1. Risk Analysis & Modelling
Lecture 3: Constrained Efficient Frontiers

3. What We Will Look At In This Lecture
The importance of the concept of return
Another look at investment ‘weights’
The concept of the efficient frontier
The calculation of the unconstrained frontier
The calculation of the constrained frontier
The use of Excel Solver to perform constrained optimisation

4. Important Concepts From Last Week
Some variables are ‘random’ and have uncertainty or ‘risk’
We decided that the mean and the variance might be good measures for risk and return
The mean and variance of a portfolio of assets can be ‘implied’ from the mean, variance and covariance of the assets it contains

5. The Return Of An Asset
Last week we talked about the return of a portfolio in terms of the return of its assets
When we talk about returns we are talking about proportional changes in the value of an asset or portfolio
Using returns allow us to simplify the problem because it allows us to think of all assets in a similar way regardless of their physical size
This size independence simplifies the problem we face in quantifying risk since we can ‘compare’ assets of different sizes

6. 2 Equally Likable Investments
Investment 1 consists of buying stocks at £10 each of which will be worth £11 in one year
Investment 2 consists of buying houses for £100,000 each of which will be worth £110,000 in one year
If we have £1,000,000 to invest should we buy 10 houses or 100,000 stocks?
What we care about is the proportional return or growth in our investments not the absolute size of their gains
The same is also true of our liabilities (like the rate of interest your bank charges you)

7. 2 Definitions of Proportional Growth
The standard way you would probably define proportion growth over a period is
Where P(0) is the price in period 0 and P(1) is the price in period 1
Using this definition the return for both the house and the stock would be 10%
Later in the course we will see why a different but related measure of proportional return is useful:

8. An Interesting Question
If we accept the basic assumptions that we can parameterise investments (and liabilities) in terms of the mean (good) and standard deviation (bad) of their returns. Then an interesting question arises:
What is the portfolio with the highest expected return for a given deviation?
As we saw in last weeks exercise the behaviour of risk is complicated by the quadratic nature of the variance of a portfolio in terms of the assets it contains. (The quadratic relationship means that more than one portfolio can have the same risk, so we have a choice)

9. The Efficient Frontier For The 2 Asset Case
E[RP]
Max Return Portfolio
StdDev[RP] WA WA x y x y
1.0
Minimum Variance
E[RP]
Max Return Portfolio
Efficient portfolio (WA = y)
Inefficient portfolio (WA = x)
StdDev[RP]

10. Observations From The 2 Asset Case
There can be 2 possible portfolios for a given level of risk due to the quadratic nature of variance
Although these portfolios have the same variance one can have a higher return than the other, and is thus more efficient
The highest return portfolio will contain 100% in the asset with the highest return
The minimum return portfolio can be determined from just looking at the variance chart
The ‘efficient’ part of the curve (the efficient frontier) is bounded between the minimum variance portfolio and the max return portfolio

11. Optimising Portfolios With More Than 2 Assets
For portfolios containing 3 assets or more there are no simple graphical representations
The two asset case is simple because we only have one choice variable (ie investment in asset A, the investment in the second asset is then implied)
We must rely entirely on mathematics to solve problems of 3 assets or more
Many of the insights we observed for the 2 assets are true for all fontiers
We will used Excel’s solver to calculate the optimal portfolios and the efficient frontier

12. An Efficient Frontier Expected Return Risk (Standard Deviation)
Maximum
Return Portfolio
Efficient Portfolios
Efficient
Frontier
Expected Return
Inefficient Portfolios
Minimum Variance Portfolio
Risk (Standard Deviation)

13. General Description Of The Problem
Minimise the Var[Rp]
Where
E[RP] = WT.R
Var[RP] = WT.C.W
Subject to
W e F
F is the set (or group) of feasible weight vectors, e comes from set notation and means belongs to. So W e F means the weight vector must belong to the set of feasible vectors (or feasible regions)

14. The 2 Asset Case In This Notation
Minimise the Var[Rp]
Where
E[RP] = WT.R
Var[RP] = WT.C.W
Subject to
W e F
where F is the set of all vectors for which WA+WB=1, 0<=WA<=1, 0<=WB<=1 and E[Rp] = T
Where T is our target return.

15. Mathematical Programming
Mathematical Programming is not about ‘programming’
Mathematical Programming is a set of algorithms used to solve problems like the minimisation of variance for a given level of return
Since variance is a quadratic function we need to use a Quadratic Programming algorithm or QP
These algorithms are fairly complex and we do not need to go into them in detail but we will look at the general ideas behind them
Excel has a built in multi purpose MP tool called Solver which can deal with QP problems containing linear constraints

16. Feasible Direction Search
To use solver effectively it is important to have some understanding of how it works
Solver is based on improving feasible directions
An improving feasible direction is a vector which alters the inputs in such a way to improve the objective function without breaking any of the constraints
By altering the inputs by a feasible direction it is possible to approach the optimal
When we reach the optimal there are no feasible directions which will give us an improvement
Calculating improving feasible directions can be achieved by using linear programming and gradient vectors
The Linear Programming Simplex Algorithm by is the basis for a number of mathematical programming algorithms

17. Feasible Solution Diagram
Improved Portfolio
Constraint Boundary
Improving Feasible Directions
Optimal
Improving
Direction
Feasible
Direction
Set
Initial Feasible Portfolio

18. Solver Flow Diagram Can I change any of the variable inputs
given the constraints? NO
YES
No Further Solutions
Do any of the possible inputs
Obtain a higher value in the objective
function? NO
YES
Change variable inputs to achieve higher
Objective output

19. What we will ask solver
Say we have a portfolio X with expected return E and standard deviation R for which all the weights are positive and sum to 1
We will ask solver if it is possible to alter the weights in portfolio X to obtain a lower standard deviation R while giving us the same expected return E and keeping all the weights positive and summing to 1
Solver will try to find a improving feasible direction from X that will obey all these rules, and will keep shifting the portfolio until there are no more improvements!

