Question

How to split portfolio into systematic and unsystematic risk? What is the formula for ri?

How is rp=rf +portfolio beta(r m-rf) + e derived?

Answer 1

Portfolio risk is the possibility that an portfolio may not achieve its objectives. There are a number of factors that contribute to portfolio risk, and while you are able to minimize them, you will never be able to fully eliminate them.

there are two type of risk  systematic and unsystematic risk.

Systematic Risk

This is the risk that highlights the possibility of a collapse of the entire financial system or the stock market causing a catastrophic impact on the entire system in the country. It refers to the risks caused by financial system instability, potentially catastrophic or idiosyncratic events to the interlinkages and other interdependencies in the overall market. Systematic risks are difficult to be mitigated since these are inherent in nature and not controlled by an individual or a group. There is no well-defined method for handling such risks but as an investor,

The sources of systematic risks can be:

• Political instability or other Governmental decision having widespread impact
• Economic crashes and Recession
• Changes in taxation laws
• Natural Disasters
• Foreign Investment Policies

The systematic risk of a security or a portfolio of securities is measured by its Beta (β). Beta measures the comovement of the security’s (or portfolio’s) return with the market. The mathematical formula for beta is as follows:

where Covariance(i, M) is the covariance of the security’s returns with the market returns and VarianceM  is a measure of the volatility of the market in general.

Unsystematic Risk

Also known as Diversifiable or Non-systematic risk, it is the threat related to a specific portfolio of securities. Investors construct these diversified portfolios for allocating risks over various classes of assets.These are risks which are existing but are unplanned and can occur at any point in causing widespread disruption

Some of the other examples of unsystematic risks are:

• Change in regulations impacting one industry
• The entry of a new competitor in the market
• A firm forced to recall one of its products (For e.g. the Galaxy Note 7 phone recalled by Samsung due to its battery turning flammable)
• A company exposed to have made fraudulent activities with its financial statements (For instance Satyam computers fudging their balance sheets)
• An employee union tactic for senior management to meet their demand

What is the formula for ri?

E(Ri) = Rf + ßi * (E(Rm) – Rf)

Or = Rf + ßi * (risk premium)

Where

E(Ri) = the expected return on asset given its beta

Rf = the risk-free rate of return

E(Rm) = the expected return on the market portfolio

ßi = the asset’s sensitivity to returns on the market portfolio

E(Rm) – Rf = market risk premium, the expected return on the market minus the risk free rate.

How is rp=rf +portfolio beta(r m-rf) + e derived?

The asset return depends on the amount paid for the asset today. The price paid must ensure that the market portfolio's risk / return characteristics improve when the asset is added to it. The CAPM is a model that derives the theoretical required expected return (i.e., discount rate) for an asset in a market, given the risk-free rate available to investors and the risk of the market as a whole. The CAPM is usually expressed:

{\displaystyle \operatorname {E} (R_{i})=R_{f}+\beta _{i}(\operatorname {E} (R_{m})-R_{f})}

• β, Beta, is the measure of asset sensitivity to a movement in the overall market; Beta is usually found via regression on historical data. Betas exceeding one signify more than average "riskiness" in the sense of the asset's contribution to overall portfolio risk; betas below one indicate a lower than average risk contribution.
• {\displaystyle (\operatorname {E} (R_{m})-R_{f})} is the market premium, the expected excess return of the market portfolio's expected return over the risk-free rate.

The derivation is as follows:

(1) The incremental impact on risk and expected return when an additional risky asset, a, is added to the market portfolio, m, follows from the formulae for a two-asset portfolio. These results are used to derive the asset-appropriate discount rate.

• Updated market portfolio's risk = {\displaystyle (w_{m}^{2}\sigma _{m}^{2}+[w_{a}^{2}\sigma _{a}^{2}+2w_{m}w_{a}\rho _{am}\sigma _{a}\sigma _{m}])}

Hence, risk added to portfolio = {\displaystyle [w_{a}^{2}\sigma _{a}^{2}+2w_{m}w_{a}\rho _{am}\sigma _{a}\sigma _{m}]}

but since the weight of the asset will be relatively low, {\displaystyle w_{a}^{2}\approx 0}

i.e. additional risk = {\displaystyle [2w_{m}w_{a}\rho _{am}\sigma _{a}\sigma _{m}]\quad }

• Market portfolio's expected return = {\displaystyle (w_{m}\operatorname {E} (R_{m})+[w_{a}\operatorname {E} (R_{a})])}

Hence additional expected return = {\displaystyle [w_{a}\operatorname {E} (R_{a})]}

(2) If an asset, a, is correctly priced, the improvement in its risk-to-expected return ratio achieved by adding it to the market portfolio, m, will at least match the gains of spending that money on an increased stake in the market portfolio. The assumption is that the investor will purchase the asset with funds borrowed at the risk-free rate, {\displaystyle R_{f}}; this is rational if {\displaystyle \operatorname {E} (R_{a})>R_{f}}.

Thus: {\displaystyle [w_{a}(\operatorname {E} (R_{a})-R_{f})]/[2w_{m}w_{a}\rho _{am}\sigma _{a}\sigma _{m}]=[w_{a}(\operatorname {E} (R_{m})-R_{f})]/[2w_{m}w_{a}\sigma _{m}\sigma _{m}]}

i.e. : {\displaystyle [\operatorname {E} (R_{a})]=R_{f}+[\operatorname {E} (R_{m})-R_{f}]*[\rho _{am}\sigma _{a}\sigma _{m}]/[\sigma _{m}\sigma _{m}]}

i.e. : {\displaystyle [\operatorname {E} (R_{a})]=R_{f}+[\operatorname {E} (R_{m})-R_{f}]*[\sigma _{am}]/[\sigma _{mm}]}

{\displaystyle [\sigma _{am}]/[\sigma _{mm}]\quad } is the "beta", {\displaystyle \beta } return— the covariance between the asset's return and the market's return divided by the variance of the market return— i.e. the sensitivity of the asset price to movement in the market portfolio's value.

This equation can be estimated statistically using the following regression equation:

{\displaystyle \mathrm {SCL} :R_{i,t}-R_{f}=\alpha _{i}+\beta _{i}\,(R_{M,t}-R_{f})+\epsilon _{i,t}{\frac {}{}}}

where αi is called the asset's alpha, βi is the asset's beta coefficient and SCL is the security characteristic line.

Once an asset's expected return, {\displaystyle E(R_{i})}, is calculated using CAPM, the future cash flows of the asset can be discounted to their present value using this rate to establish the correct price for the asset. A riskier stock will have a higher beta and will be discounted at a higher rate; less sensitive stocks will have lower betas and be discounted at a lower rate. In theory, an asset is correctly priced when its observed price is the same as its value calculated using the CAPM derived discount rate. If the observed price is higher than the valuation, then the asset is overvalued; it is undervalued for a too low price.