15 minute lesson - capm derivation

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15-Minute Lesson:
Capital Asset Pricing Model (CAPM) Derivation
Sukarnen
DILARANG MENG-COPY, MENYALIN,
ATAU MENDISTRIBUSIKAN
SEBAGIAN ATAU SELURUH TULISAN
INI TANPA PERSETUJUAN TERTULIS
DARI PENULIS
Untuk pertanyaan atau komentar bisa
diposting melalui website
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Preface
(William F.Sharpe – Father of CAPM)
CAPM is quite well known for all those students in finance class. Unfortunately, this CAPM
formula, particularly in standard corporate finance textbooks, is shown without telling us as to
how to get that mathematically.
In this article, I demonstrate the derivation of CAPM formula.
As a background, the readers are encouraged to read chapter 9 "The Capital Asset Pricing
Model" from "Investments" book, by Zvi Bodie, Alex Kane, and Alan J. Marcus (a standard
reading book for those taking exams for CFA), 10th edition, New York (USA): Mc-Graw Hill
Education. 2014. Page 292.
A word of caution from Bodie, Kane and Marcus:
We often hear that well-managed firms will provide high rates of return. We agree this is true if
one measures the firm’s return on its investments in plant and equipment. The CAPM,
however, predicts returns on investments in the securities of the firm.
Let’s say that everyone knows a firm is well run. Its stock price will therefore be bid up, and
consequently returns to stockholders who buy at those high prices will not be excessive.

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Security prices, in other words, already reflect public information about a firm’s prospects;
therefore only the risk of the company (as measured by beta in the context of the CAPM) should
affect expected returns. In a well-functioning market, investors receive high expected returns
only if they are willing to bear risk.
Karnen:
CAPM is an elegant formula, only with 2-3 pages, the formula could mathematically be derived
as shown below. Since its publication by William “Bill” F. Sharpe in 1964, CAPM has been a
topic for thousands of researches and articles and papers, and has been used by corporate,
institutional, and pension fund managers to plan and judge their investments. Bill Sharpe, then
was named winner of the 1990 Nobel Prize for Economics for his theory, called the Capital
Asset Pricing Model (At the same time, the CAPM was independently developed by John
Lintner, Jan Mossin, and Jack Treynor), in developing models to aid investment decisions. Bill
Sharpe shared the award with Harry Markowitz and Merton Miller.
CAPM is a way of matching potential gain from an investment with the potential risk. The theory
is based on the earlier work of Harry Markowitz, who with Bill Sharpe derived theory from
Stanford's Kenneth Arrow, winner of the 1972 Nobel in economics. Readers could read an
interesting interview with Bill Sharpe by Jonathan Burton in 1998 in which, Bill Sharpe
expressed his view on CAPM in his own words.
(http://web.stanford.edu/~wfsharpe/art/djam/djam.htm)
It is in its simplicity, I believe that has brought such big popularity of its use among finance
scholars and those practitioners in finance world. The idea is simple, that is let history be your
guide!
Why I said simple.....
Estimating the “expected” (note: this is ex-ante concept) return on any risky asset is only
composed of two factors:

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Expected return on risky asset = risk-free interest rate (including inflation premium) +
risk premium
The above equation just says that as the holder of a risky asset, he/she “should” (again, this is
“expected” concept) to earn a return from two sources:
1. Compensation for the opportunity cost incurred in holding the asset (this is called “the
risk-free interest rate”) and compensation for the declining purchasing power of the
currency over time (this is “the inflation premium”). These two factors are in general
equal to the expected return on a default-free bond such as a government bond.
2. Compensation for bearing the asset’s systematic risk (this is “the risk premium”).
Estimating the risk premium is quite challenging and has been a hot debate among
many practitioners, professors, economists, etc etc.
As Bodie, Kane and Marcus reminds us above, CAPM is developed particularly for estimating
the expected return of the securities. In this case, let’s start with common stock as the risky
asset.
History in this case could be our guide. From Stocks, Bonds, Bills, and Inflation 1926 -2012 (87
years), the annual return on U.S. common stocks (large stocks = 9.8%) exceeded that on U.S.
Treasury Bills (3.6%) by 6.2 percentage points. As a reward for bearing the added systematic
risk, the holders of the large common stocks earned a 6.2 percentage point, a higher annual
return than the holders of U.S. Treasury Bills. This spread which is considered as the risk
premium, could give an idea to the investors, for example, in 2013, how much risk premium they
are going to add to the 2013 U.S. Treasury Bills. Using data from U.S. Treasury
(https://www.treasurydirect.gov/govt/rates/pd/avg/2013/2013_01.htm), the U.S. Treasury Bills
average in January 2013 of 1.24 percent, added on top of this, 6.2 percent, will yield an
estimate of 7.44 percent as the expected return on common stock, or as the cost of levered
equity for a typical company with an average risk.
The big question, of course, what is the logic of adding this 6.2 percentage point “historical
excess return” as a risk premium?
The idea is again, simple. Over a very long period, in this example, 87 years of historical excess
return, the return that investors had received and what they expect to receive, should
approximate each other. It is easy to imagine a case in which investors in stock market expect a

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10-percentage point excess return on common stocks, but the actual return from the market
keeps turning out to be 5 percentage points. This fact will bring two things:
(1) Investors in stock market should lower their expectations, and
(2) Selling by disappointed investors should increase subsequent realized returns.
At the end of the day, expectations and reality “should” become a roughly close.
You are going to say, alright, history is our guide, but this expected return (or the cost of capital)
is for an average-risk company, and not all companies, probably just few, are considered having
average-risk. The next big question is how to customize or adjust our average-risk cost of
capital to reflect higher/lower risk of a specific firm.
Bill Sharpe was then genius enough, to answer such question, is to insert a “fudge factor”,
which then becomes so well know, as the company’s equity beta. CAPM then is developed into
one equation that becomes so central to the finance world:
Cost of [levered] equity capital = interest on government bill/bond + Beta_equity *
(historical excess return on common stock)
Beta_equity above could be read as a scale factor reflecting the systematic risk of a specific
company’s common stock RELATIVE TO THAT OF AN AVERAGE RISK OF COMMON
STOCKS. When the subject stock’s systematic risk equals to that of average-risk common
stocks, Beta_equity =1, and this will mean, the historical risk premium will become the equity
risk premium of that common stock.
However, for above-average-risk common stock, Beta_equity > 1, and the equity risk premium
of that stock will be higher than historical risk premium.
How about those stocks with lower-average-risk? In this case, Beta_equity < 1, and this will
bring the equity risk premium will be a fraction of the historical risk premium.
Beta is a scoring system that rates individual stocks according to their volatility relative to the
market.

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Overall, the strength of the CAPM is its ability to show the relationship between risk and return,
which is investors are not likely to take risks unless they expect returns above what they could
expect on riskless investments. It is so clear to see that the oil company will not drill for oil if all
they can expect is what a government treasury bill would provide. This strong relationship of
higher returns and riskier investments pervades all investment decisions.
In the long-run and on average, the history shows riskier investment instruments offer higher
returns. A fact that we cannot close our eyes off!
Back to Bodie, Kane and Marcus, it is said about the basic principle of equilibrium....
Here how we could get CAPM mathematically...such a beautiful and eloquent formula.

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