Modern Porfolio Theory
Markowitz Theory, also known as Modern Portfolio Theory (MPT), was developed by economist Harry Markowitz in 1952. It provides a framework for constructing an
investment portfolio to maximize returns for a given level of risk or, conversely, to minimize risk for a given expected return.
The core concept of the theory is diversification: by investing in a variety of assets that do not perfectly correlate, investors can reduce the overall risk of their portfolios. According to Markowitz, an efficient portfolio is one that offers the highest expected return for a specified level of risk or the lowest risk for a given level of return. This is often illustrated by the “efficient frontier,” a curve representing optimal portfolios that balance risk and return most effectively.
Markowitz Theory relies on key assumptions, including that investors are rational and risk- averse, and that returns follow a normal distribution. While it has faced some criticism, especially regarding its assumptions, MPT remains foundational in finance and is widely used by portfolio managers and investors for constructing diversified investment
portfolios.
The main theory says that a portfolio’s return is equal to the weighted average of the composing asset’s, while its risk (measured in variance of returns) is not linearly related to its composing assets’ risk. Formally, the theory can be defined as follows.
Expected Return of Portfolio
𝐸"𝑅!$ = & 𝑊"𝐸(𝑅")
#
"$%
where 𝐸"𝑅!$ is expected return of the portfolio that consists of n assets (i=1, 2, 3, …, n), and an asset-i weight is denoted by 𝑊" and its expected return is 𝐸(𝑅").
Portfolio’s Risk
𝜎!& = & 𝑊"&𝜎"&+ & & 𝑊"𝑊'𝜎",'
#
'$%
𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖 ≠ 𝑗
#
"$%
#
"$%
where 𝜎!& is portfolio’s variance (risk), 𝜎"& is asset-i variance and 𝜎",' is the covariance between asset’s return i and j. The covariance can be estimated by 𝜎",' = 𝜌",'𝜎"𝜎' where 𝜌",' is the coefficient of correlation between returns of asset-i and asset-j. Note that 𝜎" is simply the square root of the asset-i’s variance, it is called standard deviation.
An optimal portfolio will offer maximum 𝐸"𝑅!$ with constrained 𝜎!& OR has minimum 𝜎!&
while attaining a 𝐸"𝑅!$ target. The portfolio can be obtained by determining the weight of each asset included into it.
A symbolic example of MPT application
Supposed that we have three stocks considered as investment grade assets, let’s name them stock a, b, and c. Each expected return of the asset has been estimated as 𝐸(𝑅)), 𝐸(𝑅*), and 𝐸(𝑅+). The variance of each asset also has been estimated and denoted as 𝜎)&,
𝜎*&, and 𝜎+&. We want to form a portfolio that combines all those three stocks to achieve a
target of expected return of 𝐸"𝑅!$ = 𝑋 while minimizing its risk, 𝜎!&. The objective function:
𝑀𝑖𝑛,!: 𝜎!& = & 𝑊"&𝜎"&+ & & 𝑊"𝑊'𝜎",'
-
'$%
-
"$%
-
"$%
𝑀𝑖𝑛,!: 𝜎!& = 𝑊)&𝜎)&+ 𝑊*&𝜎*&+ 𝑊+&𝜎+&+ 2𝑊)𝑊*𝜎),*+ 2𝑊)𝑊+𝜎),++ 2𝑊*𝑊+𝜎*,+
Note that 𝜎),* = 𝜎*,) Constraints:
1. 𝐸"𝑅!$ = 𝑊)𝐸(𝑅)) + 𝑊*𝐸(𝑅*) + 𝑊+𝐸(𝑅+) = 𝑋 2. 0 ≤ 𝑊), 𝑊*, 𝑊+ ≤ 1 (no borrowing)
3. 𝑊)+ 𝑊*+ 𝑊+ = 1 (no short selling) Assignment (Real World Case)
As a personal financial planner for a client, you are considering recommending investment in Bank Mandiri (BMRI) and Adaro, a well-known coal mining company in Indonesia
(ADRO). Furthermore, a fund manager provides a well-diversified portfolio that aligns with the IDX composite index (IHSG) constituents. A financial analyst has recently published a stock/fund report on the aforementioned assets, which has been summarized as follows:
Note:
The return and the standard deviation are estimated on daily basis Numbers in the lower left table are the coefficient correlation of returns.
Tasks
1. Construct a portfolio with a maximum expected return (E(Rp)) while its risk
(variance) is constrained to the following levels: 0.007%, 0.010%, 0.040%, 0.090%, 0.100%, and 0.160%. Plot the portfolios on a two-dimensional graph, with E(Rp) on the vertical axis and standard deviation (𝜎!) on the horizontal axis. Interpret the graph and provide insights gained from its analysis.
2. If your client agrees that those three assets align with his risk appetite but desires to allocate at least 10% of their portfolio to each asset, will this requirement
necessitate a revision of the portfolio you previously constructed?
