of our client, including risk-free assets, whose expected return is 0.8% and standard deviation is 0.0356，the market appears to be 0.0365% more profitable but has much more higher standard deviation (0.0468 compares to
0.0356).
To be more specific, if our desired return equals to the average historical return(0.8365%) of the marketo, our best capital allocation line of the final portfolio in Part One could still produce a lower level of risk(standard deviation of 0.0375), which is almost 0.01 smaller than that of the market(0.0468). This means that if the client wants to get the same return as the market, he would always choose our portfolio for lower risk. Alternatively, if we repeat the above procedure using the risk (standard deviation) of the market which is 0.0468, we can get a return of 1.0204% which is 0.1839% more than that of the market, meaning that clients can benefit more under the same risk by simply investing in our portfolio.
To sum up, our portfolio seems to outperform the market. But the targeting return and its related risk will be decided by our client’s preference.
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2.2.1 Default Warning and related advice
1. Our model is totally based on historical performance of individual stocks. There is no
evidence that future price will move completely in the same way that they did in the past.
2. Industry/sector risks. Four of the chosen firms in the portfolio are from property market.
If those companies face an unexpected market condition such as a total property market
meltdown that the past five years had never experienced, your realized return is likely to
drop rapidly. Different shares carry different level of risk in their field of investing, you
should make yourself aware of those risks prior to investing.
3. The expected return calculated in the portfolio is 0.8% which is your requirement, but the
realized number would not likely to be the same. According to the risk calculation, your
final return could be lower or higher than 0.8% under different market situations. It is
even possible to have a negative return under extremely terrible market circumstances.
4. Stock prices were market information formed by people’s belief on the true value of the
company, but there is no guarantee that the firms had published all the truthful
information.
5. The risk exposure were calculated using a five-year investment period. If the customer’s
desired investing period is shorter than five years, the risk involved would be higher.
Thus it is encouraged for the investor to undertake longer investing period for lower time
risks.
2.2.2 Additional Survey and Risk Aversion
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To help you evaluate your risk-bearing and loss-bearing standards, we can use two scientific concepts, utility and risk aversion. Utility is a measurement of comparing investment portfolios on the basis of the expected return and risk, reflecting the satisfaction level of you towards a portfolio. The formula is equivalent to Expected return of the portfolio minus half of your Risk aversion multiplied by Variance of the portfolio according to CFA regulation. Your utility, on the other hand, depends on your ability to bear risk, which is also demonstrated by risk aversion.
In Part One, to maximize your utility, we had done some mathematical calculation to come up the final portfolio for you assuming your risk aversion is (Expected return of the tendency portfolio minus Risk-free rate) divided by (the weight of the tendency portfolio multiplied by variance of the tendency portfolio), 0.3599. This means that you are likely to be a risk lover to some extent. However, you might not like this portfolio if you are a conservative investor. The simplest method is to fill in a questionnaire and it will show your risk preference so that we can do a risk assessment for you, such that further adjustment on the portfolio can be made.
Following are two sample questions:
1) The chart below shows the greatest one-year loss and the highest one-year gain on three different hypothetical investments of $10,000.* Given the potential gain or loss in any one year, where would you invest your money?
● Fund A
● Fund B
● Fund C
2)My current and future income sources (such as salary, Social Security, pension plans) are . . .
● Very unstable
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● Unstable
● Somewhat stable
● Stable
● Very stable
2.2.3 Investment Time Span
What’s more, you also have to tell us the approximate length of time you desire your investment to hang on. Our calculation is based on the outcome of the 5-year investment period, which means shorter period of time might suffer from more temporary fluctuations and you might not be satisfied with the final result. This will be a problem if you need liquidity back, meanwhile we will fail to keep our promise to provide you with an ideal return.
Part Three
3.1 Modification One
As we mentioned in the third question, an serious weakness in our selection model is that there is no guarantee our future expected return can be simulated by historical data. Therefore it is extremely crucial to incorporate newly occurred market conditions. A recommended solution is to include a three-scenario analysis to modify the nine stocks’ expected returns before step one.
Detailed explanation is provided below:
1. Gather individual stocks and HSI monthly historical return data from the last 10 years.
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2. Use benchmarks to define time periods in that 10 years that is undergoing recession (two
consecutive quarters of decreasing GDP)and booming session (such as months with the
unemployment rate under 5%). The rest of the time would be defined as normal sessions.
3. Calculate the frequencies of those three market situations, use the percentage as the the
probability of that market state happening.
For example:
Probability Market State s.d. Average return
20% Boom a x%
60% Normal b y%
20% Recession c z% 4. The average return of x%, y% and z% and standard deviation data is calculated through
the data points within their market state’s period. For instance, if stock A yields an
average of x% return when market is Booming, y% when market is Normal and z% when
market is in recession, then the final average return would be
20%*x%+60%*y%+20%*z%. The return of the nine stock will not change at this
moment yet, not before we include the latest news.
5. A possible event to incorporate is the trade war, which has been looming for half a year
since the beginning of this year. We will assess news like this and re-determine the
likelihoods for possible market states. A reduction of almost 25% in A-shares this year in
China has also raised concerns for a market meltdown.Those news tends to raise the
probability of the recession state in the near future, for example, from 20% to 40%, while
decreasing the possibility of other two states respectively. Since it’s only the raw
parameters of the 9 stocks that has been changed, no further hassle is needed for the rest
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of the calculation. We believe that a scenario analysis like this would help the portfolio to
better match the future trend.
3.2 Modification Two
The above method is to adjust the value of all stocks’ parameters based on the whole market’s situation. To make our portfolio better reflecting future’s expectation, we can demonstrate another method to add weighting on individual stocks. Following will show the procedures of this adjustment:
1. Single out the 9 stocks based on their industries. For example, Cathay Pacific Airways
belongs to the transportation industry, CITIC belongs to the banking industry, while
PCCW belongs to the telecom service industry.
2. Gather news on certain industries. For example, according to a market analysis done by
Bank of America not far from now, the oil price is likely to rise sharply to $100/barrel
next year, depending on political elements and production goals in the middle-east and
Russia. This could have a negative impact on the transportation industry, since a major
cost for airline companies is fuel cost.
3. Such industry analysis can then be applied to single stock such as Cathay Pacific Airways
in average return estimation. Instead of using the monthly average of the five year data,
we could extract the lower 40th percentile, or less, depending on the severity of the
negative news. Therefore, the calculation of single asset average return, along with
Modification one, would be:
Er = P boom * x% + P Neutral * y% + P Recession * z%
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Where x%, y% and z% is the 40th percentile of the sample distribution, instead of exactly the mean. This method could work together with the first one to better summarise industry specific news and has a better ability in adjusting individual assets. Together, we believe that by using those two methods, we can obtain a wiser choice for our client’s investment.
3.3 Modification Three(Can be used with Modification Two but not Modification One )
We can use a method to endow the obtained data a “time value”, which is increasing within time as well. Following will show the procedures of this adjustment.(Assume the adjustment is based on annual base for simplification).
1. Give distinguished weights of different-year expected returns by timeline.Define that the
data from 5 years ago only distributes W1(5% for example ) for today ,that of this year
occupies W5(60% for example) and vice versa.
2. Come out with more reliable historical returns and standard deviations of each stocks and
the whole market by calculation.
Here ,Wi indicates the “time values”(in percentage) of different years, Ri indicates the average returns of every year.
3. Update all the results in our excel sheets, and repeat the rest calculations.