20. The Solver Model Check to see if Risk can be reduced further
Solver Algorithm
Alter the weights
given the constraints
Model
Outputs
Portfolio
Variance
Var[RP]
Variable Inputs
WT.VA.W
Weight Vector: W
WT.E[RA]
Portfolio
Exp Return
E[RP]
Fixed Inputs
Variance Matrix: V
Return Vector: R

21. Calculating the Efficient Frontier using Solver
When calculating the efficient frontier using Excel’s Solver we must first calculate the range of expected returns over which it spans
The lowest return on the efficient frontier is the expected return related to the minimum standard deviation portfolio
The highest return on the efficient frontier is the expected return related to the maximum return portfolio
The efficient frontier lies between these 2 values
All these are, of course, restricted by any constraints we impose on the frontier

22. Bounding the Efficient Frontier
Maximum Return
Expected Return
Efficient Frontier
Return on minimum risk portfolio
Risk (Standard Deviation)

23. Calculating Points on the Frontier
To graph out the efficient frontier we need to calculate point between these minimum and maximum returns
In general it is best to take evenly spaced returns between these two extremities and calculate the minimum risk for that return a point on the efficient frontier:
Min Var[RP] subject to E[RP] = X where X is
our return for which we wish to generate the corresponding efficient frontier point.

24. Diagram of Calculation Method
Evenly spaced returns used to calculate the
points on the frontier
Expected Return
Points on the
Frontier
Calculated
Minimum Variances
Risk (Standard Deviation)

25. Where do I Find Solver
Solver comes as standard with Excel and is an Excel Add-in
It is found under Tools->Solver
If it is not available then you might need to tell Excel to load it by going to Tools->Addins and making sure the “Solver Add-in” is checked
Solver is programmed in VBA!

26. Using Solver
By changing cells specifies the inputs solver will alter to find the minimum or maximum
Target Cell Specifies
The output we wish to
minimize/maximize
Do we
wish to
Min or Max?
Constraints specify the conditions that must
Be met in a feasible solution
Press to Solve!

27. Solver Step 1: Calculate Minimum Variance Portfolio
Target Cell is Portfolio variance
We want to minimize the
Target Cell
We tell Solver it Can change Portfolio
weights
This is subject to the constraints that each of the weights must be greater than or equal to 0 ($B$12:$B$13 >= 0) and the sum of the weights must equal 1 ($B$15 = 1)

28. Step 2: Calculate Maximum Return Portfolio
Target Cell is Portfolio
Return
We want to maximize the
Target Cell
We tell Solver it Can change Portfolio
weights
This is subject to the constraints that each of the weights must be greater than or equal to 0 ($B$14:$B$15 >= 0) and the sum of the weights must equal 1 ($B$17 = 1)

29. Step 3: Calculate Points on the frontier for a set level of return
We want to minimize the
Target Cell
Target Cell is Portfolio variance
We tell Solver it can change Portfolio
weights
This is subject to the constraints that each of the weights must be greater than or equal to 0 ($B$14:$B$15 >= 0) and the sum of the weights must equal 1 ($B$17 = 1)
and that the return must be equal to some level ($F$14 = $G$42)

30. Our Efficient Frontier Algorithm
Use solver step 1 to calculate the minimum variance portfolio and save its risk and return values (by copy and paste)
Use solver step 2 to calculate the maximum return portfolio and save its risk and return values
Take evenly spaced returns between the values calculated in 1) and 2) and use solver step 3 to calculate their associated efficient frontier values saving each risk and return pair
Copy and Paste the mean and variance of portfolio return into a table (using Paste Special, Values)
Graph out all risk return pairs to view the efficient frontier

31. Sector Constraints
Often we wish to optimise portfolios with complex sets of constraints such as limiting the amount we invest in any given sector
We can quite easily add Sector Constraints such as no more than 20% of the portfolio can be invested in a given sector
This would take the form:
WA+WB+WC < 20%
where A,B and C stocks in a given sector
We will see in later lectures how we can build an Asset-Liability model of an insurance company and use linear constraints to model regulation constraints

32. What if you like risk?
An interesting question is does the efficient frontier analysis apply if you like risk?
Would you still want to minimize risk for a given level of return?
The answer is yes: “Minimise Risk for a given level of return” has an equivalent “Optimise return for a given level of risk”
This is called the ‘Dual’
So for a given level of risk the efficient frontier also gives us the best return
Efficient portfolios are said to be stochastically dominant to inefficient portfolios

33. The Dual
For every constrained maximisation problem there is an equivalent constrained minimisation problem called the Dual
So far we have been saying maximise the risk for a given level of return
The dual of this is to maximise return for a given level of risk
Investors often have target returns and want to know how much risk they must incur
But they could equally well have a risk threshold and want to know what the maximum return they could obtain