The APT model in investment management

Executive summary

The Capital Asset Pricing Model (CAPM) is considered a pivotal model in the computation of investment risk and the expected return on the investment. CAPM provides a way of ascertaining the expected return for stocks and estimating the required return. The single-index model (SIM) also aids in measuring the return and risk of a stock. It assumes that there is only one macroeconomic factor that brings about systematic risk influencing all stock returns. The APT model proposes that the return on financial security has a linear relationship with H systematic risk factors. The assertion made is that investors want to be given compensation for all of the risk factors that have a systematic impact on a security return. The Fama-French (FF) three-factor model divides the fundamental factors into three factors comprising the value factor, market factor, and scale factor for a more improved expounding influence of excess return. The model’s rationale is that firms with high value and small-capitalization repeatedly outperform the overall market. The Black-Scholes formula expresses the current value of a European call option on a stock that does not pay any dividends before the option's expiration. As computed, the call option is 11.06, whereas the put option is 3.93. As computed, the Sharpe ratio is 0.68, the Treynor Measure is 11.97, Jensen's Alpha is 2.05, and the information ratio is 0.098.

Table of contents

Introduction 4

CAPM and extensions 4

APT and extensions 6

Economic indicators and the business cycle 8

Option pricing 8

Forward and Futures pricing 10

Performance evaluation models 11

Recommendations 14

Conclusions 14

References 15

Introduction

Investment security analysis encompasses the valuation of certain securities that might be incorporated into the portfolio. This report conducts an extensive examination of investment and portfolio analysis. Specifically, the report will examine the capital asset pricing model, the single-index model, the arbitrage pricing theory, and the Fama-French three-factor models. Secondly, the report will discuss the Black-Scholes formula and compute call and put options. Also, considering the suitable performance measure relies on the portfolio's role to be assessed, the report will delve into the discussion of Sharpe ratio, information ratio, Treynor measure, and Jensen alpha.

CAPM and extensions

Irrespective of the extent to which investments are diversified, there is always some risk level that will be existent. Bearing this in mind, proper investment management necessitates seeking out a return rate that aids in compensating for such a risk. The Capital Asset Pricing Model (CAPM) is a pivotal model in the computation of investment risk and the expected return on the investment. Any investment faces two types of risk, including the systematic risk and the unsystematic risk. On the one hand, systematic risk refers to market risks related to matters affecting the market, such as a financial recession. In contrast, unsystematic risk refers to the specific risk that is linked to individual stocks. CAPM is employed to examine this specific risk and uses the following formula:

In this case:

R? = Expected return to a stock

Rrf = the risk-free rate

Rm = the return to the market

? = Beta of the stock

(Rm - R rf ) = Equity market premium

The single-index model (SIM) is a relatively basic model for financial asset pricing that is largely employed in measuring the return and risk of a stock. In particular, the model for this particular model is delineated as follows:

In this case:

rit = return to a stock i in period t

rf = the risk-free investment return rate, for instance, interest rate from the United States Treasury Bills

rmt = the return to the market portfolio in period t

?i = the Alpha for the stock. This also refers to the abnormal return of the stock

?i = the beta for the stock. This also refers to the responsiveness of the stock to market return

?it = this refers to the unsystematic or diversifiable risk of the stock (Abildtrup et al., 2011).

The rationale underpinning the single-index model is that the return of a stock is impacted by the market beta but has a firm-specific expected value, also termed as the Alpha, in addition to a firm-specific unanticipated component, which is the residual. In simpler terms, this particular model supposes that there is solely one macroeconomic factor that brings about systematic risk influencing all stock returns. It is imperative to note that this one factor can be captured by the market index's return rate, for instance, the S&P 500 (Tarantino, 2010).

By employing the single index model, the return of a stock can be categorized further into its expected excess returns as a result of firm-specific factor, for the most part, also known as its alpha coefficient (?), expected returns as a result of macroeconomic forces influencing the market as a whole, and anticipated microeconomic forces influencing the company (Tarantino, 2010) solely.

There are three key suppositions made by the single-index model, including the following:

1. The majority of firms react similarly to macroeconomic factors, and as a result, have a positive covariance

1. Several firms have greater sensitivity to these macroeconomic factors in comparison to other firms, thereby bringing about a firm-specific variance that is referred to as its beta (?)

