(p.331) Appendix III Growth Asset Portfolio Construction Principles
(p.331) Appendix III Growth Asset Portfolio Construction Principles
Optimizing an investment portfolio, that is, maximizing the yield and simultaneously minimizing the risk is a major area in finance.
In the last forty years plenty of good books and research have been published on that topic; the objective of this appendix is not to write a new one but to present briefly various key points which could support thoughts and help understand the consequences of investment decisions.
In 1952, Harry Markowitz (Nobel Prize in Economic Sciences in 1990) published the article ‘Portfolio Selection’ in the Journal of Finance on the maximization of investment portfolio management. According to the theory, it’s possible to construct an ‘efficient frontier’ of optimal portfolios offering the maximum possible expected return for a given level of risk.
The idea is pretty simple: diversification reduces risks.
A3.1 Risk-Adjusted Return Portfolio Maximization
Assumptions:
• Every investment is risky: the return of an investment is unknown; the probability of getting a specific return is based on an historical analysis of the returns.
• Investors are risk-averse.
A3.1.1 Rate of Return of Portfolio
For an underlying asset (stock, currency, option, bond, or portfolio) that one buys at time zero and sells at a fixed time (T≥ 0), during this single investment period, the total return is:
Where,
X_{0}_{:} amount of money at time 0
$Xt$: amount of money at time T
The rate of return is:
Let us suppose that a portfolio of n assets is built and each asset has its own return of investment (where n: various assets in the portfolio).
(p.332) An amount of money X_{0 I} is selected to invest in each asset of the portfolio.
At this point, we introduce the weight per asset in the portfolio w_{i;} the sum of the weight is 1.
Consequently, the return of the portfolio is:
The sum of the weight is 1,
Where,
r_{i}_{:} total return of the investment.
A3.1.2 Expected Return and Variance of a Return of Investment of a Portfolio
The objective is to build a portfolio with the highest expected return and the lowest risk (one simple measure of risk is variance, see Appendix II, ‘Introduction to Statistics’).
When you buy an asset, the return r is uncertain; r is a random variable.
If we know the mean ($\stackrel{\u2015}{{r}_{l}}$) for each asset (i) of a portfolio that we could call $\stackrel{\u2015}{{r}_{l}}=1,\mathrm{\dots},n$, the variance of asset (i) , ${\sigma}_{i}^{2}$ and the covariance between assets (i), ${\sigma}_{ij}$, the problem can be formulated in two ways:
• For a risk-averse investor, to minimize the variance in accordance with a given expected return $\stackrel{\u2015}{r}$
Where,
${W}_{i}$ and ${W}_{i}$ are the weight of amount invested in assets i and j
${\sigma}_{\text{ij}}$ is the covariance between i and j
Subject to:
• To maximize the expected return in accordance with a given variance:
Subject to:
The objective is to maximize the risk reward:
Subject to:
Φ: parameter that corresponds to the objective of the investor in terms of return and the risk associated with this return.
Portfolio return is the proportion-weighted combination of the constituent assets’ returns.
Where,
X_{i} : amount invested in asset i; the sign can be positive as an investor buys an asset or negative as he sells it.
n: number of securities within the portfolio $\text{E}({\tilde{R}}_{n})$: expected return of asset i
Regarding a portfolio P with n assets, the formula of the variance of the return is:
Where,
i is different from j
and the standard deviation of the return of a portfolio is:
A3.2 Illustration
A3.2.1 Modern Portfolio Theory: The Efficient Frontier with No Risk-free Asset
Every combination of the risky assets can be plotted in a risk-expected return space (see Figure A3.1).
The hyperbola is the efficient frontier if no risk-free asset is available. It consists of the set of all portfolios, either a minimal variance subject to a fixed expected return, or a maximal expected return subject to a fixed variance.
The upper edge of this region is the efficient frontier if no risk-free asset is available.
The straight line is the efficient frontier with a risk-free asset. (p.334)
Table A3.1. Calculation of an efficient frontier of a portfolio of two assets
Asset B |
Asset A | ||
---|---|---|---|
Expected return |
14% |
Expected return |
7% |
Standard deviation |
7% |
Standard deviation |
3% |
Correlation between B and A = ρ AB = 1 |
Portfolios located on the efficient frontier offer the best possible expected return for a given risk level.
Now, let us examine two case studies to understand how to build an efficient frontier with two assets.
Regarding the first case study (Table A3.1), the correlation between two assets is 1; the correlation between the two assets is –0.50 for the second case study.
A3.2.2 Case Studies
A3.2.2.1 Case Study 1
Based on Table A3.2, let us build the curve of the portfolios of assets A and B with invested amounts from 0 to 100 per cent for each of them (see Figure A3.2).
In Practice, How do we Build this Curve?
For the portfolio analysis where the weight of A is 100 percent of the amount invested and consequently, the weight of B is nil, the formulae are:
Expected return:
Table A3.2. Expected return, standard deviation, and correlation of a portfolio of two assets
Asset B |
Asset A |
||
---|---|---|---|
Expected return |
14% |
Expected return |
7% |
Standard deviation |
7% |
Standard deviation |
3% |
Correlation between B and A = ρ AB = 1 |
Amount invested in % in |
Portfolio | ||
---|---|---|---|
Asset B |
Asset A |
Standard deviation |
Expected return |
100% |
0% |
7.0% |
14.0% |
90% |
10% |
6.6% |
13.3% |
80% |
20% |
6.2% |
12.6% |
70% |
30% |
5.8% |
11.9% |
60% |
40% |
5.4% |
11.2% |
50% |
50% |
5.0% |
10.5% |
40% |
60% |
4.6% |
9.8% |
30% |
70% |
4.2% |
9.1% |
20% |
80% |
3.8% |
8.4% |
10% |
90% |
3.4% |
7.7% |
0% |
100% |
3.0% |
7.0% |
Standard deviation:
and so on to calculate the other portfolios.
Notice number ‘1’ at the end of the equation; this is the correlation between A and B.
As an illustration, notice on the graph in Figure A3.2, portfolios ‘A’ and ‘B’.
(p.336) A3.2.2.2 Case Study 2
In case study 2 the expected returns and risks of assets A and B are the same as found in case study 1. However, the correlation is different (see Table 3.3).
Notice in Figure A3.3 the impact of the correlation between case studies 1 and 2 which is different.
Table A3.3. Expected return, standard deviation, and correlation of a portfolio of two assets
Asset B |
Asset A | ||
---|---|---|---|
Expected return |
14% |
Expected return |
7% |
Standard deviation |
7% |
Standard deviation |
3% |
Correlation between B and A = ρ AB = –0.40 |
Amount invested in % in |
Portfolio | ||
---|---|---|---|
Asset B |
Asset A |
Standard deviation |
Expected return |
100% |
0% |
7.00% |
14.00% |
90% |
10% |
6.19% |
13.30% |
80% |
20% |
5.39% |
12.60% |
70% |
30% |
4.61% |
11.90% |
60% |
40% |
3.88% |
11.20% |
50% |
50% |
3.21% |
10.50% |
40% |
60% |
2.65% |
9.80% |
30% |
70% |
2.30% |
9.10% |
20% |
80% |
2.24% |
8.40% |
10% |
90% |
2.50% |
7.70% |
0% |
100% |
3.00% |
7.00% |
(p.337) As noticed in the previous example, portfolio A is the portfolio with 100 per cent of the amount invested in this asset and 0 per cent invested in asset B and portfolio B is the portfolio with 100 percent of amount invested in this asset and 0 per cent in asset A.
A3.2.2.3 Conclusion
The aim of the efficient frontier (the Markowitz model) is to minimize risk subject to a given expected return, that is, to find the optimal amount of money W_{i} to invest in each asset (i).
The most important part of this model is the selection of the assets, the risk factors (standard deviation) and the correlation between the assets.
Other interesting models can be considered too; they are extensions to the Markowitz model.
A3.3 Overview of the Markowitz Model
Portfolio return is the proportion-weighted combination of the constituent assets’ returns and its volatility is a function of the correlations Cov _{ij} of the component assets (i, j).
Expected return:
Where,
R: return on the portfolio,
R_{i}: return on asset i
W_{i}: weighting of component asset i
Portfolio return variance:
Where,
${\rho}_{ij}$: correlation coefficient between the returns on assets i and j.
The expression can be written as:
Where,
${\rho}_{ij}=1$
Portfolio return volatility (standard deviation)
(p.338) For a two assets portfolio:
Portfolio return:
Portfolio variance:
For a three assets portfolio:
Portfolio return:
Portfolio variance: