## 18 Feb A Look at Risk Premium Evaluation Models

*This article is written under the assumption that the reader is aware of the basic risk premium evaluation models and theories such as the Modern Portfolio Theory and the Capital Asset Pricing Model. This article explains why there was a need for such evaluation mechanisms and why, in some way shape or form, these models were flawed and hence there was and is a need for new mechanisms for evaluating risk premiums.*

## Evolution of models to calculate Risk Premiums

In the realm of corporate finance, investments and valuations–the central pillar of all estimates is the risk premium associated with an asset class. Over the years, there have been many models that have been used to calculate the risk premium, each with its own assumptions and restrictions. First, let us understand why there is a need for a risk premium or for such models as a whole. When compared to the hypothetical risk free investment, each alternate investment asset has a degree of risk and an amount of return, to compensate for that risk, associated to it. There are two major factors that an investor takes into account before making an investment decision, risk aversion and macro-economic perception.

In order to understand the genesis of the models and the underlying set of issues they solve and simultaneously have, let us consider a novice investor. The investor is looking at the US stock market and, for the sake of simplicity, wants to invest only in stocks. The primary question for the investor is which stock(s) to invest in. A logical first step might include collating a list of, say ten, stocks that have had the highest return for the past year in the, investor’s favorite, airline industry.

Now, investing in just one stock would not be a good idea because if the particular airline company performs poorly (drunk pilots flying the plane), that might affect the stock price and the returns can suffer. To tackle this issue, the investor should invest in multiple firms. That way she is spreading the onus of returns and hence the risk across multiple companies in the airline industry. Now consider the scenario where oil prices surge to very high levels and the airlines are forced to increase ticket prices, lowering the demand for air travel. To tackle this possible turn of events, the investor should invest in multiple industries. But what if those industries too are negatively affected by increased oil prices? To answer this question, let’s look at our first model.

## Modern Portfolio Theory

The modern portfolio theory or MPT was developed by **Dr. Harry Markowitz** in 1952 and it laid the foundation of modern portfolio management. MPT suggests that in order to construct an efficient portfolio, the risk and return parameters of the individual equities should not be considered for addition to or removal from the portfolio in silo. Instead, there should be a relative comparison of the security under consideration with the overall portfolio of the investor. He suggested that it is a security’s covariance with the portfolio that determines the incremental risk of the portfolio and hence the incremental returns. This was is what we call diversification to minimize risk.

Here is the mathematical representation of what MPT posits:

The expected return of a portfolio of N stocks is given by:

**E[r] = w _{1} * r_{1} + w_{2} * r_{2} + … + w_{N} * r_{N}**

And the variance of the portfolio is given by:

**Var _{p} = Σw^{2} * σ^{ 2} + Σ w_{i} * w_{j} * σ_{i,j}**

If N is sufficiently large in this case then the first half of the term will tend towards zero. What this implies to our investor is that by investing in multiple stocks one can reduce the some of the risk. This is called the idiosyncratic risk. Otherwise also known as diversifiable, unique or firm-specific risk. The above equation also shows that the second half of the equation does not become small as the number of stocks in the portfolio increase. Thus, diversification cannot eliminate the systematic, or the undiversifiable or the common risk. Our investor is now equipped with the knowledge of diversification. But can he expect to be a successful investor based on just this knowledge. What if everyone was using the same principles of optimization?

To include a few other factors to answer the problem, let us look at few of the assumptions that the MPT makes:

- Returns on an asset are normally distributed
- All investors are rational in that they tend to minimize risk
- The MPT assumes a frictionless market i.e. there are no transactions costs associated with buying and selling of securities
- It also assumes a perfect market where all investors have all the same information required to make investment decisions
- There is unlimited supply of both funds to buy securities and that of the securities itself

These assumptions fail in the real world. There are always transaction costs, fees and taxes, associated with any trade. There is not a perfect distribution of information in the market. There definitely is not an unlimited supply of funds and securities in the market. To further this, consider a scenario where our investor decides that he wants to buy more of Alaska Airlines shares and there is not enough sellers in the market. And then there a few others with a similar need. This bumps up the share price and the interested parties have to change their estimates of return and variance (risk) because the weight of the individual security in the portfolio changes.

