The near universally adopted modern portfolio theory (MPT) put forward by Nobel laureate Harry Markowitz in 1952 is blind to the effect of portfolio investment on the capital markets’ overall risk/return profile and on the macro systems upon which the market relies for stability.

Modern portfolio theory is based on several key concepts, some of which have been recognized with the Nobel Prize in economics. If modern portfolio theory is the thesis, then behavioral finance is the Hegelian antithesis, which has also garnered a couple of Nobel Prizes. The synthesis is post-modern portfolio theory.

#### Mean-Variance Analysis

Put all your eggs in one basket — and watch that basket!

Mark Twain,The Tragedy of Pudd’nhead Wilson

Before modern portfolio theory, investors had a common sense notion of risk and return. The basic idea was to select individual stocks that had the lowest expected risk and the highest expected return. In order to mitigate the risk associated with any single stock, the idea was to diversify the portfolio by selecting a number of these low-risk, high-return stocks.

In practice, this approach might mean identifying railroad stocks as low risk and high return and therefore constructing a portfolio consisting only of railroad stocks. This notion of diversification left something to be desired.

In 1952, Markowitz suggested a different, more principled solution. Markowitz proposed a mathematical theory of diversification whereby an investor did not simply pick individual stocks, but instead fashioned an entire portfolio, according to quantitative principles of reward and risk.

Markowitz’ theory rests on the following assumptions:

- Investors want to maximize returns for a given level of risk. If an investor is given a choice of two assets with equal expected levels of risk, she will choose the asset with the higher expected rate of return.
- Investors are generally risk-averse. If an investor is given the choice of two assets with equal expected rates of return, then risk aversion results in the investor’s selecting the investment with the lower perceived level of risk. This creates a positive relationship between expected return and expected risk.
- Each investment alternative has a probability distribution of expected returns over a given holding period.
- Investors maximize their expected utility for any one period and experience diminishing marginal utility of wealth.
- Risk is measured by the volatility of expected returns.
- Investors base decisions only on expected return and risk.
- If risk is constant, then higher returns are preferred to lower returns. If the returns are constant, then the lower risk is preferred to higher risk.

We view the returns for a security as a random variable, which can have expected values, variances, and correlations. A portfolio is a collection of these random variables, for which we may calculate the cumulative expected return, variance and volatility.

- Expected Portfolio Return (where R is Return, w is the respective weight))

- Portfolio variance (ρ is the correlation)

- Portfolio volatility

- Portfolio return: (two-asset portfolio)

- Portfolio variance: (two-asset portfolio)

- Portfolio variance: (three-asset portfolio)

**Inputs to Portfolio Optimization**

- The expected return of each asset
- The standard deviation of the returns of each asset
- The covariance (or correlation) in the movements of returns for every pair of assets in the portfolio.
- The percentage of the total portfolio’s value invested in each asset.

**The Markowitz Efficient Frontier**

The concept of Efficient Frontier was also introduced by Markowitz and is easier to understand than it sounds. It is a graphical representation of all the possible mixtures of risky assets for an optimal level of Return given any level of Risk, as measured by standard deviation.

The chart above shows a hyperbola showing all the outcomes for various portfolio combinations of risky assets, where standard deviation is plotted on the X-axis and Return is plotted on the Y-axis.

The straight line (Capital Allocation Line) represents a portfolio of all risky assets and the risk-free asset, which is usually a triple-A rated government bond.

Tangency portfolio is the point where the portfolio of only risky assets meets the combination of risky and risk-free assets. This portfolio maximizes return for the given level of risk.

Portfolio along the lower part of the hyperbole will have a lower return and eventually higher risk. Portfolios to the right will have higher returns but also higher risk.

**Sharpe Ratio**

Nobel Laureate William F. Sharpe has derived a formula that helps to measure the risk-adjusted performance. As per definition, Sharpe Ratio helps in arriving at an answer which helps us analyze the risk that can be, and allowing you to make decisions on investments and also helps analyze the performance of a group.

Returns: Returns could be measured in any frequency, like daily, weekly or annually as long as the distribution of returns is normal. The major flaw of the formula lies here in the returns which can never be ascertained as normal but are indeed fluctuating.

Risk-free security: Risk-free security used ought to go with the period of the investment it is being compared against. You have to deduct the amount of returns from the risk-free returns so that it can be divided by the standard deviation of the returns.

Standard Deviation: after calculating the excess returns by deducting risk-free security and returns, we need to divide the answer with standard deviation to arrive at the answer.

**An example to explain the Sharpe Ratio**

Say there are two workers A and B. If A generates returns worth 30% and worker B generates returns worth 25% by looking at the returns you would analyze that A is a better worker. But if worker B has a manager to take lesser risks in attaining the returns while worker A has taken major risks, the returns would be calculated otherwise.

Risk free rate is ascertained as 5% and if worker A’s standard deviation is about 7% and worker B’s is 5% then the Sharpe Ratio calculated for worker A would be 3.5 while for worker B it would be 4, which shows that worker B is a better worker and has managed a better risk-adjusted return. The higher the value of the Sharpe ratio, the better it is for the business.

Although the Sharpe ratio is quite simple to use, which is one the reasons why it is so popular, there can be certain flaws that may not always be overlooked. Based on three major components, the returns, the risk-free returns and standard deviation the Sharpe Ratio is an easy method to find out the additional returns you can get by holding a risky asset over a risk-free asset. The concept is higher the better.

#### Asset-liability Management

Elegant solutions aren’t always ideal, and by neglecting one side of an individual or pension plan’s balance sheet, MPT is only looking at half the problem.

Hence the recent focus on liability-driven investment (LDI) strategies, otherwise known as asset-liability management (ALM). More complete and holistic than MPT, LDI explicitly includes an investor’s current and future liabilities.

LDI requires more expertise in fixed income than the traditional approach, as the liabilities could be considered fixed-income instruments. This may be intimidating for pension fund trustees and advisers. After all, fixed-income investment and analysis can often seem counterintuitive: Many government securities yield negative returns, and most investment-grade bonds are expected to lag inflation, implying an expected loss of value. The need for rising equity markets is far easier to understand than fluctuating bond yields.

Nevertheless, there is a growing consensus in the wholesale capital markets that LDI creates better portfolios, particularly when it comes to retirement needs. I believe LDI should filter down to the retail market, both through LDI-style passive investing and active multi-asset class funds and through adviser and user education.

LDI requires accurate cash flows. A defined-benefit (DB) pension scheme is a classic example. But similar LDI frameworks are also being used for defined-contribution (DC) plans, where cash-flow needs are implicit rather than explicit.

The idea is that rational investors would not look to maximize real assets, subject to risk constraints — usually defined as volatility with total portfolio value. Rather they’d seek to maximize the economic surplus — subject to constraints on the variability of the surplus. This makes a difference because the volatility of the present value of the liabilities is related to the volatility of the funding status. It also shifts the focus of investors from equity prices to long-term treasury or government yields.

For investors with similar objectives, an LDI framework is a useful risk management tool. A better understanding of ALM will lead to enhanced risk-adjusted outcomes for more investors.

The popular MPT framework of expected value optimization given a risk constraint is ripe for disruption. Digital asset management or robo-advice can help distribute LDI technology to the mass market, and we can expect the industry to move in this direction.