Investors’ goals Inputs and output Minimizing risk Minimizing risk subject to a target return Maximizing expected returns subject to a
target level of risk
The optimal portfolio: Maximizing risk-adjusted returns
Restrictions
The big picture
Excel section
Challenge section
I
n Chapter 4 we discussed how to calculate the risk and return of a portfolio. In this chapter we’ll discuss how to obtain optimal portfolios. More precisely, we’ll see how to minimize risk; how to minimize risk for any desired level of return; how to maximize expected return for any target level of risk; and how to maximize risk-adjusted returns.And we’ll discuss in detail how to do all this in Excel. (Before reading this chapter, it is essential that you become familiar with all the concepts discussed in Chapter 4. It is also essential that you both read the Excel section and work out the problems in the Challenge section.)
Investors’ goals
We all invest for different reasons and we may all have different goals. Some people save for retirement, others to go to college, some to eventually buy a home, others attempt to become rich quickly. The goals are endless, and yet we can group most of the different reasons for investing into four main goals, all of which we’ll discuss below.
Each of these goals can be stated formally as a mathematical problem. And in all these cases, investors face some restrictions (sometimes given, sometimes self-imposed) that must also be incorporated into the mathematical problem.
Although all these problems are different, they do, however, share some common characteristics.
First, all problems have the final goal of either maximizing or minimizing some target magnitude, generally called the objective function. Second, the maximization or minimization of the objective function is subject to at least one restriction, and often to more than one. Third, the common restriction to all problems is to invest all the capital that has been allocated to the portfolio. This means that optimization problems do not determine how much capital to invest;
rather, given the capital to be invested, they determine how to optimally allocate it among the assets considered.
So, what are the four major problems? Investors are usually interested in (1) minimizing the risk of their portfolio; or (2) minimizing the risk of their portfolio subject to a target return; or (3) maximizing the expected return of their portfolio subject to a target level of risk; or, the ultimate goal, (4) maximizing risk-adjusted returns.
We’ll discuss all these problems below, but a bit of notation first. We’ll call Ep and SDpthe expected return and risk of a portfolio, respectively. We’ll call Rfthe risk-free rate. And we’ll call xithe proportion of the portfolio invested in asset i, that is, the amount of money invested in asset idivided by the amount of money
invested in the portfolio. Finally, as discussed in Chapter 4, the expected return and risk of a portfolio are respectively given by
Ep= x1· E(R1) + x2· E(R2) + . . . + xn· E(Rn) (11.1)
(11.2)
where E(Ri) denotes the expected return on asset i, Covij denotes the covariance between assets iand j, and nis the number of assets in the portfolio.
Inputs and output
All optimization problems require some inputs in order to yield an output. What are the inputs and output in our problems? The inputs consist of expected returns, variances (or standard deviations), and covariances (or correlations).
More precisely, for each asset we need to input its expected return, its variance, and its covariances to the rest of the assets in the portfolio. For a portfolio of n assets, this implies nexpected returns, nvariances and (n2– n)/2 covariances.
(Recall that Covij= Covji.)
How to estimate these parameters, however, is controversial. We could base our estimates on historical (ex-post) returns or on forward-looking (ex-ante) returns. The main problem with historical estimates is that means, variances, and covariances tend to change over time and their past values may or may not reflect their expected values, which are the ones we really need for portfolio optimization. The problem with forward-looking estimates, on the other hand, is how to estimate them properly without resorting to historical data, or how to adjust the historical estimates to reflect changing expectations.
The output of all these problems is a set of weights x1*, x2* . . . xn* that achieve the goal stated in the objective function subject to the restrictions of the problem. (In finance and economics the ‘*’ symbol is typically used to denote optimality.) Having obtained these optimal weights, we can then plug them back into the objective function in order to determine its optimal value.
SDp=
{ Σ
n n xi· xj· Covij}
1/2i=1 j=1
Σ
Minimizing risk
Let’s start with the simplest of all problems, which consists of finding the combination of assets that yields the portfolio with the lowest possible risk.
Formally, this problem is stated as Min x1,x2, . . . ,xn
Subject to →x1+ x2+ . . . + xn= 1
The first line states the goal, which is to minimize the risk of the portfolio as measured by its standard deviation of returns. The second line is the ‘allocation restriction’ that we mentioned above, which states that, given the capital to be invested in the portfolio, we need to find how to optimally allocate it among all the assets considered.