1. The existing covariance amongst the different stocks contained within a portfolio can be computed through the multiplication of their market variance and their betas (Tarantino, 2010; Levy, 2011).

An approach of examining diversification through portfolio assets encompasses examining risk and return characteristics in the investment using the CAPM and the SIM models. Notably, in SIM's case, the sole source of correlation between the asset returns is the market portfolio for the significance of asset returns taken into consideration. In contrast, multi-index models make the supposition that there exist numerous sources of systematic risk overseeing the anticipated asset returns (Abildtrup et al., 2011).

APT and extensions

The arbitrage pricing theory (APT) model hypothesizes that an asset's expected return is influenced by a wide range of risk factors, contrasted with market risk as presumed by the CAPM simply. In particular, the APT model asserts that the return on financial security has a linear relationship with H systematic risk factors (Brigham and Ehrhardt, 2013). Nonetheless, the model does not postulate the systematic risk factors, but the assumption made is that there is a linear correlation between asset returns and risk factors. As provided, the APT model gives the assertion that investors want to be given compensation for all of the risk factors that have a systematic impact on the security return. Imperatively, this reimbursement is the summation of the products of systematic risk for every risk factor and the risk premium apportioned to it by the capital market (Fabozzi, 2015).

Supporters of the APT model argue that it comprises numerous significant advantages over the CAPM model. First, the model makes less limiting suppositions regarding the preferences and inclinations of the investor toward risk and return. Notably, the CAPM theory makes the supposition that investors have a trade-off between risk and return only on the foundation of the expected returns and standard deviations of potential investments (Fabozzi, 2015). On the other hand, the APT model necessitates that some relatively inconspicuous limits be positioned on prospective investor utility functions. One more advantage of this model is that there are no suppositions made regarding the distribution of asset returns (Focardi and Fabozzi, 2004).

Furthermore, there is the advantage that is owing to the reason that the APT model is not dependent on the ascertainment of the true market portfolio, the theory is prospectively testable (Focardi and Fabozzi, 2004). In essence, the model assumes that it is not conceivable to have arbitrage. Employing no extra funds and devoid of increasing risk, it is not conceivable for an investor to generate a portfolio to increase return. The APT model offers theoretical backing for an asset pricing model where that are several risk factors. Consequently, models of this kind are considered multifactor risk models and apply to portfolio management (Focardi and Fabozzi, 2004).

The Fama-French (FF) three-factor model was developed by Eugene F. Fama and Kenneth R. French to challenge the CAPM theory comprehensively. In particular, these scholars established that CAPM's beta value could not elucidate the variances of excess return. Consequently, they propositioned a three-factor model that splits the fundamental factors into three factors: the value factor, market factor, and scale factor, for a more improved expounding influence of excess return. To ascertain whether the model applies to stock markets existent in other nations, Fama and French examined the stock returns and pricing factors in various nations. They asserted that the Fama-French (FF) three-factor model is better than the CAPM (Fama and French, 2012).

Scholars and academics employ this particular model as it delineates the returns of stocks or security in terms of three different factors comprising the market risk, how firms with small capitalization outperform firms with large capitalization, and how firms with high book-to-market value outperform firms with low book-to-market value. It is imperative to note that the rationale of the Fama-French (FF) three-factor model is that firms with high value and small-capitalization tend to outperform the overall market repeatedly.

There are updates made to the original FF three-factor model. In recent times, scholars have stretched out this particular model to incorporate other factors comprising quality, low volatility, and momentum (Karp and van Vuuren, 2017). Consequently, Fama and French adapted their original three-factor model to comprise five different factors. On top of the initial three factors, this extension includes the notion of profitability, in the sense that firms reporting greater future earnings have greater returns in the stock market. The additional factor incorporated in the extension is an investment, which suggests that firms directing their profit towards significant growth projects have a likelihood of facing losses in the stock market (Fama and French, 2015).

