Question 1:
After consider the requirements, we decide to use Mean-Variance preferences method to calculate the utility we can get from investing four stocks: DBS, SPH, GE, SIA.
We know the equation:
Ur= μ-γ2 Varri=μ- γ2σi2
We can decide weight of four stocks by maximizing the utility of the portfolio. (N=4, T=232)
So we suppose:
1. The best portfolio is the one that satisfy owner most, which means have the maxi utility.
2. The owner is afraid of risk, and the risk aversion is measured by γ, γ=3
3. The portfolio will be hold for one year.
From equation:
Rate=R1-R2R1
We can get return rates of four stocks in different periods since Jan-96 (See sheet 1), and the mean return rates μ for the four stocks are:
DBS=0.10814638, SPH=0.0733264, GE=0.12002196, SIA=0.06982988
And we can get the Covariance Matrix:
0.105279399 0.041817263 0.05022109 0.0565126810.041817263 0.062596535 0.025656805 0.0373403140.05022109 0.025656805 0.109059667 0.0386166250.056512681 0.037340314 0.038616625 0.065556537
A= 21.372. B= 1.72756669. C= 0.18175538, D= 0.90015379
And we get σ σDBS=0.32446787, σSPH=0.250193,σGE=0.33024183, σSIA=0.25604011
From the equation: σportfolio2=wi2σi2+wiwjσi,j (i≠j)
We can get the σ with different weights (See sheet)

Since efficient frontier is equal to the minimum-variance frontier for expected returns larger than the expected return of the global minimum-variance portfolio

We also know the equation wμp=a+bμp Expected portfolio return: μp1=μTw(μp) Variance αp12=wT⋁w=a+bμpT⋁(a+bμp)
We can get the highest utility after calculation: U= 0.0176662, with mean return is 0.095

REBALANCE:
In order to make the profit better, we decide to use Trend Analysis to rebalance the weight of our portfolio.
The first way we can use is just use the trend theory to decide the timing of rebalance, whenever the trend line is crossed, we will consider to rebalance using the method we used at the beginning of the report.
The second way we can use is to not only decide the timing, but also deciding the buy or sell after the trend line is crossed.
The lowest point two years from now is point A, and we suppose the net lowest price is point B; the point C is the price at Apr-15. We connect the point A and point B and get a line, which indicates a trend. From the picture we can know that the trend line is upward and from the trend theory in stock market, the line is a support line, and if the stock price of one-month period is below or go down cross the line, we will consider to sell off the stock. And also at the same time, we connect the first peak two years from now and the second peak (not the second highest) and then connect the two points, we can get another line which is the resistance line and if the stock go across the line, it will indicate that the stock price will increase in the future.
If the first support line is crossed by the stock price, we can reconnect the two new lowest points (from the time that support line is crossed by the stock price) and have a new support line.
If the first resistance line is crossed by the stock price and we reconnect the two new highest points (from the time that support line is crossed by the stock price) and have a new resistance line. The following is the picture of DBS’s stock price, since DBS is in bank industry and the risk is not as high as other three companies.

The following is the data sheet:

So if the stock [rice of DBS goes down and crosses the trend line, the support line for one period (one month), we will consider sell off the stock and buy other stocks, and if the resistance line of DBS crossed by stock price, we will consider to buy more DBS stock.

Question 2:
BACKGROUND
Risk is defined as the probability of unacceptable loss and is equal to the annualized variance for returns that are normally distributed. There is normally a tradeoff between risk and return. A risk adverse investor shall take on more risk only if the return he is promised is more. With the help of utility theory, this trade off was given a more robust explanation. Daniel Bernoulli argued that the price an individual would be willing to pay for a risky strategy would be its utility. He also stated that utility resulting from an increase in wealth is inversely proportional to the assets previously owned by the investor. Also, risk averseness of each individual is different.
The Effective Utility hypothesis assumes that every individual investor captures his wealth into utility, through a utility function that has 2 main features. First, an investor always prefers more to less. The implication is that a graph of utility against wealth is upward sloping always. The second feature is that utility resulting from an increase in wealth is inversely proportional to the previous level of wealth. This provides us with a useful technique on which we can base our portfolio calculations.
Gamma is used to measure the tradeoff that we should make between risk and return for a particular investor. Thus, setting a gamma allows us to calculate an investor’s optimal portfolio by defining the penalty or adjustment, which is equal to half the product of variance and gamma.

METHODOLOGY
We first calculated the mean and variance of returns for each of the funds provided. The mean measures the average return of the funds while the variance measures dispersion of returns. By using this, we excluded funds that were mean variance dominated by other funds (refer to graph 1). Funds B, C, D, E mean –variance dominate funds A, F, H, I and J.

We then utilized the mean variance preference theory. Mean Variance preference method of portfolio optimization is a popular way to simulate portfolio decisions simply because it is easy to understand and apply: it is assumed that investors maximize their preferences that are based directly on the mean and variance of their portfolios.
For the simple decision problem, the assumptions are: * Single-period model * Preferences depend only on the mean and variance of payoffs * At a given mean, lower variance is preferred * At a given variance, a higher mean is preferred * Price-taking with no taxes or transaction costs
Thus, instead of defining utility over consumption or wealth, Mean-Variance preferences are defined over moments of return uMVri=Eri-γ2Varri=μi-γ2σi2 where γ captures risk aversion and γ/2 σ^2 (i) applies a penalty to the variance in returns. Risk Aversion is a concept that implies that investors would prefer lower returns with known risks than higher returns with unknown risks. We have used γ=3 as we have assumed that we are risk averse albeit moderately.
Justification (A): If returns R(i) are normally distributed then R(i)W (W is a given level of wealth), is also normally distributed, and we can rewrite any utility function of R(i)W as a function of the expectation (or mean) and variance of r(i).

RESULT
Fund B has the highest utility of 0.007734445 among funds B, C, D, and E | Fund B | Fund C | Fund D | Fund E | Mean | 0.010157353 | 0.009077206 | 0.005903676 | 0.0068125 | Variance | 0.001615272 | 0.001124715 | 0.000853344 | 0.001002221 | Volatility | 0.04019045 | 0.033536775 | 0.029212051 | 0.031657878 | Utility | 0.007734445 | 0.007390133 | 0.004623661 | 0.005309168 |

Since the target weight of the selected fund is 20% and our initial portfolio mix of equities and bonds is 60:40, we recalculate the weights and allocate 80% of the portfolio between equities and bonds in the ration of 60:40 resulting in new portfolio weights of 48:32:20 for equities, bonds and the fund. Portfolio | plus Fund B | plus Fund C | plus Fund D | plus Fund E | Return | 0.006417178 | 0.006201149 | 0.005566443 | 0.005748208 | Variance | 0.000691858 | 0.000564125 | 0.000469789 | 0.000530029 | Utility | 0.005379392 | 0.005354961 | 0.00486176 | 0.004953164 | MDD | -0.240128759 | -0.264549079 | -0.266234811 | -0.25165422 |

We calculated the portfolio return, the portfolio variance and the portfolio utility by using the weights specified above and by including each of the funds- B, C, D and E individually in the new portfolio.
The portfolio variance is = σ^2(rp) = w^2(e)σ^2(re) + w^2(b)σ^2(rb) + w^2(f)σ^2(rf ) + 2w(e)*w(b)*cov(r(e),r(b)) + 2w(e)*w(f)*cov(r(e),r(f) ) + 2w(b)*w(f)*cov(r(b),r(f) ).
The portfolio mean is calculated as R(p) = w(e)*r(e) + w(b)*r(b) + w(f)*r(f ) where e= equity, b= bonds, and f= fund.
We also used the maximum drawdown method in our analysis. Maximum drawdown measures the single largest drop from the highest point to the bottom in the value of a portfolio. The formula for maximum drawdown is as follows:
Maximum Drawdown = (Peak value before largest drop - Lowest value before new high established) / (Peak value before largest drop)
It is a good indicator of potential losses the fund can experience. The percentage drop in cumulative returns for fund B is the lowest (-0.2401) compared to fund’s C, D, and E.

CONCLUSION
Based on the findings above, Fund B is our chosen Fund on an individual basis as well as when including in the given portfolio.