This document summarizes critiques of the Capital Asset Pricing Model (CAPM) and presents alternative models. It discusses empirical studies from the 1980s and 1990s that found variables other than beta help explain stock returns, contradicting CAPM. Fama and French's 1992 study found firm size and book-to-market ratio better predict returns than beta. Their three-factor model and the Arbitrage Pricing Theory were proposed as alternatives to CAPM. Overall, the document outlines major empirical challenges to CAPM and influential models that improved on its ability to explain stock returns.
Determination of exchange rate chapter 6Nayan Vaghela
Determination of exchange rate, mint par theory, balance of payment theory, Purchasing power parity theory, Absolute version and relative version, Criticisms
Capital Asset Pricing Model (CAPM)
A model that describes the relationship between risk and expected return. The general idea behind CAPM is that investors need to be compensated in two ways: time value of money & risk. The time value of money is represented by the risk-free (rf) rate in the formula and compensates the investors for placing money in any investment over a period of time. The other half of the formula represents risk and calculates the amount of compensation the investor needs for taking on additional risk. This is calculated by taking a risk gauge (beta) that compares the returns of the asset to the market over a period of time and to the market premium (Rm-rf).
Because of the risk-return tradeoff, you must be aware of your personal risk tolerance when choosing investments for your portfolio. Taking on some risk is the price of achieving returns; therefore, if you want to make money, you can't cut out all risk. The goal instead is to find an appropriate balance - one that generates some profit, but still allows you to sleep at night.
Determination of exchange rate chapter 6Nayan Vaghela
Determination of exchange rate, mint par theory, balance of payment theory, Purchasing power parity theory, Absolute version and relative version, Criticisms
Capital Asset Pricing Model (CAPM)
A model that describes the relationship between risk and expected return. The general idea behind CAPM is that investors need to be compensated in two ways: time value of money & risk. The time value of money is represented by the risk-free (rf) rate in the formula and compensates the investors for placing money in any investment over a period of time. The other half of the formula represents risk and calculates the amount of compensation the investor needs for taking on additional risk. This is calculated by taking a risk gauge (beta) that compares the returns of the asset to the market over a period of time and to the market premium (Rm-rf).
Because of the risk-return tradeoff, you must be aware of your personal risk tolerance when choosing investments for your portfolio. Taking on some risk is the price of achieving returns; therefore, if you want to make money, you can't cut out all risk. The goal instead is to find an appropriate balance - one that generates some profit, but still allows you to sleep at night.
Prepared by Students of University of Rajshahi
Shahin Islam
Aslam Hossain
Shahidul Islam
Amy Khatun
Sohanuzzaman Sohan
MD. Rehan
Bikash Kumar
Rahid Hasan
Ali Haider
Uttam Kumar
MD. Abdullah AL Mamun
Mamunur Rahman
presented by Mango squad
For downloading this contact- bikashkumar.bk100@gmail.com
Testing and extending the capital asset pricing modelGabriel Koh
This paper attempts to prove whether the conventional Capital Asset Pricing Model (CAPM) holds with respect to a set of asset returns. Starting with the Fama-Macbeth cross-sectional regression, we prove through the significance of pricing errors that the CAPM does not hold. Hence, we expand the original CAPM by including risk factors and factor-mimicking portfolios built on firm-specific characteristics and test for their significance in the model. Ultimately, by adding significant factors, we find that the model helps to better explain asset returns, but does still not entirely capture pricing errors.
Dividend portfolio – multi-period performance of portfolio selection based so...Bogusz Jelinski
What will happen to your investment if you ignore change of share prices while calculating both returns and risk during stocks portfolio selection? Is it profitable in long term to
sell a share when its price has started to skyrocket?
The CBS Television surprise hit “Undercover Boss” has aired for six consecutive seasons and
features publicly traded firms, closely-held corporations, and in some instances not-for-profit institutions. While
there has been much analysis on the ethical dilemmas faced by the undercover CEO or other executive, no
practical analysis of a firm‟s profitability has been conducted on any of the firms featured on the show.
Conventional wisdom would suggest that financial performance of a featured firm would improve after the initial
airing date, as the show typically ends on a „feel good‟ note and most often places the executive, as well as the
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how to sell pi coins in all Africa Countries.DOT TECH
Yes. You can sell your pi network for other cryptocurrencies like Bitcoin, usdt , Ethereum and other currencies And this is done easily with the help from a pi merchant.
What is a pi merchant ?
Since pi is not launched yet in any exchange. The only way you can sell right now is through merchants.
A verified Pi merchant is someone who buys pi network coins from miners and resell them to investors looking forward to hold massive quantities of pi coins before mainnet launch in 2026.
I will leave the telegram contact of my personal pi merchant to trade with.
@Pi_vendor_247
What price will pi network be listed on exchangesDOT TECH
The rate at which pi will be listed is practically unknown. But due to speculations surrounding it the predicted rate is tends to be from 30$ — 50$.
So if you are interested in selling your pi network coins at a high rate tho. Or you can't wait till the mainnet launch in 2026. You can easily trade your pi coins with a merchant.
A merchant is someone who buys pi coins from miners and resell them to Investors looking forward to hold massive quantities till mainnet launch.
I will leave the telegram contact of my personal pi vendor to trade with.
@Pi_vendor_247
how to sell pi coins at high rate quickly.DOT TECH
Where can I sell my pi coins at a high rate.
Pi is not launched yet on any exchange. But one can easily sell his or her pi coins to investors who want to hold pi till mainnet launch.
This means crypto whales want to hold pi. And you can get a good rate for selling pi to them. I will leave the telegram contact of my personal pi vendor below.
A vendor is someone who buys from a miner and resell it to a holder or crypto whale.
Here is the telegram contact of my vendor:
@Pi_vendor_247
how to sell pi coins in South Korea profitably.DOT TECH
Yes. You can sell your pi network coins in South Korea or any other country, by finding a verified pi merchant
What is a verified pi merchant?
Since pi network is not launched yet on any exchange, the only way you can sell pi coins is by selling to a verified pi merchant, and this is because pi network is not launched yet on any exchange and no pre-sale or ico offerings Is done on pi.
Since there is no pre-sale, the only way exchanges can get pi is by buying from miners. So a pi merchant facilitates these transactions by acting as a bridge for both transactions.
How can i find a pi vendor/merchant?
Well for those who haven't traded with a pi merchant or who don't already have one. I will leave the telegram id of my personal pi merchant who i trade pi with.
Tele gram: @Pi_vendor_247
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• Real GDP growth slowed down due to problems with access to electricity caused by the destruction of manoeuvrable electricity generation by Russian drones and missiles.
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• The USA has finally approved an aid package for Ukraine, which includes USD 7.8 bn of budget support; however, the conditions and timing of the assistance are still unknown.
• As in March, annual consumer inflation amounted to 3.2% yoy in April.
• At the April monetary policy meeting, the NBU again reduced the key policy rate from 14.5% to 13.5% per annum.
• Over the past four weeks, the hryvnia exchange rate has stabilized in the UAH 39-40 per USD range.
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what is the best method to sell pi coins in 2024DOT TECH
The best way to sell your pi coins safely is trading with an exchange..but since pi is not launched in any exchange, and second option is through a VERIFIED pi merchant.
Who is a pi merchant?
A pi merchant is someone who buys pi coins from miners and pioneers and resell them to Investors looking forward to hold massive amounts before mainnet launch in 2026.
I will leave the telegram contact of my personal pi merchant to trade pi coins with.
@Pi_vendor_247
Turin Startup Ecosystem 2024 - Ricerca sulle Startup e il Sistema dell'Innov...Quotidiano Piemontese
Turin Startup Ecosystem 2024
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The unveiling of the IndusInd Bank Poonawalla Fincorp eLITE RuPay Platinum Credit Card marks a notable milestone in the Indian financial landscape, showcasing a successful partnership between two leading institutions, Poonawalla Fincorp and IndusInd Bank. This co-branded credit card not only offers users a plethora of benefits but also reflects a commitment to innovation and adaptation. With a focus on providing value-driven and customer-centric solutions, this launch represents more than just a new product—it signifies a step towards redefining the banking experience for millions. Promising convenience, rewards, and a touch of luxury in everyday financial transactions, this collaboration aims to cater to the evolving needs of customers and set new standards in the industry.
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In India, financial inclusion remains a critical challenge, with a significant portion of the population still unbanked. Non-Banking Financial Companies (NBFCs) have emerged as key players in bridging this gap by providing financial services to those often overlooked by traditional banking institutions. This article delves into how NBFCs are fostering financial inclusion and empowering the unbanked.
Currently pi network is not tradable on binance or any other exchange because we are still in the enclosed mainnet.
Right now the only way to sell pi coins is by trading with a verified merchant.
What is a pi merchant?
A pi merchant is someone verified by pi network team and allowed to barter pi coins for goods and services.
Since pi network is not doing any pre-sale The only way exchanges like binance/huobi or crypto whales can get pi is by buying from miners. And a merchant stands in between the exchanges and the miners.
I will leave the telegram contact of my personal pi merchant. I and my friends has traded more than 6000pi coins successfully
Tele-gram
@Pi_vendor_247
US Economic Outlook - Being Decided - M Capital Group August 2021.pdfpchutichetpong
The U.S. economy is continuing its impressive recovery from the COVID-19 pandemic and not slowing down despite re-occurring bumps. The U.S. savings rate reached its highest ever recorded level at 34% in April 2020 and Americans seem ready to spend. The sectors that had been hurt the most by the pandemic specifically reduced consumer spending, like retail, leisure, hospitality, and travel, are now experiencing massive growth in revenue and job openings.
Could this growth lead to a “Roaring Twenties”? As quickly as the U.S. economy contracted, experiencing a 9.1% drop in economic output relative to the business cycle in Q2 2020, the largest in recorded history, it has rebounded beyond expectations. This surprising growth seems to be fueled by the U.S. government’s aggressive fiscal and monetary policies, and an increase in consumer spending as mobility restrictions are lifted. Unemployment rates between June 2020 and June 2021 decreased by 5.2%, while the demand for labor is increasing, coupled with increasing wages to incentivize Americans to rejoin the labor force. Schools and businesses are expected to fully reopen soon. In parallel, vaccination rates across the country and the world continue to rise, with full vaccination rates of 50% and 14.8% respectively.
However, it is not completely smooth sailing from here. According to M Capital Group, the main risks that threaten the continued growth of the U.S. economy are inflation, unsettled trade relations, and another wave of Covid-19 mutations that could shut down the world again. Have we learned from the past year of COVID-19 and adapted our economy accordingly?
“In order for the U.S. economy to continue growing, whether there is another wave or not, the U.S. needs to focus on diversifying supply chains, supporting business investment, and maintaining consumer spending,” says Grace Feeley, a research analyst at M Capital Group.
While the economic indicators are positive, the risks are coming closer to manifesting and threatening such growth. The new variants spreading throughout the world, Delta, Lambda, and Gamma, are vaccine-resistant and muddy the predictions made about the economy and health of the country. These variants bring back the feeling of uncertainty that has wreaked havoc not only on the stock market but the mindset of people around the world. MCG provides unique insight on how to mitigate these risks to possibly ensure a bright economic future.
Even tho Pi network is not listed on any exchange yet.
Buying/Selling or investing in pi network coins is highly possible through the help of vendors. You can buy from vendors[ buy directly from the pi network miners and resell it]. I will leave the telegram contact of my personal vendor.
@Pi_vendor_247
how can i use my minded pi coins I need some funds.DOT TECH
If you are interested in selling your pi coins, i have a verified pi merchant, who buys pi coins and resell them to exchanges looking forward to hold till mainnet launch.
Because the core team has announced that pi network will not be doing any pre-sale. The only way exchanges like huobi, bitmart and hotbit can get pi is by buying from miners.
Now a merchant stands in between these exchanges and the miners. As a link to make transactions smooth. Because right now in the enclosed mainnet you can't sell pi coins your self. You need the help of a merchant,
i will leave the telegram contact of my personal pi merchant below. 👇 I and my friends has traded more than 3000pi coins with him successfully.
@Pi_vendor_247
how can i use my minded pi coins I need some funds.
Multi factor models in asset pricing
1. Prepared By
Anuj Vijay Bhatia
F1401 (FPRM 14)
Theory of Finance
Institute of Rural Management Anand (IRMA)
2. CAPM: A Short Critique
• The CAPM model assumes a linear relationship between the expected return in a
risky asset and its β. Assets can only earn a high average return if they have a
high market β.
• Fama and French (2000) summarize the popularity of the CAPM by their
statement:
“The attraction of the CAPM is that it offers powerful and intuitively pleasing
predictions about how to measures risk and the relation between expected return
and risk.”
• Fama and French (2000) also offer their opinion on its relevance:
“Unfortunately the empirical record of the model is poor – poor enough to
invalidate the way it is used in applications.”
3. Empirical Studies on CAPM
• During the 1980’s several studies resulted in the identification of additional factors that provide
explanatory power other than β for average stock returns.
• Variables that have no special standing in asset pricing theory were shown to have reliable power in
explaining the cross section of returns (these variables are referred to as anomalies by Fama and French)
• Banz (1981) finds that Market Equity (ME) adds to the cross section of expected returns provided by the
market β.
• Basu (1983) finds that low earnings-price ratios (E/P) stocks help explain the cross section of US stocks
returns while high (E/P) stocks experiencing lower returns could be explained by the CAPM.
• DeBondt and Thaler (1985) find that stocks with abnormally low long term returns (average returns in
three years) experience abnormally high long term future returns (average returns in the next three years)
and vice versa.
• Bhandari (1988) finds a positive relationship between leverage and the cross section of average return.
• Rosenberg, Reid and Lanstein (1985) find a positive relationship between the average return and the ratio
of a firm’s book value to market equity (BE/ME).
• Lakonishok, Sheifer and Vishny (1994) find a strong positive relationship between average returns and
BE/ME and cashflow/price ratio (C/P). These relationships could not be explained by the CAPM.
4. Fama & French (1992)
• Fama & French in their paper “The Cross-Section of Expected Stock
Returns” presented one of the major empirical arguments against the
CAPM model.
• Their data comprised of individual stock returns from 1963 to 1990 for all
NYSE and AMEX listed stocks, then adding all NASDAQ-listed stocks
from 1973 to 1990.
• They found that there is a strong negative correlation between a firms size
(measured by Market Equity ME) and beta (-0.988).
• They designed an empirical test that carefully separated the two.
5. Fama & French (1992)
• The above Panel shows simple sorts based on size measured by the natural log of equity
market capitalization of a company.
• When the portfolios are arranged by size there is a visible relationship between monthly
returns and both size and beta.
6. Fama & French (1992) [cont..]
• In this Panel the portfolios are sorted on the basis of Betas.
• The relationship between monthly returns and Betas is tenuous.
7. Fama & French (1992) [cont..]
• Fama and French ran a two-pass sort to attempt to separate the effect of beta from size.
We can see the relationship between return and company size in the above table.
• If we read across the rows we can see relationship between returns and betas, keeping
size constant.
• For the small and large equity rows, the relationship between beta and return goes in the
wrong direction and it is not very strong in the remaining rows.
8. Fama & French (1992) [cont..]
• Fama and French ran multiple regressions with the individual stock returns
as dependent variables.
• Beta was not statistically significant related to the average monthly returns
of individual securities, either by itself or when combined with the size in
multiple regression.
• The strongest model did not include the beta at all.
• It explained return as a negative function of size( Log ME) and positive
function of the log of ratio of book to the market value of equity (BE/ME).
• However, when they exclude the NASDAQ stocks and extend their data
back to period 1941-1990, then beta is significant and positively related to
returns both for portfolios and for individual stocks.
9. Research Post Fama-French (1992)
• Roll and Ross (1994) showed that even small departures of the index
portfolio from the ex post market efficiency can easily result in empirical
results that show no relationship between beta and average cross-sectional
returns.
• Kothari, Shanken and Sloan (1995) study the same relationship between
beta, size and the ratio of BE to ME. They conclude that “examination of
cross-section of expected returns reveal economically and statistically
significant compensation (6-9% p.a.) for beta risk.
• They note that the annual returns are used to avoid the significant
seasonality of returns, there is a significant linear relationship between
returns and beta between 1941-1990, size also relates to returns but the
incremental economic contribution is small (<1%).
10. Fama and French (1996): The Three Factor Model
• Fama and French (1996) update their work and present a three-factor model that, when
fit to data for NYSE, AMEX and NASDAQ stock returns for 366 months from July
1963- December 1993, provides best explanation of excess returns.
• The model can be expressed as:
E(𝑅𝑖 ) − 𝑅 𝑓 = 𝑎𝑖 + 𝑏𝑖 [𝐸(𝑅 𝑀) − 𝑅 𝑓 ] + 𝑠𝑖 𝐸[𝑆𝑀𝐵] + ℎ𝑖 𝐸[𝐻𝑀𝐿]
• E(𝑅𝑖) is the expected return on asset i. 𝐸(𝑅 𝑀) is the expected return on the value-weight
market portfolio. 𝑅 𝑓 is the risk-free interest rate of 1-month Treasury bills. The
coefficients 𝑏𝑖 , 𝑠𝑖 and ℎ𝑖 are the βs of the stock on each of the three factors.
• The model says that the excess returns of the ith security over risk free rate is a linear
function of three factors:
1. The excess of the market portfolio over the risk-free rate
2. The difference between the returns on a portfolio of small stocks and the large stocks,
SMB; and
3. The difference between the returns on a portfolio of high and low book-to market
stock, HML.
11. The Arbitrage Pricing Theory: Ross [1976]
• Formulated by Ross [1976], the arbitrage pricing theory (APT) offers a testable
alternative to the capital asset pricing model. The CAPM predicts that security rates of
return will be linearly related to a single common factor—the rate of return on the
market portfolio. The APT is based on similar intuition but is much more general.
• It assumes that the rate of return on any security is a linear function of k factors as
shown below:
• ෩𝑹𝒊 = 𝑬(෩𝑹𝒊) + 𝒃𝒊𝟏
෩𝑭 𝟏+. . . +𝒃𝒊𝒌
෩𝑭 𝒌 +∈𝒊 (i = 1, . . . , n) ….. (1)
෩𝑹𝒊 = random rate of return on the ith asset
𝑬(෩𝑹𝒊)= expected return on the ith asset
෩𝑭 𝒌 = kth factor common to the returns of all assets under consideration with a mean of zero, common
factors that in essence capture the systematic component of risk in the model
𝒃𝒊𝒌 = a coefficient called a factor loading that quantifies the sensitivity of asset i’s returns to the
movements in the common factor (and is analogous to the beta in the CAPM)
∈𝒊 = an error term, or unsystematic risk component, idiosyncratic to the ith asset, with mean zero (a
random zero mean noise term for the ith asset)
• The CAPM may be viewed as a special case of the APT when the market rate of return
is assumed to be the single relevant factor.
12. APT: Assumptions
• The APT is derived under the usual assumptions of perfectly competitive and
frictionless capital markets (No Arbitrage Opportunities)
• Furthermore, individuals are assumed to have homogeneous beliefs that the random
returns for the set of assets being considered are governed by the linear k-factor model
given in equation.
• The theory requires that the number of assets under consideration, n, be much larger
than the number of factors, k, and that the noise term, Ei, be the unsystematic risk
component for the ith asset. It must be independent of all factors and all error terms for
other assets.
• APT does not require the following CAPM assumptions:
1. Investors are mean-variance optimizers in the sense of Markowitz.
2. Returns are normally distributed.
3. The market portfolio contains all the risky securities and it is efficient in the mean-
variance sense.
13. Derivation of APT
• In equilibrium all portfolios that can be selected from among the set of assets under
consideration and that satisfy the conditions of (a) using no wealth and (b) having no risk must
earn no return on average. These portfolios are called arbitrage portfolios.
• To see how they can be constructed, let wi be the change in the dollar amount invested in the ith
asset as a percentage of an individual's total invested wealth. To form an arbitrage portfolio that
requires no change in wealth, the usual course of action would be to sell some assets and use the
proceeds to buy others.
• Mathematically, the zero change in wealth is written as:
σ𝑖=1
𝑛
𝑤𝑖 = 0 ……………..(2)
• If there are n assets in the arbitrage portfolio, then the additional portfolio return gained is:
෨𝑅 𝑝 = σ𝑖=1
𝑛
𝑤𝑖
෨𝑅𝑖 ………………………..(3)
σ𝑖=1
𝑛
𝑤𝑖 𝐸( ෨𝑅𝑖) + σ𝑖=1
𝑛
𝑤𝑖 𝑏𝑖1
෨𝐹1 + ⋯ + σ𝑖=1
𝑛
𝑤𝑖 𝑏𝑖𝑘
෨𝐹𝑘 + σ𝑖=1
𝑛
𝑤𝑖 ∈𝑖 ………(4)
14. Derivation of APT [Cont..]
• To obtain a riskless arbitrage portfolio it is necessary to eliminate both diversifiable (i.e.,
unsystematic or idiosyncratic) and undiversifiable (i.e., systematic) risk.
• This can be done by meeting three conditions: (1) selecting percentage changes in
investment ratios, wi, that are small, (2) diversifying across a large number of assets, and
(3) choosing changes, wi, so that for each factor, k, the weighted sum of the systematic
risk components, bk, is zero. Mathematically, these conditions are:
𝑤𝑖 ≈
1
𝑛
, …. (5)
n chosen to be a large number, ….(6)
σ 𝑤𝑖 𝑏𝑖𝑘 = 0 for each factor….(7)
• Because the error terms, 𝜀𝑖, are independent the law of large numbers guarantees that a
weighted average of many of them will approach zero in the limit as n becomes large.
• In other words, costless diversification eliminates the last term (the unsystematic or
idiosyncratic risk) in Eq (1). Thus, we are left with:
෨𝑅 𝑝 = σ𝑖=1
𝑛
𝑤𝑖 𝐸( ෨𝑅𝑖) + σ𝑖=1
𝑛
𝑤𝑖 𝑏𝑖1
෨𝐹1 + ⋯ + σ𝑖=1
𝑛
𝑤𝑖 𝑏𝑖𝑘
෨𝐹𝑘 … (8)
15. Derivation of APT [Cont..]
• At first glance the return on our portfolio appears to be a random variable, but we have
chosen the weighted average of the systematic risk components for each factor to be
equal to zero (σ 𝑤𝑖 𝑏𝑖𝑘 = 0 ).
• This eliminates all systematic risk. One might say that we have selected an arbitrage
portfolio with zero beta in each factor. Consequently, the return on our arbitrage
portfolio becomes a constant. Correct choice of the weights has eliminated all
uncertainty, so that Rp is not a random variable.
• Therefore Eq. 4 becomes: 𝑅 𝑝= σ 𝑤𝑖 𝐸 ( ෨𝑅𝑖) …….. (9)
• Recall that the arbitrage portfolio, so constructed, has no risk (of any kind) and requires
no new wealth. If the return on the arbitrage portfolio were not zero, then it would be
possible to achieve an infinite rate of return with no capital requirements and no risk.
Such an opportunity is clearly impossible if the market is to be in equilibrium. In fact, if
the individual arbitrageur is in equilibrium (hence content with his or her current
portfolio), then the return on any and all arbitrage portfolios must be zero. In other
words,
𝑅 𝑝= σ 𝑤𝑖 𝐸 ( ෨𝑅𝑖) = 0 …….. (10)
16. Derivation of APT [Cont..]
• Eqs. (2), (7) and (10) are really statements in linear algebra. Any vector that is
orthogonal to the constant vector*, i.e.,
σ𝑖=1
𝑛
𝑤𝑖 . 𝒆 = 0
and to each of the coefficient vectors, i.e.,
σ 𝑤𝑖 𝑏𝑖𝑘 = 0 for each k
must also be orthogonal to the vector of expected returns, i.e.,
σ𝑖=1
𝑛
𝑤𝑖 𝐸( ෨𝑅)𝑖 = 0
[*Note that Eq. (2) says that the sum of the investment weights equals zero. This is really
a no-wealth constraint. No new wealth is required to take an arbitrage position. Recall that
e is a column vector of ones.]
17. Derivation of APT [Cont..]
• An algebraic consequence of this statement is that the expected return vector must be a
linear combination of the constant vector and the coefficient vectors. Algebraically, there
must exist a set of k + 1 coefficients, 𝜆0, 𝜆1, 𝜆2,…. 𝜆 𝑘 such that
𝐸( ෨𝑅)𝑖 = 𝜆0 + 𝜆1 𝑏𝑖1+……+ 𝜆 𝑘 𝑏𝑖𝑘 ……………..(11)
• Recall that the 𝑏𝑖𝑘 are the "sensitivities" of the returns on the ith security to the kth
factor.
• If there is a riskless asset with a riskless rate of return, Rf , then 𝑏0𝑘 = 0 and Rf = 𝜆0
• Hence Eq. (11) can be rewritten in "excess returns form" as
E (𝑅𝑖) - 𝑅 𝑓 = 𝜆1 𝑏𝑖1+……+ 𝜆 𝑘 𝑏𝑖𝑘 ……………..(12)
18. The Arbitrage Pricing Line
• This figure illustrates the arbitrage pricing
relationship (12) assuming that there is only a single
stochastic factor, k.
• In equilibrium, all assets must fall on the arbitrage
pricing line. A natural interpretation for 𝜆 𝑘 is that it
represents the risk premium (i.e., the price of risk), in
equilibrium, for the kth factor.
• Because the arbitrage pricing relationship is linear we
can use the slope-intercept definition of a straight line
to rewrite Eq. (12) as:
• E (𝑅𝑖) = 𝑅𝑓 + [ ҧ𝛿 𝑘 − 𝑅𝑓 ] 𝑏𝑖𝑘 , where ҧ𝛿 𝑘 is the
expected return on a portfolio with unit sensitivity to
the kth factor and zero sensitivity to all other factors.
• Therefore the risk premium, 𝜆 𝑘 , is equal to the difference between (1) the expectation of a portfolio that has
unit response to the kth factor and zero response to the other factors and (2) the risk-free rate, 𝑅 𝑓 :
𝜆 𝑘 = ҧ𝛿 𝑘 − 𝑅 𝑓
• In general the arbitrage pricing theory can be rewritten as:
E (𝑅𝑖) - 𝑅 𝑓 = [ ҧ𝛿1 − 𝑅 𝑓 ] 𝑏𝑖1 + ….. + [ ҧ𝛿 𝑘 − 𝑅 𝑓 ] 𝑏𝑖𝑘 … (13)
19. Superiority of APT over CAPM
1. The APT makes no assumptions about the empirical distribution of asset
returns.
2. The APT makes no strong assumptions about individuals' utility functions (at
least nothing stronger than greed and risk aversion).
3. The APT allows the equilibrium returns of assets to be dependent on many
factors, not just one (e.g., beta).
4. The APT yields a statement about the relative pricing of any subset of assets;
hence one need not measure the entire universe of assets in order to test the
theory.
5. There is no special role for the market portfolio in the APT, whereas the CAPM
requires that the market portfolio be efficient.
6. The APT is easily extended to a multiperiod framework (Ross [1976]).
20. Summary
• Both models CAPM and the APT, that enable us to price risky assets in equilibrium.
• Within the CAPM framework the appropriate measure of risk is the covariance of returns
between the risky asset in question and the market portfolio of all assets. The APT model is
more general.
• Many factors (not just the market portfolio) may explain asset returns. For each factor the
appropriate measure of risk is the sensitivity of asset returns to changes in the factor. For
normally distributed returns the sensitivity is analogous to the beta (or systematic risk) of
the CAPM.
• The CAPM was shown to provide a useful conceptual framework for capital budgeting and
the cost of capital. It is also reasonably unchanged by the relaxation of many of the
unrealistic assumptions that made its derivation simpler.
• Finally, although the model is not perfectly validated by empirical tests, its main
implications are upheld—namely, that systematic risk (beta) is a valid measure of risk, that
the model is linear, and that the trade-off between return and risk is positive.
• The APT can also be applied to cost of capital and capital budgeting problems. The earliest
empirical tests of the APT have shown that asset returns are explained by three or possibly
four factors and have ruled out the variance of an asset's own returns as one of the factors.