The fluctuation in price continues till it an equilibrium point is reached and there is no surplus demand or supply in the market. What can be inferred about the relationship between risk and return? Let us take a look at our next model for that.

## Capital Asset Pricing Model (CAPM: Birth of the Beta)

The capital asset pricing model or the CAPM describes the relationship between expected returns and the systematic risk and is given by:

**E[r] = R _{f} + β_{i} * (E[r_{m}] – R_{f})**

Where

R_{f} is the risk-free rate of return

β_{i} is the beta value of the security_{
}E[r_{m}] is the average return on the reference market

E[r_{m}] – R_{f} is the excess return over the risk free asset

The beta value of the security is its sensitivity relative to the market portfolio and can be calculated by running a regression analysis. Thus, a beta of 1 would imply that the security’s price will move with that of the market, less than one would mean it (security) will be less volatile and greater than 1 means that it’ll be more volatile than the market.

The CAPM makes a few assumptions as well; some of them are:

- There is an infinite supply of risk free asset and the investor can lend and borrow at will
- Investors are diversified. This means that the investors are seeking a return only for the systematic risk as they have nullified the specific risk by diversification
- The capital markets are perfect

Our investor’s expertise is now upgraded to consider market swings as well whilst selecting a particular security or stock to invest in. But is this still reliable? CAPM assumes a well-functioning market and thus there are certain risk factors that it does not account for, stock market irregularities for instance. Another example is the irrational behavior heavily influences the supply and demand of a security and thus, its price. Moreover, not all investors are diversified and thus, they are holding a sub-optimal portfolio.

The MPT and CAPM both suffer from the same major drawback, they are backward looking. Considering that we always try to project the expected return for an asset this seems to be counter intuitive. In such a case, it makes sense to consider models (to calculate equity risk premiums) that are forward looking.

Another key logical gap in these models is the survivorship bias. The US market was the most successful market of the 20^{th} century. When these models calculate the risk premium on the data based on this market, there is an assumption made that this trend is sure to continue, which might or might not be the case. In addition, since the models give the statistical mean of the variables, they are pretty static and might not be the best representations of the current economic activity.

Considering these two factors, we look at our final model.

## Implied Equity Risk Premium

Since past data is not the best measure of future performance, we try to look forward and estimate the returns that our investor can expect over a period of time. We take the simple dividend discount model and apply it to our case. The DDM states that the current value of a stock is the present value of the dividends (in perpetuity). Mathematically it looks like:

Present Value of Stock = Dividend_{t+1} / r

Where ‘r’ is the discount rate

If we consider a growth firm:

Present Value of Stock = Dividend_{t+1} / (r – g)

Where ‘g’ is the rate at which the firm is expected to grow

Given an assumption of the firm’s growth rate, and the knowledge of the current share price and the expected future dividend payouts on the basis of previous year, our investor can calculate the expected return on the stock.

There are certain explicit benefits to this model:

- It is forward looking. Irrespective of a firms or a market’s performance in the past this model tries to objectively look at the future performance to gauge returns
- It is dynamic. Since we are not averaging out data collected for a period of many years, this model presents a dynamic view day after day of the expected returns
- The calculations can be tailored. In case we have companies in different phases of their growth we can modify our assumption of the growth rate of the company to come to a number that is best for the company. For instance, for any blue chip company we might want to consider a conservative growth rate (perhaps equal to the GDP) and for a startup we can assume a high growth rate.

Is this now the best way forward for our investor?

## In conclusion

Over the years there have been many models that have tried to accurately establish and predict the risk & return relationship. There have been multiple evolutionary models that sprang out of the MPT and CAPM that take factors such as taxes, liquidity, and dividend yield into account to try and present a more accurate picture. Since the market is an overwhelmingly dynamic mixture of elements it is difficult to do so. Add to it the fluctuations of the human nature, impulse trades and gut-feel investing and the task of presenting a coherent picture only gets harder. On the flipside, if successful investing relied only on one single formula everyone would be an active investor but as they say there is no such thing as free lunch!