The solution to this problem is a set of weights x1*, x2* . . . xn* that determines the portfolio with the lowest risk (measured by its standard deviation of returns). We can then plug these optimal weights (together with the inputs of the problem) into equations (11.1) and (11.2) to determine the expected return and risk of this portfolio which, as discussed in Chapter 4, is called the minimum variance portfolio(MVP).
Minimizing risk subject to a target return
Investors often have a target return they want to achieve, and they obviously want to achieve it bearing the lowest possible risk. Formally, this problem can be stated as
Min x1,x2, . . . ,xn
Subject to →Ep= x1· E(R1) + x2· E(R2) + . . . + xn· E(Rn) = ET
→x1+ x2+ . . . + xn= 1
The first line states the goal, which is (as in the previous problem) to find the portfolio with the lowest risk. The second line states the restriction that the
SDp=
{ Σ
n n xi· xj· Covij}
1/2i=1 j=1
SDp=
{ Σ
n n xi· xj· Covij}
1/2i=1 j=1
Σ
Σ
portfolio must have an expected return of ET(the target return). And the third line is the allocation restriction we already discussed.
The solution to this problem is a set of weights x1*, x2* . . . xn* that determines the portfolio with an expected return ETthat has the lowest risk. We can then plug these optimal weights (together with the inputs of the problem) into equation (11.2) to determine the risk of this portfolio. (Its expected return is predetermined in the first constraint and is equal to ET.)
Maximizing expected returns subject to a target level of risk
Some investors may have a maximum level of risk they are willing to bear, and want to find the portfolio that yields the highest expected return for that level of risk. Formally, this problem can be stated as
Max x1,x2, . . . ,xn Ep= x1· E(R1) + x2· E(R2) + . . . + xn· E(Rn) Subject to →
→x1+ x2+ . . . + xn= 1
The first line states the goal, which is to find the portfolio with the highest expected return. The second line states the restriction that the portfolio must have a target level of risk of SDT. And the third line is the allocation restriction.
The solution to this problem is a set of weights x1*, x2* . . . xn* that determines the portfolio with a risk of SDTthat has the highest expected return.
We can then plug these optimal weights (together with the inputs of the problem) into equation (11.1) to determine the expected return of this portfolio.
(Its risk is predetermined in the first constraint and is equal to SDT.)
The optimal portfolio: Maximizing risk-adjusted returns
All the previous problems state different goals (and restrictions) that investors may have. However, finance theory suggests that the ultimate goal of a rational investor should be to find the portfolio that optimally balances risk and return.
SDp=
{ Σ
n n xi· xj· Covij}
1/2= SDTi=1 j=1
Σ
In other words, the ultimate goal of the rational investor is unequivocal: to maximize risk-adjusted returns.
The issue of risk-adjusted returns is explored in detail in the previous chapter. For our current purposes, it suffices to highlight two issues. First, that the ‘best’ portfolio is not the one that maximizes the expected return. If that were the case, we’d put all our money in the one asset with the highest expected return. But that is not what we usually do. We care also about risk and therefore we diversify. In other words, we care both about returns and sleeping soundly at night too.
Second, although there are many ways of defining risk-adjusted returns (as discussed in the previous chapter), perhaps the most widely used definition is the relatively simple Sharpe ratio (Sp)which is given by
(11.3)
Note that an increase in the expected return of the portfolio or a decrease in its risk will increase the Sharpe ratio.
We can now restate the goal of maximizing risk-adjusted returns as finding the portfolio that maximizes the Sharpe ratio. Formally,
Max x1,x2, . . . ,xn =
Subject to x1+ x2+ . . . + xn= 1
The first line states the goal, which is to find the portfolio with the highest Sharpe ratio, and the second line is the allocation restriction. The solution to this problem is a set of weights x1*, x2*. . . xn*that determines the portfolio with the highest risk-adjusted return. We can then plug these optimal weights (together with the inputs of the problem) into equations (11.1), (11.2), and (11.3) to determine the expected return, risk, and Sharpe ratio of this portfolio.
Sp=Ep– Rf SDp
Sp=Ep– Rf SDp
{ Σ
n n xi· xj· Covij}
1/2i=1 j=1
x1· E(R1) + . . . + xn· E(Rn) – Rf
Σ
Restrictions
Finally, a quick comment on the restrictions of all the problems we discussed.
Besides the allocation restriction (and the other two we considered), we can add to these problems as many constraints as necessary. We could, for example, restrict short-selling by adding the restriction
x1≥ 0, x2≥ 0 . . . xn≥ 0
Or we could limit ourselves to invest not more than 20% of the capital in the portfolio in any single asset by adding the restriction
x1≤ 0.20, x2≤ 0.20 . . . xn≤ 0.20
The possibilities are, of course, endless. The portfolio-optimization program in Excel discussed below can solve all the problems we discussed above and handle as many restrictions as necessary.
