# Portfolio Management

**Portfolio management**is a process of making investment decisions corresponding to financial objectives.

Top synonyms for portfolio management are asset management and investment management.

**Objective of Portfolio Management**is to create an efficient portfolio providing the best risk–return opportunities available given an investment horizon and a set of risk constraints.

### Parameters important in portfolio management

**Investment horizon**is the planned time for holding asset portfolio. This parameter is important to establish the risk related to classes of assets. For example, stocks are less risky when held in portfolio for 1+ year.

**Risk**is a probability of an adverse outcome. Risk of an asset is a probability of an asset to provide a loss or a return lower than expected. Specific level of

**risk tolerance**established between investor and his portfolio manager is an important part of portfolio calculation.

**Expected Asset Return**– is an expected profit (value of the return) on the investment, measured as a mean of the distribution of the return variable.

**Asset Volatility**– is the variation of a trading prices on asset over time measured by the

__standard deviation__(sigma σ) of logarithmic function of the returns.

**Liquidity**–is a degree to which an asset can be quickly traded or sold. Keeping certain level of liquidity is an important strategy in portfolio management that allows flexibility in seizing new investment opportunities.

**Marketability of an asset**– an asset being listed and actively traded and therefore easy to trade and exchange quickly, as opposed to illiquid and semi-liquid securities or toxic assets.

### Portfolio Management Theories

Portfolio Management Theoriesare theoretical models attempting to calculate the optimal asset allocation depending on various parameters.**1.**

**Modern Portfolio Theory**(MPT) is a mathematical model first introduced by an economist and Nobel prize winner,

__Harry Markowitz__in 1952.

Model describes risk and return of any portfolio asset or a group of portfolio assets in relation to each other, and their percentage allocation in the portfolio.

**Some assumptions of MPT theory**

- Investors are risk-averse (they will always choose a lower uncertainty investment given the same returns)

- Investors goal is to maximize economic returns

- Investors are rational and share the same investment views

- Investors do not influence the market prices

- Crowd psychology does not influence the market

- Every investor has access to the same information about the assets and the market

- There are no taxes, trading fees, dividends or capital gains that could influence investing decisions

- Assets returns are normally distributed

- All assets and infinitely liquid and instantly tradable

- Risk and returns need to have the same time horizon

- The model works within one single investment period

**The expected return of an asset**

E(R)=p1R1+p2R2+.....+pnRn,

Pn = the probability the return actually will occur in state n,

Rn = the expected return for state.

**The expected return of a portfolio**

,

w

_{i}= share of i asset in the portfolio.

**Asset’s variance (mean-variance)**– is an approach to measuring the investment risk of an asset. It measures a spread of asset’s expected returns based on the historical data.

,

Pn = probability of occurrence,

Rn = return in n occurrence,

E(R) = expected return.

**Two asset’s covariance**– is a statistical measure of how 1 asset moves in relation to another.

.

**Correlation coefficient**– describes how two assets move relatively to each other on the market.

.

Figure 1: Assets class correlations coefficient from 1.0 to -1.0.

__– a sum of each individual variance and cross-asset correlation__

**General Portfolio variance**.

__is a curve of efficient portfolios that represent the best ratio of an Average of Annual Returns (based on historical data) against the Risk measured as a standard deviation of annual returns available. Anything below the curve is inefficient. Anything above the curve is impossible.__

**Efficient Frontier**Figure 2: Markovitz Efficient Frontier.

Source [The efficient frontier].

**The drawbacks of MPT**

Modern Portfolio Theory’s assumptions do not correspond to market reality.

- Asset returns often do not follow a normal distribution,

- Expected Returns and Risks are impossible to calculate accurately based on historical data,

- Investors seek risks and act irrationally,

- Investors do not have access to all the information,

- Psychology plays an important role in the market volatility,

- Etc.

**2.**

**Post-modern portfolio theory**(PMPT) is a model created by Brian M. Rom and Kathleen Ferguson in 1991 to solve certain aspects of practical application of MPT by introducing additional variable called Internal Rate of Return.

**Internal Rate of Return**

**(IRR)**is an annualized effective compounded return rate that being used as a discount rate causes the net present value of future cash flows from an investment to equal zero.

,

NVP - Net Present Value

r – IRR

C0- Initial investment at the start higher or equal to zero

Cn - Cash flow at the period n

n- non negative integer, a period of the cash flow (in years or months)

N- number of periods (in years)

Drawbacks of PMPT

- Many false assumptions similar to MPT.

**3. Black-Litterman Model**was published in 1992, by Fisher Black, Robert Litterman, and Goldman Sachs. It makes MPT more applicable in practice by incorporating investor opinions in the new variable E[R]– the new Combined Return Vector.

**Combined Return Vector Equation**

,

E[R] – the new Combined Return Vector (N x 1 column vector)

τ – a scalar

Σ – a covariance matrix of excess returns (N x N column vector)

P – a matrix which identifies assets involved in the views (K x N matrix or N x 1 row vector in the special case of 1 view)

Ω – a diagonal covariance matrix of error terms from the expressed views which represents the uncertainty of each view (K x K matrix).

Π – the Implied Equilibtium Return Vector (N x 1 column vector)

Q – the View Vector (K x 1 column vector)

The Black-Litterman Model also assumes

**equilibrium**(Π) meaning that the allocation of assets is proportional to the market values of those assets.

**Resources:**

The efficient frontier

Jack Mayer, The theory of risk and risk aversion. Chapter 3.

Ketan Vaischa. What is the difference between covariance and variance

Robert Shiller. Financial Markets, Yale Course

AMP Capital. What is goal based investing?

Asset Dedication. Goal-driven investment approach.