Economic indicators and the business cycle

For the most part, economic indicators are employed in the prediction of the business cycle. Also, the business cycle impacts portfolio management as it influences the determination of the selection of assets from cyclical and defensive industries. Significantly, an expansion in the business cycle brings about higher interest rates and a surplus of capital, which consequently instigates a decline in investment. On the other hand, a contraction in the business cycle brings about lower interest rates and a deficiency of capital that instigates a rise in investment (Laopodis and Laopodis, 2012). An investor cannot control economic cycles. However, it is conceivable to personalize investment practices concerning the cycles. For example, in stocks for companies in the cyclical industries, there is a high level of volatility and a tendency to go in line with economic trends. On the other hand, in stocks for companies in the defensive industries, there is a low level of volatility and a tendency to outperform the market during a recession in the economic cycle. Also, the stock prices for defensive companies will rise less compared to the stock prices of cyclical companies in the course of an expansionary period because they are less responsive to expansionary cycles (Laopodis and Laopodis, 2012).

Option pricing

The Black-Scholes formula expresses the current value of a European call option on a stock that does not pay any dividends before the option's expiration. The formula for the call option is as follows:

In this case:

C = current value of the call

S = current value of the stock = 84

r = rate of interest = 0.029

t = time remaining to the option’ expiration = 0.5

X – exercise price = 78

Therefore, the call option is 11.06

The formula of the put option is as follows:

Therefore, the put option is 3.93

Forward and Futures pricing

The basis alludes to the variance between the spot price and the futures price. Notably, when a contract comes to maturity, the basis must be equivalent to zero. In this case, any gains or losses incurred on the futures and the commodity's position will fully cancel themselves. That is:

FT – PT = 0.

Nonetheless, before maturity, the futures price to be delivered later may have substantial dissimilarity from the current spot price. When these contracts are held until the maturity date, the hedger does not usually bear any risk. Nonetheless, if the contract faces liquidation before the date of maturity, the hedger has to bear basis risk. This is owing to the reason that the futures price and spot price may not move in perfect lockstep before the delivery date.

In this case, it is provided that the current futures price for gold for delivery ten days from July 1, 2020, is US$1,730.60 per ounce. Also, the prices of gold are indicated below:

Date

Futures Price

July 1

July 2

July 3

July 4

July 5

July 8

July 9

July 10

July 11

July 12

July 15

It is also assumed that one futures contract consists of 100 ounces of gold and that the maintenance margin is 5%, and the initial margin is 10%. Taking this into consideration, the daily mark-to-market settlements for each contract held by the short position is computed as indicated below:

Day

Profit (Loss) per Ounce x 100 Ounces / Contract

Daily proceeds

1,742.15 - 1,730.60 = 11.15

1,700.50 - 1,705.35 = - 4.85

1,683.45 - 1,700.50 = - 17.05

1,690.35 - 1,683.45 = 6.9

1,713.50 - 1,690.35 = 23.15

1,722.50 - 1,713.50 = 19

1,738.25 - 1,722.50 = 15.75

1,733.50 - 1,738.25 = - 4.75

1,710.45 - 1,733.50 = - 23.05

10 (Maturity)

1,730.60 - 1,733.50 = - 2.9

Summation

Performance evaluation models

The fund portfolio is demonstrated as follows:

Fund portfolio

Market

Average return

Beta

Standard deviation

Tracking error (nonsystematic risk)

Risk-free rate

Sharpe ratio

For the most part, investors are usually concerned with the expected excess return that can be generated by supplanting treasury bills with a risky portfolio, in addition to gaining insight regarding the risk that they would subsequently face. Characteristically, aside from T-bills, other investments comprise of risk in return for the opportunity of earning more compared to the low rate of the T-Bill. Mostly, investors partake in the pricing of risky assets for the risk premium to be proportionate or in line with the risk of the expected excess return. This implies that an ideal way of measuring risk is using the standard deviation in excess returns. Bearing this in mind, the Sharpe ratio refers to the reward-to-volatility measure concerning the trade-off between the reward, which is the risk premium, and the risk as explained by the standard deviation:

The Sharpe Ratio is computed using the following formula:

Sharpe Ratio = Risk Premium / Standard Deviation of Excess Return

Fund portfolio = (16% - 1.75%) / 21%

Treynor measure

In numerous instances, as an investor, one might opt for a portfolio or a fund that will comprise a mixture of securities to create the investor's risky general portfolio. When using several managers, the non-systematic risk is significantly diversified away, and as a result, systematic risk ends up being the pertinent measure of risk. The Treynor measure is defined to be the risk-to-reward ratio that divides the expected excess return by systematic risk, which is the beta. The following formula is employed: