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Department of Business FINANCE I (102329) – Group 3 – 2012 - 13 Study Guide. Dr. Maria-Antonia Tarrazon & Dr. Joan Montllor
TOPIC 3: PORTFOLIO THEORY (II): MARKOWITZ’S PORTFOLIO SELECTION MODEL
Contents: Summary (II)
4. Markowitz’s portfolio selection model^1
4.1 Markowitz ‘s model: Setting out
Markowitz’s portfolio selection model deals with the capital allocation decision or, in other words, with how an investor chooses between the risk-free asset (or Treasury bills in real life) and an optimal portfolio of risky assets (later called “market portfolio” in the CAPM - the next model that we will consider in Topic 4 - or stock index in real life).
In the portfolio selection model (and also in the CAPM in Topic 4), investors are risk-averse, which means that:
- if two investments offer the same expected rate of return, they choose the one with less risk (lower p), and
- if two investments have the same risk, they choose the one with higher expected return (higher E(Rp) ).
Markowitz’s model is an optimization model that
- maximizes the expected return^2
Max E(R ) x E(Rj)
n
j 1
p ^ j
- subject to the following restrictions:
- Any target risk level: j' jj' j ' j j'
n
j 1
n
j' 1
j
2 j
n
j 1
2 j
2 * σ (^) p x σ xxρ σ σ j
(^) ( 14 )
(where (^) σ^2 p*means any target risk level).
- Investment budget fully vested: x 1
n
j 1
j
(former equation 3)
- No short sales allowed (in this version of the model)^3 : (^) x (^) j 0 j ( 15 )
(^1) Harry M.Markowitz: “Portfolio selection”, Journal of Finance , 7 (1), March 1952, 77-91.
Harry M.Markowitz: Portfolio selection , 1st^ edition, John Wiley & Sons, New York, 1959, and 2nd^ edition, Basil Blackwell, Oxford, 1991. Harry M.Markowitz, William F.Sharpe (CAPM), and Merton H.Miller were awarded the 1990 Nobel Prize for Economics. (^2) Altenatively, the optimization problem can be set out as the minimization of the variance for any target
expected return.
Department of Business FINANCE I (102329) – Group 3 – 2012 - 13 Study Guide. Dr. Maria-Antonia Tarrazon & Dr. Joan Montllor
4.2 The efficient frontier of risky assets (FIRST STAGE OR SETTING)
The feasible set of risky assets is determined:
- by all stocks, each of them considered as an individual asset or isolated investment (=no diversification), and
- by all possible portfolios - or combinations of these stocks, with higher or lower degree of diversification.
The portfolio selection problem is solved with quadratic linear programming. Given the expected return and standard deviation for each stock, as well as the correlation coefficient between each possible pair of stocks, the set of efficient portfolios is calculated.
This outcome of the optimization problem is also known as the efficient frontier of risky assets:
E(R p )fσ^ p. ( 16 )
Selecting an investment on the efficient frontier of risky assets:
As a result of this setting out, the efficient frontier of risky assets E(R p )fσ^ p is a concave
function. In other words, efficient portfolios lay on the upward sloping part of the frontier, while inefficient portfolios are to be found on the downward sloping part of the frontier.
The main idea behind the efficient frontier of risky assets is that, for any risk level, investors are interested only in that portfolio with the highest expected return. An for any expected return, investors will only choose that portfolio with the lowest standard deviation.
This stage of the model is called by us in the classroom the FIRST STAGE OR SETTING. In this stage, there are only risky assets (stocks or risky portfolios).
For drawings of this setting, see, for example, Bodie/Kane/Marcus, Figures 8.10, 8.12, 8.13, and 8.15. Or Brealey/Myers, Figures 8.4 and 8.5.
(^3) If short sales are allowed, the third restiction disappears.
Department of Business FINANCE I (102329) – Group 3 – 2012 - 13 Study Guide. Dr. Maria-Antonia Tarrazon & Dr. Joan Montllor
The line between rf on the vertical axis and the tangency point (P*) on the concanve part of the frontier of risky assets determines the new efficient frontier with lending, as expressed by the following equation:
p P*
P* f p f σ σ
E(R ) r E(R ) r
This stage of the model is called by us in the classroom the SECOND STAGE OR SETTING. In this stage, agents can invest in the risk-free asset (lending) and in efficient risky portfolios. Nevertheless, borrowing is still not allowed.
For drawings of this setting, see, for example, Bodie/Kane/Marcus, Figure 8.16.
5.3 Lending and borrowing (THIRD STAGE OR SETTING)
The third and final stage of this model introduces a new possibility, this time affecting investors who, keeping always a risk-averse attitude, are willing to assume high risk. Investors can now borrow money to invest more than 100% of their budget in the optimal risky portfolio P*.
In this stage, two settings are possible:
- a more realistic one, where agents lend and borrow at differential rates, being the borrowing rate (k) higher than the risk-free interest rate (rf): k > rf
- and an alternative setting, the one that will be kept until the end of this portfolio selection model (and afterwards also assumed by the CAPM), where investors can lend and borrow money at the risk-free interest rate: k = rf.
Department of Business FINANCE I (102329) – Group 3 – 2012 - 13 Study Guide. Dr. Maria-Antonia Tarrazon & Dr. Joan Montllor
When lending and borrowing at rf are allowed, four types of investors arise in the model:
▪ Type I: 100% of the budget invested in the risk-free asset:
- Expected return on his/her portfolio: E(RpI) = rf
- Risk associated: pI = 0
- Budget distribution: =0 and (1-)= 1. ▪ Type II: 100% of the budget distributed between the risk-free asset and the optimal risky portfolio P*:
- Expected return on his/her portfolio: E(RpII) = (1-)·rf + ·E(RP*)
- Risk associated: pII = ·P*
- Budget distribution: 0< <1 and 0<(1-)< 1. ▪ Type III: 100% of the budget invested in the optimal risky portfolio P*:
- Expected return on his/her portfolio: E(RpIII) = E(RP*)
- Risk associated: pIII = P*
- Budget distribution: =1 and (1-)=0.
▪ Type IV: more than 100% of the budget invested in the optimal risky portfolio P*:
- Expected return on his/her portfolio: E(RpIV) = (1-)·rf + ·E(RP*) Important: In this case, (1-)·rf < 0 and denotes payment of interests.
- Risk associated: pIV = ·P*
- Budget distribution: >1 and (1-)< 0.
Notice that all four equations for the expected portfolio return are variations of equation (20), depending on the values taken by (fraction of the budget invested in the optimal risky portfolio P*) and [1-] (part of the budget invested in the risk-free asset).
This stage of the model is called by us in the classroom the THIRD STAGE OR SETTING.
For drawings of this setting, see, for example, Bodie/Kane/Marcus, Figures 8.17, 8.18, 8.19 and 8. (with differential rates for borrowing and lending). Or Brealey/Myers, Figure 8.6 (with lending and borrowing at the same rate).