The big picture
The optimization of portfolios cannot be implemented without the aid of spreadsheets or specialized software packages. Even when considering just a few assets, the problems are usually too daunting to solve by hand. All programs used to optimize portfolios, however, require the same inputs, which basically consist of expected returns, variances or standard deviations, and covariances or correlations. Given those inputs, the program will provide as output the optimal weights, as well as the risk and return of the optimal portfolio.
Some investors may want to minimize risk. Others may want to minimize risk subject to a target return. Others may want to maximize returns subject to a target level of risk. And all of them want, at the end of the day, to maximize risk- adjusted returns. The Excel program discussed below will help you to solve all these problems.
Excel section
We’ll discuss in this section a rather simple Excel program to optimize portfolios. The discussion is based on a four-asset portfolio but you should have little trouble in adapting the program to any number of assets. It is also based on the problem of maximizing risk-adjusted returns, but once again you should
have little trouble in adapting the program to solve the problems in the other sections of the chapter. In fact, in the Challenge section you will be asked to do both things.
The program makes use of the Solver in Excel, which means that it finds numerical (rather than analytical) solutions. Far from being a weakness, this makes it easy handle as many restrictions as desired by simply making slight changes in the Solver dialogue box.
The program basically works in three steps. First, we input the required parameters (a risk-free rate, expected returns, standard deviations, and covariances); then, we make some calculations based on those inputs; and finally we use the Solver to find the optimal solution. The output of the program consists of the set of optimal weights and the risk, return, and Sharpe ratio of the optimal portfolio.
TABLE 11.1
A B C D E F G H I
1
2 A portfolio-optimization program in Excel
3 Chapter 11
4 5 6 7
8 Rf Weights
9
10 1 2 3 4
11 ERs
12
13 SDs
14 Ep
15 Covs
16
17 SDp
18 19
20 Weights Sp
21 Sum
22
23 ER Vector
24
25 SD Matrix
26 27 28 29
Take a look at Table 11.1, which contains the set-up of the model. The cells shaded in light gray are the ones in which we’ll input the parameters; the cells shaded in darker gray are the ones in which we’ll perform some calculations.
This is, step-by-step, what we need to do to find an optimal portfolio using this program:
■ Inputs:
● Enter the risk-free ratein cell C8.
● Enter the expected returnsin cells C11:F11.
● Enter the standard deviationsin cells C13:F13.
● Enter the covariancesin cells C15:F18.
● Enter the weights (25% in our case) to initialize the Solver in cells C20:F20.
■ Calculations:
● Enter ‘=sum(C20:F20)’ in cell F21.
● Block cells H9:H12, enter ‘=transpose(C20:F20)’ and hit ‘Ctrl+
Shift+Enter’ simultaneously.
● Enter ‘=C20*C11’ in cell C23 and copy this expression to cells D23:F23.
● Enter ‘=sum(C23:F23)’ in cell H15.
● Enter ‘=C$20*C15*$H9’ in cell C25, copy this expression to cells D25:F25, and then copy the range C25:F25 to the range C26:F28.
● Enter ‘=sqrt(sum(C25:F28))’ in cell H18 and hit ‘Ctrl+Shift+
Enter’ simultaneously.
● Enter ‘= (H15-C8)/H18’ in H21.
■ Solver:
● Target Cell: H21
● Equal to: Max
● By Changing Cells: C20:F20
● Subject to Constraints: F21=1
That’s it! Not too bad, is it? Before you rush to implement this program, however, let’s highlight a few things. The program requires us to calculate all the necessary inputs beforehand. In other words, before initializing the program, we need to have numbers for the risk-free rate, the expected returns, the standard deviations, and all covariances. That means that first we have to calculate these parameters from the relevant data (or come up with our own forward-looking best guesses), and only then move on to work with the program.
In order to start looking for the optimal solution Solver needs to be initialized with a set of arbitrary weights. The actual value of these weights is largely irrelevant as long as they add up to 1. A good rule-of-thumb for the initial values is to enter weights equal to 1/n (where n is the number of assets in the portfolio) in all the relevant cells. In other words, we initialize Solver by giving it the values of an equally weighted portfolio (that is, weights of 50% in the two- asset portfolio, 25% in the four-asset portfolio, 20% in the five-asset portfolio, and so on).
