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Arbitrage Pricing Theory and Fama and French Model

Portfolio management entails the creation and management of a variety of assets with the objective of achieving a long-term goal for an investor depending on their risk appetite. The goal of this paper is to emphasize the relevance of several portfolio theories such as the Arbitrage Pricing Model and the Fama-French three component model, as well as the differences between them. The second section of the paper discusses economic indicators that are used to forecast an industry's business cycle. The banking and finance industry in the United States has been chosen as the industry to study the economic cycle and forecast the industry's performance over the next five years. Black Scholes model is an option pricing model which is utilized by investors to price options. The Black-Scholes model is used to price call and put options in the third section of the report. The Black Scholes Model's assumptions and limitations in forecasting the price of call and put options have been thoroughly studied. The report's last section covers and emphasises several portfolio performance evaluation metrics such as the Sharpe ratio, Treynor ratio, Jensen's alpha, and information ratio. All of the above-mentioned assessment approaches' calculations have been given, together with their consequences for the AlphaWell fund.

Asset pricing models are an essential part of investment management as it allows an investor to assess the true worth of an asset. This information would assist the investor in deciding an optimal asset allocation as it gives an idea to the investor regarding the risk-return trade off the investments. There are several asset pricings models prevalent in the financial world including the Arbitrage Pricing Theory and the Fama-French model. Usage and benefits of the two of the models are discussed in detail below:

It's a multi-factor asset pricing model that assumes asset returns can be predicted using a range of economic variables. The theory also believes that markets are efficient, making it difficult to make above-average profits in the market. As a result, a linear relationship between the asset and several macroeconomic parameters that reflect systematic risk can be used to estimate an asset's return. The following formula represents the arbitrage pricing theory:

Systematic risk is one of the APT's elements, and it cannot be mitigated by diversifying the portfolio across asset classes. Gross National Product (GNP), corporate bond spreads, and yield curve volatilities have historically been the most profitable macroeconomic variables in predicting asset returns.

The APT is increasingly popular among investors in the financial industry because it evaluates an asset's necessary rate of return by attributing various elements to it. The APT focuses on risk variables that affect asset returns rather than asset attributes, making the model more accurate than single factor models like the CAPM. The Arbitrage Pricing Theory has a number of flaws, one of which is that it presupposes a linear link between macroeconomic variables and asset returns. As a result, the model would be useless for adding variables with non-linear characteristics and predicting outcomes. The model has shown to be useful in predicting returns over the medium to long term, but not so much in the near term.

Arbitrage Pricing Theory

The Fama-French model is also known as three-factor model as it expands the Capital Asset Pricing Model by adding two additional risk factors apart from systematic risk factor measured by beta. The risk factors that are added to the CAPM model are size risk and value risk which explains the return of an asset or a portfolio more efficiently. The Fama French Model has three primary factors which includes size of the firms, values of book to market of stocks and excess return on the market measured by deducting the risk-free rate by the portfolio’s return. Small minus big (SMB) is the factor which represents the size factor and it represents public limited companies with small market caps generating higher returns than large market cap companies. The factor of High minus low (HML) represents stocks which are value stocks having high book to market ratios and which generates returns which are higher than the overall markets. The model assumes that most of the movements in an asset’s return can be attributed to the above-mentioned factors and any unexplained expected return may be attributed to unsystematic risk.

Small companies outperform large companies over time, according to the model, while value companies outperform growth companies in the long run. According to the model, 90 percent of a diversified portfolio's performance can be explained using the model and its assumptions. The Fama-French Model is used by investors all over the world because it allows them to underweight or overweight individual elements and target different levels of expected return from their portfolio.

The business cycle is characterised as periodic instability in economic activity, such as the country's employment level and production capacity utilisation. A typical business cycle entails an increase in production and employment activities until they reach a peak, after which they decline until the economy reaches a low point in terms of output and employment. Business cycles can be described as a very complicated and difficult to anticipate occurrence. Traders, economists, investors, and governments use a variety of measurements and indicators to forecast the movement of economic activity and the current stage of the business cycle.

Leading indicators, trailing indicators, and coincident indicators are three types of economic indicators that are supposed to anticipate the peaks and troughs of a business cycle. The following section highlights the importance of all the indicators for predicting the business cycle of a country:

These indicators track economic activity and movements in order to predict the start of a new business cycle. Leading indicators include average weekly hours worked in manufacturing companies, factory orders of items and raw materials, and stock prices on the company's financial markets. These indicators are examined for increases and declines, which aids in determining when a specific type of economic cycle is about to begin. Investors all across the world place a premium on leading indicators because they provide an early indication of the business cycle. Consumer expectations, unemployment compensation claims, and the interest rate spread are among the key leading indicators.

Fama-French Model

After leading indicators have anticipated a certain trend, trailing indicators play a vital role in confirming it throughout the economic cycle. Economic activity that rise or fall in response to the business cycle's trend. Longevity of unemployment, labour costs, CPI levels, commercial loan activities, and other trailing indicators are examples. Various investors regard trailing indicators to be unimportant since they alter in economic variables after the business cycle trend has set established. In evaluating and recognising structural issues in an economy, lagging indicators are frequently beneficial.

Personal level of income, level of industrial production and unemployment rate are some examples of coincident indicators which changes with time as the trend in business cycle of a country picks up. These indicators also help to identify and confirm a trend in business cycle. All of the indications discussed above are critical in determining the future trend of an economy's business cycle. Leading indicators are given greater weight in forecasting business cycles, but trailing indicators are given less weight because their primary function is to confirm a trend.

Following the outbreak of the Covid-19 pandemic in 2020, central governments all around the world engaged in quantitative easing to promote demand and improve liquidity. Interest rates were lowered over the world to encourage more credit lending. As a result of the additional capital available in the US economy, inflation has begun to sneak in, and prices of goods and commodities have begun to rise. In order to counter the inflation levels in the country, US Fed has decided to increase the interest rate to fight the inflation for the future years to come until the inflation stabilizes (Saphir and Dunsmuir 2022). The banking industry would benefit from a rising interest rate environment because banks would be able to borrow at cheaper rates and lend at greater rates. Due to the minimal risk of default, high yield bonds from the banking and financial industry are predicted to perform well (Martin 2022).

The banking industry has regained its lost ground during the pandemic times as world economy have been rising steadily and recovering from the impacts of pandemic. According to (Mckinsey 2021) fifty one percent of the banks in US have return on equity less than the cost of equity of the bank but this data should change as the interest rate environment around the world including US is expected to rise. Banking and financial industry have increased their level of surveillance and prudence of future risks by building up capital and reserves specially after the impact of the Covid 19 pandemic. Stress testing processes adopted by the banks going forward in the future would ensure efficient credit flows in all times. Multiple cyclical variables impact the performance of banks and the financial industry, including supportive and accommodating monetary policy. In the long run, in the lack of a large number of profitable choices, banks and financial institutions may be enticed to take more risks. Due to cyclical causes and strong bank balance sheets, banks and financial institutions may experience lower profitability; as a result, banks and financial institutions must adjust to the new normal. Lower profitability situations may drive company management to take on higher risks and leverage, potentially causing financial system instability ( 2022).

Economic indicators

Black Scholes model is an option pricing model and is of greater importance in the financial world. The model is used to price option expecting it to expire in the money on the basis of certain assumptions (Anwar and Andallah 2018). The idea is premised on the idea that the fundamental stock prices dictate the value of a call option, with the value increasing as the stock price increases and reducing as the stock price falls. The following are the assumptions of the Black Scholes Model:

  • The assumption of the model is that the stock does not pay dividends.
  • The model assumes that markets are random and the prediction of its movement is not possible.
  • Absence of transaction costs while transacting in the option.
  • The primary assumption of the model is that the risk-free rate of return and volatility of the underlying stock are known and constant.
  • The model's fundamental assumption is that the option is European and can only be exercised at the contract's end. The asset's returns are also expected to be dispersed logarithmically (Dar and Anuradha 2018).

The following are the inputs necessary to value an option according to the Black Scholes Merton model:

Share Price (S), Strike Price (K), Expiration Date (t), Interest Rate (r), and Volatility (sigma).

The following is the formula for calculating call option price using the above-mentioned variables:

S * N(d1 ) – Ke(–rt) * N(d2 ) C = S * N(d1 ) – Ke(–rt) * N(d2 )


d1 = [ln (S/K) + (r + 2/2) * t]

d2 = d1 -(sigma* t^0.5) – d1

The D1 and D2 were calculated to be equal to 0.159099026 and -0.088388348 respectively. Using the NORMDIST function of the excel the ND1 and ND2 were calculated to be equal to 0.563204572 and 0.464784011 respectively. The call option calculated using the Black Scholes model was equal to $7.69.

Using the NORMDIST function of the excel the N-D1 and N-D2 were calculated to be equal to 0.436795428 and 0.535215989 respectively. The put option value calculated using the Black Scholes model was equal to $7.03.

The buyer of a call option has the opportunity to purchase the underlying asset at a present price at any time before the expiration date. The fundamental disadvantage of call options is that they may expire worthless, resulting in the loss of the price paid. Put options provide an investor the right to sell an underlying asset at a certain price. If the put option is sold and the price goes in the other direction, the option writer may lose a lot of money.

Portfolio performance evaluation may be described as a word that refers to measurements that aid in determining how effectively a money manager achieves a balance of high return and acceptable risk by evaluating portfolio return and then adjusting it for portfolio risk (Banihashemi and Navidi 2017). The success of a portfolio manager in satisfying an investing mandate is determined in part by how well he or she performs. The fundamental goal of reviewing a portfolio's performance is to see if investing goals are being reached. The goal of performance evaluation is to compare performance to a baseline, which aids in determining management compensation and salary. Investors may also use performance evaluation to figure out how to enhance their portfolio's performance and take the appropriate measures in that regard. Past performance may not be a good predictor of future performance; therefore, performance evaluation may provide an inaccurate image of portfolio performance. If the performance review process is repeated several times within a year, it may be deceptive and lead to short-termism. If the assessment procedure is not followed on a regular basis, there is a danger of stale and delayed portfolio performance metrics (Ensslin et al, 2017). The performance review procedure is expensive, and it may be argued that risk adjusted portfolio measurements do not provide a clear and simple image of the portfolio. There are various performance evaluation measures which include Sharpe Ratio, Jensen Alpha, Treynor ratio and Information Ratio all of which are discussed below in detail:

Leading indicators

Sharpe ratio is a performance evaluation method which is used to measure the risk adjusted return potential of an investment or a portfolio. Risk adjusted return can be defined as the additional average return received over the risk-free rate of return. By subtracting the risk-free rate of return from the mean return, the performance of risk-taking activity may be determined (Benhamou et al, 2019). A greater Sharpe ratio implies a better investment, whereas a lower Sharpe ratio shows that the portfolio is not earning as much as it should base on the portfolio's standard deviation. The portfolio's Sharpe ratio is positive, but it's below 1 at 0.86, which is lower than the benchmark's Sharpe ratio of 0.95. As a result, on a risk-adjusted basis, it may be argued that the fund has underperformed the benchmark.

The Treynor ratio is a performance measure matrix which measures the risk-adjusted return of an investment or a portfolio with respected to the systematic risk inherent in the portfolio which is measured by beta. Because there is no risk of default, the return from a risk-free investment is usually the return given by U.S. Treasury securities. The formula for Treynor ratio includes deducting the expected return of the portfolio by the risk-free rate of return and dividing it by the beta of the portfolio. The Treynor ratio of the portfolio was around 0.14 which is less than the Treynor ratio of the benchmark indicating that the portfolio has underperformed the benchmark.

Jensen Alpha is a portfolio evaluation method focused on measuring the risk-adjusted performance of the portfolio by comparing it with the level of expected return predicted with the Capital Asset Pricing Model (CAPM). CAPM is an asset pricing model which utilizes the beta and the average market return to predict an appropriate required rate of return. When a manager performs extraordinarily, he/she is said to have been generated alpha for the clients. Using the risk-free rate, beta of the portfolio and the average market return, the CAPM expected return was calculated to be equal to 16.99%. The fund exceeded the returns provided by the market by 101 basis points which indicates a good performance.

Information ratio measures the returns of a portfolio beyond the returns of the benchmark comparing the volatility of the portfolio. The benchmark is represented by a broader market index for the particular investment type. The information ratio measures the level of skill that a manager possesses which is essential for generating excess returns compared to the benchmark the portfolio is mandated to follow (Lattimore and Gyorgy 2021). The tracking error is used as an input in the calculation of information ratio and it measures the consistency in which the portfolio manager tracks the index. The information ratio of the portfolio was around 2.5 which is considered to be a good indicator.


The purpose of the report was to highlight the importance of various aspects of portfolio management explaining multiple theories. The first portion of the research focuses on the significance of several portfolio theories, such as the Arbitrage Pricing Model and the Fama-French three component model, as well as their differences. The Fama-French model is the most widely used model for calculating the needed rate of return on an asset since it incorporates various elements in addition to systematic factors when evaluating the asset's returns. Leading indicators, trailing indicators, and coincident indicators are three separate sorts of economic indicators that are designed to anticipate business cycle peaks and troughs, and they are discussed in the second half of the study. The banking and financial industry was studied, and future trends were forecasted based on current economic conditions and financial scenarios. Black Scholes model is the most used option pricing model in the financial world and the third part of the report discusses on the same. Option price based on hypothetical details was estimated with the help of multiple variables. The uses and limitations of call options and put options were also explained. The report's last section focuses on evaluating the AlphaWell fund's performance using several performance measures such as the Sharpe Ratio, Treynor Ratio, Information Ratio, and Jensen's Alpha. The fund surpassed the benchmark in terms of information ratio and Jensen's alpha, but underperformed by a modest margin in terms of Sharpe ratio and Treynor ratio.


(2022) Available at: (Accessed: 14 April 2022).

Anwar, M.N. and Andallah, L.S., 2018. A study on numerical solution of Black-Scholes model. Journal of Mathematical Finance, 8(2), pp.372-381.

Banihashemi, S. and Navidi, S., 2017. Portfolio performance evaluation in Mean-CVaR framework: A comparison with non-parametric methods value at risk in Mean-VaR analysis. Operations Research Perspectives, 4, pp.21-28.

Benhamou, E., Saltiel, D., Guez, B. and Paris, N., 2019. Testing Sharpe ratio: luck or skill?. arXiv preprint arXiv:1905.08042.

Dar, A.A. and Anuradha, N., 2018. Comparison: binomial model and Black Scholes model. Quantitative finance and Economics, 2(1), pp.230-245.

Ensslin, L., Ensslin, S.R., Dutra, A., Nunes, N.A. and Reis, C., 2017. BPM governance: a literature analysis of performance evaluation. Business Process Management Journal.

Lattimore, T. and Gyorgy, A., 2021, July. Mirror descent and the information ratio. In Conference on Learning Theory (pp. 2965-2992). PMLR.

Martin, C., 2022. 2022 Corporate Bond Outlook: Focus on Income. [online] Schwab Brokerage. Available at: <> [Accessed 14 April 2022].

Saphir, A. and Dunsmuir, L., 2022. Fed officials nod to big rate hikes to fight 'inflation, inflation, inflation'. [online] Reuters. Available at: <> [Accessed 14 April 2022].

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