## Portfolio Analysis & Risk Management

**Notes on
the Theory and Mechanics of Building Optimal Portfolios**

**Introduction**

Published research indicates that of the three broad types
of investment strategies the **asset
allocation** process, that is the apportionment of funds across asset
classes, accounts for roughly 92 to 95 percent of the variability in portfolio
returns. **Security selection**, that is
which securities/asset classes to include and not include in the portfolio, and
**market timing,**, that is when to buy
and sell securities, account for only 3 – 8 percent of portfolio returns.[1]

In particular read https://blogs.cfainstitute.org/investor/2012/02/16/setting-the-record-straight-on-asset-allocation/

Given the overwhelming importance of the asset allocation process in generating returns it is well worth our time to learn the asset allocation process of building portfolios.

These notes apply to both ‘Strategic’ as well as ‘Tactical’ asset allocations (recall what we mean by strategic and tactical asset allocation). The objective is to produce weights (or ranges of weights) for the apportionment of the investors’ funds among the various markets/asset classes.

The following are “steps” as in a cookbook to building an optimal portfolio. Some are highly technical in nature. We want a good conceptual understanding of the portfolio building – asset allocation process as well as an understanding of the mechanics involved in building portfolios for the case of two risky asset classes. Beyond two risky asset classes the mechanics become very technical and we leave it to programmers to write the software and algorithms to solve for the optimal weights.

The concept is *straightforward*; namely to match the investor’s return – risk
tradeoffs which are *‘subjective’* with
the *‘objective’* return – risk
tradeoffs based on expected returns, risks and return correlations among the
candidate asset classes.

The subjective (personal) tradeoffs to the investor are: (a) degree of risk tolerance; (b) return objective(s); (c) terminal wealth/value of fund; (d) internal constraints. The objective return-risk tradeoffs are: (a) market returns; (b) market risks, skewness, kurtosis, etc.; (c) return co-variances, correlations, co-integration,

What I want each of you to
understand is the *logic* in the steps
of building a portfolio and the *mechanics
*for the case where we have only two risky asset classes.

**Foundations**

Conceptually, the process of building portfolios for
investors is grounded in micro economic theory. Economic theory provides a
framework for maximizing consumer satisfaction/utility subject to budget
constraints. *Modern* *Portfolio Theory (MPT) provides a framework
for maximizing investor returns/utility subject to return-risk and any other
constraints across asset classes*.

*In essence, a good
portfolio is one where the subjective return-risk tradeoffs of the investor are
matched with the objective return-risk tradeoffs across asset classes as
determined by market forces. *

The approach is a ‘top-down’ one. The goal is to build a
portfolio containing the weights (or proportions of each asset class) that are
the best (optimal) for the investor. The best portfolio is one where the
optimal weights represent *objective* market
return-risk characteristics that are the same as the *subjective* return – risk characteristics of the investor.

**The Recipe**

**Ingredients**- One investor (with monies to invest ) – can be either individual or institutional.
- Team of ethical, hardworking, knowledgeable, dedicated , eager, smart investment professionals with a range of state-of-the art skills.
- State of the art computer hardware and software for understanding client needs, analyses of data and for solving dynamic optimization problems.
- Information and data on a range of securities markets/asset classes. Ideally, the data should be highest frequency possible.

**Cooking Directions**

**Step 1: Know thy client!**

Specify Investor Objectives.

This includes objectives for the Purpose(s) and Goals of the Portfolio, Investment Horizon, Return Requirements, Risk Tolerance, any Internal or External Constraints on Designing, Implementing and Managing the Portfolio, the pursuit of Active, Passive or Mixed Investment Strategies, and portfolio Governance issues.

** Step 1A. A Note and Digression on
Investor Utility Functions**

Importantly, we assume that it is not the dollar amount of investor terminal value or wealth that drives investment decisions but, instead, the “want fulfilling satisfaction” that the investor receives from wealth. This assumption allows us to recognize the subjectivity of value; the notion that investors place different degrees of subjective value on the same amount of wealth. Simply put, investor A may place a different value on having a $X portfolio than investor B.