The calculations we’re required to perform, and the commands we’re required to use, are all very simple. When transposing the weights (second step in the required calculations) and calculating the risk of the portfolio (sixth step), you must remember to hit ‘Ctrl+Shift+Enter’ simultaneously. This is because both these calculations are arrays and that’s just the way they’re entered into Excel.
To activate the Solver we simply go to the ‘Tools’ menu and select ‘Solver’
from the choices. When in the Solver’s dialogue box, the target cell we enter is the one with the value of the Sharpe ratio, and we ask Solver to maximize this value. The cells that Solver will adjust in order to find the portfolio with the highest Sharpe ratio are the portfolio weights, taking into account the constraint that these weights add up to 1. In order to input this restriction, we first need to click ‘Add’ in ‘Subject to Constraints,’ then fill the three required boxes (the cell reference, the sign, and the numerical value), and click ‘OK.’ Once we’re done with these steps, we click ‘Solve’ and then (when asked whether we want to keep the solution) ‘OK.’
Finally, adding more constraints when necessary is very easy. If we wanted to restrict short-selling (that is, if we wanted Solver to restrict the solution to only positive weights), for example, all we need to do is, in the Solver’s dialogue box, click ‘Add’ in ‘Subject to Constraints’ and enter ‘C20:F20’ in ‘Cell Reference,’
select ‘>=’ in the choice of signs, enter ‘0’ in ‘Constraint,’ and finally hit ‘OK’
and ‘Solve.’ If we wanted instead to restrict the weights to being no larger than 20%, we would click ‘Add’ in ‘Subject to Constraints’ and enter ‘C20:F20’ in
‘Cell Reference,’ select ‘<=’ in the choice of signs, enter ‘0.2’ in ‘Constraint,’
and finally hit ‘OK’ and ‘Solve.’
If the whole thing looks a little messy, don’t worry; this is a typical case of
‘easier done than said.’ Once you run the program a couple of times, you will see that you can obtain solutions very quickly and effectively. That’s why the one below is a Challenge section that you can’t skip. Get to work on it then!
Challenge section
1 Consider Table 11.2, which contains annual summary statistics for four emerging markets, Argentina, Brazil, Chile, and Mexico (all summarized by MSCI indices, in dollars, and accounting for both capital gains and dividends) between 1988 and 2003. Panel A reports the mean return (AM), standard deviation (SD), maximum return, and minimum return; panel B reports the variances and covariances.
TABLE 11.2
Panel A
Argentina Brazil Chile Mexico
(%) (%) (%) (%)
AM 40.9 33.6 25.1 31.3
SD 105.1 62.0 38.9 46.9
Maximum 405.0 172.2 116.1 126.0
Minimum -50.5 -61.6 -28.5 -40.6
Panel B
Argentina Brazil Chile Mexico
Argentina 1.1039 0.5186 0.3240 0.3557
Brazil 0.5186 0.3850 0.1943 0.1673
Chile 0.3240 0.1943 0.1515 0.1291
Mexico 0.3557 0.1673 0.1291 0.2203
Using the portfolio-optimization program discussed in the Excel section, and a risk-free rate of 5%, for each of the questions below find the set of optimal weights, as well as the expected return, risk, and Sharpe ratio of the optimal portfolio. (Note: It is implicit in all questions that the restriction x1+x2+x3+x4= 1 must apply.)
(a) Find the portfolio with the highest risk-adjusted return.
(b) Find the portfolio with the highest risk-adjusted return, subject to the restriction that no short-selling is allowed (that is, restricting all weights to be positive).
(c) Find the portfolio with the highest risk-adjusted return, subject to the restrictions of no short-selling and that no asset can take more than 30% of the money invested in the portfolio (that is, restricting all weights to be no larger than 30%).
(d) Find the minimum variance portfolio (MVP).
(e) Find the portfolio with the lowest risk given a target return of 30%.
(f) Find the portfolio with the highest expected return given a target risk (standard deviation) of 40%.
2 Go back to panel A of Table 5.2 containing the annual returns of Disney and Microsoft between 1994 and 2003 and calculate the relevant covariance. Then, using the portfolio-optimization program discussed in the Excel section, and a risk-free rate of 5%, for each of the questions below find the set of optimal weights and the expected return, risk, and Sharpe ratio of the optimal portfolio. (Note: It is implicit in all questions that the restriction x1+x2= 1 must apply.) (a) Find the minimum variance portfolio (MVP).
(b) Find the portfolio with the lowest risk given a target return of 15%.
(c) Find the portfolio with the highest expected return given a target risk of 26%.
(d) Find the portfolio with the highest risk-adjusted return.