Solved by verified expert:As a financial risk manager (FRM) working for an asset management firm, you are facing the taskof constructing a set of optimal portfolios, which eventually will be proposed to the clients. Ideally,you should deliver a summary of different strategies stating the risk and return of each. In doingso, you wanna make sure that each strategy is delivering the best risk-return trade-off.In technical terms, you are asked to construct a Mean-Variance Efficient Frontier (MVEF) giventhe universe of the 12 stocks. To do so, you need to solve the following optimization problem for a given m Your final goal is to provide a list of (m, p(m)) for a set of m values. After deriving this list, you need to: plot each m against its corresponding p(m) – the plot should deliver a concave function representing what is known as the MVEF. highlight the point on the MVEF with the maximum Sharpe-ratio (SR) – in fact, this point refers to the SR portfolio (recall Portfolio 2) split the data into two periods, one covering the years 2015 and 2016 (Period 1) an done covering the more recent years 2017 and 2018. Repeat the above and compare between the two periods MVEF. Provide a number of insights.

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Mean-Variance Efficient Frontier (20 Points) – Bonus

As a financial risk manager (FRM) working for an asset management firm, you are facing the task

of constructing a set of optimal portfolios, which eventually will be proposed to the clients. Ideally,

you should deliver a summary of different strategies stating the risk and return of each. In doing

so, you wanna make sure that each strategy is delivering the best risk-return trade-off.

In technical terms, you are asked to construct a Mean-Variance Efficient Frontier (MVEF) given

the universe of the 12 stocks. To do so, you need to solve the following optimization problem for a

given m:

min σp2 = w0 Σw

(7)

w0 1 =1

(8)

w0 µ =m

(9)

w

subject to

where

1. w denotes a d × 1 vector of portfolio weights

2. Σ is the covariance matrix of the asset returns

3. µ is the vector of mean returns, while m denotes the mean target

For each given m, there is an optimal portfolio w(m) that has a mean return of µp (m) = m and

volatility of σp (m). Your final goal is to provide a list of (m, σp (m)) for a set of m values. After

deriving this list, you need to:

• plot each m against its corresponding σp (m) – the plot should deliver a concave function

representing what is known as the MVEF.

• highlight the point on the MVEF with the maximum Sharpe-ratio (SR) – in fact, this point

refers to the SR portfolio (recall Portfolio 2)

• split the data into two periods, one covering the years 2015 and 2016 (Period 1) an done

covering the more recent years 2017 and 2018. Repeat the above and compare between the

two periods MVEF. Provide a number of insights.

Proposed Solution

1. We covered this problem in both the class and the handouts. In particular, there is a closed

form solution for the above optimization problem that can be represented as a function of

A, which represents the risk aversion of the client. According to Equation (9) from Class 1

8

handouts, it follows that the optimal portfolio is a combination of two portfolios (funds):

w = w0 +

1

w1

A

(10)

where w0 is the global minimum variance portfolio (GMV) and w1 = Bµ is the more “aggressive” portfolio (recall Equation (9) from Handouts 1 handouts). The weight allocated to

the latter is determined by A, which is proxies the risk-aversion of the investor. In particular,

under certain assumptions, it follows that

A=

m − w00 µ

w01 µ

−1

(11)

2. Using the stock returns data, estimate the vector of mean returns µ and the covariance matrix

Σ. Then do the following computations:

(a) Given Σ, compute the GMV portfolio, i.e. w0 from Equation (11) from Class 1 handouts

(b) Given µ and Σ, compute w1 = Bµ – see Equation (10) from Handouts 1

(c) Compute the mean return on each fund, i.e. µ0 = w00 µ and µ1 = w01 µ

(d) For a given m, there is a unique value A as described in Equation (11). In particular,

you wanna set the values of m to range between µ0 and 5×max(µi ) ∀i = 1, .., 12. You

need to do this for at least 20 unique values of m in such a range.

(e) Ideally, you should write a function that takes m, µ, and Σ and returns the optimal

portfolio w(m) and its optimal volatility σp (m) where

σp (m) =

q

w(m)0 Σw(m)

(12)

Note: the above task requires matrix multiplication only, such that you do not need to

perform any numerical optimization. However, if you wish to consider additional constraints

or risk measures (e.g., Value-t-Risk), a numerical solution is needed.

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