We do make the critical assumption that all investors are subject to the “Law of Diminishing Marginal Utility of Wealth.” How, then, do we represent the want satisfying power investors receive from wealth? We do this via “utility functions.” These functions just relate the utility of an investor to their wealth from investing. What are some common forms for these utility functions?

- Log utility: U = ln(W). This function just says that an investor’s utility is equal to the natural log of his/her/its wealth.

- Reciprocal: U = -(1/W). This function just says that the investor’s utility is equal to the negative of the reciprocal of wealth.

- Root: U = (W)
^{1/2}. This function just says that the investor’s utility is equal to the square root of wealth.

**U = E(R) – .5A****s**This utility function simply says that investor utility: (1) increases with expected return; (2) decreases with risk (NOTE THE NEGATIVE SIGN BEFORE THE .5 TERM). It is known as a ‘constant relative risk aversion’ (CRRA) utility function.^{2}- The “A” term is the investor’s subjective degree of risk tolerance. 1 < A < 10.
- Neuroscience is helping us to better understand the factors behind a client’s “A” and the changes (at times sudden and sharp) in it. See the following link for one such article: http://rstb.royalsocietypublishing.org/content/365/1538/331.short
**From the paper Abstract: “acutely elevated steroids may optimize performance on a range of tasks; but chronically elevated steroids may promote irrational risk-reward choices. We present a hypothesis suggesting that the irrational exuberance and pessimism observed during market bubbles and crashes may be mediated by steroid hormones. If hormones can exaggerate market moves, then perhaps the age and sex composition among traders and asset managers may affect the level of instability witnessed in the financial markets.”**- s
^{2}is the variance/risk. We could (AND SHOULD) also use the semi-variance. - .5 is a scaling term.
- The above utility function is the one used in
the Bodie, Kane, and Marcus
*Investments*text.

*All of the Above Items Should Be Embedded in the Client’s
Investment Policy Statement**.*

**Step 2. Identify
and evaluate asset classes/securities for possible inclusion in the client’s portfolio.
**

- Identification based on: (1) Investor Objectives: (2) Analysis of Markets/Asset Classes; (3) Capital Market Expectations; and (4) Portfolio Manager’s Insights
- ‘Sharpe’ ratio based formula for potentially adding an asset class.
- Add asset class if: E(Rnew –Rf)/snew > {E(Rexist – Rf)/sexist.}Corr(Rnew,Rexist.) How much of the ‘new’ asset class to add needs to be determined.
- The index for each asset class also serves in building a benchmark for performance evaluation.
- Evaluation based on estimation and analysis of historic geometric and arithmetic return measures, several risk measures including downside ones, measures of the distribution of returns such as existence of skewness and leptokurtosis, and covariances/correlations of returns (Assignments 3 and 5!)

**Step 3.Form Capital Market
Expectations (use the annual data)**

- For each asset class/market/security estimate
a Capital Market Expectation for: (1) Expected Return; (2) Risk
**s**; (3) Co-variances/Correlations of Return with every other asset class/market/security. - These data, based on the analysis from Step 2, will be used to construct Expected Returns and Risks for a large number of portfolios. One of these portfolios should be the best one for the investor.

**Step 3A. Allocation
of Portfolio between Risk – Free and Risky Assets. (A slight digression)**

- Using data on: (1) Risk-Free Return; (2) Return on Risky Assets as a Group/Class; (3) Risk/ Variance and Standard Deviation (or Semi Variance and Semi Standard Deviation) of Risky Asset Return as a Group/Asset Class;
- Generate
**Capital Allocation Line**showing Expected Returns and Risks as Weights between Risk – Free and Risky Assets are varied : - Formula Are:

E(R_{c}) = R_{F } +
W_{R}[(E(R_{R}) – R_{f})]

s_{c} = W_{R }x sR

- Substitute into CRRA Utility Function and Solve for Optimal Allocation

W^{* = }[(E(R_{R})
– R_{f})]/ A x s^{2}_{R}

W* is the percent of the client’s portfolio invested in Risky assets; (1- W*) is invested in Risk Free assets.

**Step 4. Generate the
Investment (Portfolio) Opportunity Set.**

- Use only the risky asset classes.
- For each Risky Asset Classes we need: (1) an Expected Return; (2) Measure(s) of Risk; (3) Covariance return matrix or Pairwise Correlations of Returns with every other asset class.
- The Expected Returns are either the arithmetic annual average returns (why arithmetic and not geometric?); derived from the basic formula; or based on Valuation and Multiple- such as P/E formula.

- E(R
_{Asset Class}) = Risk Free Return + Risk Premia (factor model based). - E(R
_{Asset Class) }= E(Cash Flow, e.g. FCF or EPS) x Market Multiple. - E(R
_{Asset Class) }= Earnings Yield = Earnings/Price = Inverse of P/E. - Measures of Risk (standard deviation, semi-standard deviation, etc. derived from historic data assuming returns are normally distributed (why?)
- Covariance/Correlation matrices derived from return data for each asset class.
- See http://mathworld.wolfram.com/Covariance.html for discussion of covariance/correlation matrices.

**Step 4A**. **Use Fundamental Equations 1 and 2 and the
Methods below**.

FE 1: Expected Return on a Portfolio.

E(R_{p})
= S_{i=1}^{N }W_{i } * E(R_{i})

FE 1 simply states that the expected return on a portfolio is the sum of the expected return on each security in the portfolio multiplied by its weight – where the weight for each security is its percent of the total value of the portfolio.

FE 2: Portfolio Risk (for the case where there are only 2 securities in the portfolio)

Version 1: s_{p}^{2 }= W_{i}^{2} s^{2}_{i
}+W_{j}^{2} s_{j}^{2 } + 2W_{i}W_{j}
Cov_{ri ,rj}

Recalling
r_{ri,
rj } = Cov/si sj;
and -1<r<
+1

Version 2: s_{p}^{2 }= W_{i}^{2} s^{2}_{i
}+W_{j}^{2} s_{j}^{2 } + 2W_{i}W_{j}
s_{i}s_{j}r_{rirj}

Version 2 of Fundamental Equation 2 simply states that the risk of a portfolio depends on four items:

- The weights given to each security represented by the W (weight) terms.
- The
riskiness of each security in the portfolio given by the s
^{2}variance terms. - The riskiness of each security in the portfolio given by the s standard deviation terms.
- The
covariances/correlations of returns given by the Cov/s
^{2}terms._{}

Note the following:

- There are “N” weight terms where N is the number of asset classes/securities in the portfolio.
- There are “N” variance and standard deviation terms, one for each asset class/security.
- There are N(N-1) covariance/correlation terms.

The point is the covariance/correlations of returns is the most important determinant of portfolio risk!

- Method
1: Vary the weights for each asset class/security (make sure the weights sum to
100% – even where there are short sales with negative weights these will be
offset by long positions with higher weights). Use Fundamental Equations (1)
and (2) to generate Expected Returns and Risks for “many” different portfolios.
That is, each time the weights are changed a “new” portfolio with a different
expected return and risk is generated.
- Method 2: Specify a target expected return for the portfolio. Determine the weight for each asset class/security that provides the target return with the lowest variance/standard deviation. Repeat this process many times.

- Variation on Method 2 – Method 3: Specify a target risk – standard deviation for the portfolio. Determine the weight for each asset class/security that provides the target risk with the highest expected return. Repeat this process many times.

**Step 6. Determine the Efficient Frontier/Minimum
Variance Frontier (see charts)**

- Step 5 provides a large scatter plot of portfolios, each with different weights and different expected returns and risks.
- Eliminate the inefficient portfolio, that is, portfolios dominated in either return or risk by another/other portfolios.
- If either Method 2 or the Variation on Method 2
are used then via the step above the Minimum Variance Frontier is determined.
*The Global MV Portfolio is the one with the smallest variance/standard deviation.* - Eliminate so-called inefficient portfolios. These are portfolios where, for the same risk, they have a lower return, or for the same return they have higher risk.
- After all the inefficient portfolios have been
eliminated we are left with the
*Efficient Frontier of Portfolios*. One of these portfolios should be the best risky portfolio for our client.

**Step 7. Determine
the Best Risky Portfolio from all the Risky Portfolios on the Efficient
Frontier. Case of No Risk-Free Asset Class.**

- The solution for the optimal weights for each asset class is a standard constrained maximization problem in calculus (formulas). When there are more than 2-3 asset classes/securities the mathematics becomes very laborious, and thank goodness for writing the algorithms and the software.
- For the case of only two (2) asset classes the
formula are derived and presented on pages 201 – 206 of your text
*Investments,*11^{th }editionby Bodie, Kane and Marcus. - The formulas above give the optimal or best weights for each asset class of risky assets. Use these weights, substitute into Fundamental Equations (1) and (2), to determine the Expected Return and Risk of the best portfolio.

**Step 8. Determine
Best Risky Portfolio and Allocation to Risk Free. Case of Risk-Free.**

- If there is a risk free asset class, such as T-Bills, into which some funds will be invested then to determine the optimal allocation between risky and risk free assets:
- The optimal risky portfolio has the highest return-to-risk/Sharpe ratio.
- We seek to maximize the return – to – risk ratio subject to the constraint that the sum of all the weights must be 100%.
- Maximize S
_{p}= E(R_{pi}– R_{f})/ s_{p}

Subject to: S_{wi }= 1.

- Now, allocate the fund between Risky and
Risk-Free Assets using the above and the
formula from Step 4.: W
^{* = }[(E(R_{R}) – R_{f})]/ A x s^{2}_{R}

W* = weight of entire portfolio in the optimal risky portfolio

R_{p} = expected return
on optimal risky portfolio.

R_{f} = risk free rate of
return.

A = client coefficient of risk aversion.

s^{2} = variance of return on optimal
risky portfolio.

**Step 9. Build the Portfolio
and Monitor Performance**

- Execute the Trades Necessary to Construct the Portfolio.
- Monitor the Portfolio Performance on a Regular and Frequent Basis to evaluate whether or not it is meeting Investor Objectives from Step 1, and to determine the sources of its Returns and Risk. Is performance due to Market Events, Manager Decisions, Luck?

**Step 10. Repeat steps 1-9 as warranted**.

- Changes in client objectives.
- Changes in number and types of asset classes/securities.
- Changes in Expected Returns, Risks, Correlations, Capital Market Expectations, etc.

**Some additional issues to consider**

- The process of building portfolios is the same regardless of the client! This is known as the Separation Property.
- Results are highly(?) sensitive to CME of E(Rs). Small changes in E(Rs) may lead to large changes in optimal asset class weights.
- Roy Safety First Criterion: A form of “Shortfall Risk.”
- SF Ratio = E(R
_{p}– R_{L})/s_{p}, where R_{L }minimum/threshold level of return.

- Black Theorem as a Shortcut: “weights of any MVP are a simple linear combination of weights of any two other MVPs.” If we find optimal weights for any 2 portfolios, then optimal weights for ‘N’ portfolios can be easily found.
- Corner Portfolios: Part of the Efficient Frontier where portfolios hold the same asset classes and the asset weights among portfolios change at the same rate. Use to obtain optimal weights for E(Rs) between adjacent corner portfolios. Highest E(R) asset class is usually the first corner portfolio; GMV portfolio is the last.

[1] See Gary Brinson, Brian Singer, and Gilbert
Bebower, “Determinants of Portfolio Performance.” *Financial Analyst Journal, *May/June 1991.