28
3 Portfolio Theory and CAPM
2. Unexpectedly, you have inherited some money from an unknown aunt and you
want to invest it in the stock market. To make the problem manageable, you decide
to invest in two items, Apple
® Inc. and an index fund, based on a mix of SP500
companies. A table of closing values at the end of every trading day for one year
(27.3.2018–27.3.2019) is available in the file stocks.dat from this book’s
web site. This text file contains three columns: the first one indicates the trading
day t d , the second contains the Apple data S 1 , and the third the SP500 data S 2 .
There is a short MATLAB script get_stocks.m available to load this file.
a. Plot the stock values S versus the trading day to obtain an impression of how
the stocks evolve.
b. Calculate the day-to-day returns r k = (S k+1 − S k )/S k , plot the values and calculate their average. Reminder: scale the average daily returns with the trading
days to obtain the annual returns.
c. Then you decide to use Markowitz’s theory to calculate how to partition your
money between the two stocks in order to obtain an annual return of 5, 10, and
15%. Interpret the results and how to implement them.
3. For the highest annual return of ρ = 0.15%, repeat the analysis from Exercise 2
with a risk-free asset a) with r f = 0%, b) with r f = 3% included in the portfolio.
4. Based on the data in stocks.dat, use the SP500 data in the third column
as a proxy for the market as a whole and assume an annual risk-free rate of
r f = 5%. Use CAPM to determine whether the stock in the second column is
over or undervalued.
5. Before buying a company, you analyze their financial reports, which state that the
capital base is 16 × 10
6
e of which 7 × 10
6
e is financed by debt at an interest
rate of 5%. The tax rate is t = 30%. The annual rate of return to the investors
historically was about 12% and the annual cash flow was typically 1.7 × 10
6
e.
The asking price that the present owner demands is 8.1 × 10
6
e. Can you expect
to recover your investment over 6 years?
References
1. H. Markowitz, Portfolio Selection. J. Finance 7, 77 (1952)
2. L. Landau, E. Lifschitz, Lehrbuch der theoretischen Physik, Band I: Mechanik (Akademie
Verlag, Berlin, 1979)
3. H. Goldstein, J. Safko, C. Poole, Classical Mechanics (Pearson, Harlow, 2014)
4. J. Tobin, Liquidity preference as behavior towards risk. Rev. Econ. Stud. XXVI, 65 (1958)
5. W. Sharpe, Capital asset prices: a theory of market equilibrium under conditions of risk. J.
Finance 19, 425 (1964)
6. J. Lintner, The valuation of risk assets and the selection of risky investments in stock portfolios
and capital budgets. Rev. Econ. Stat. 47, 13 (1965)
7. E. Fama, K. French, The capital asset pricing model: theory and evidence. J. Econ. Perspect. 18,
25 (2004)
3 Portfolio Theory and CAPM
2. Unexpectedly, you have inherited some money from an unknown aunt and you
want to invest it in the stock market. To make the problem manageable, you decide
to invest in two items, Apple
® Inc. and an index fund, based on a mix of SP500
companies. A table of closing values at the end of every trading day for one year
(27.3.2018–27.3.2019) is available in the file stocks.dat from this book’s
web site. This text file contains three columns: the first one indicates the trading
day t d , the second contains the Apple data S 1 , and the third the SP500 data S 2 .
There is a short MATLAB script get_stocks.m available to load this file.
a. Plot the stock values S versus the trading day to obtain an impression of how
the stocks evolve.
b. Calculate the day-to-day returns r k = (S k+1 − S k )/S k , plot the values and calculate their average. Reminder: scale the average daily returns with the trading
days to obtain the annual returns.
c. Then you decide to use Markowitz’s theory to calculate how to partition your
money between the two stocks in order to obtain an annual return of 5, 10, and
15%. Interpret the results and how to implement them.
3. For the highest annual return of ρ = 0.15%, repeat the analysis from Exercise 2
with a risk-free asset a) with r f = 0%, b) with r f = 3% included in the portfolio.
4. Based on the data in stocks.dat, use the SP500 data in the third column
as a proxy for the market as a whole and assume an annual risk-free rate of
r f = 5%. Use CAPM to determine whether the stock in the second column is
over or undervalued.
5. Before buying a company, you analyze their financial reports, which state that the
capital base is 16 × 10
6
e of which 7 × 10
6
e is financed by debt at an interest
rate of 5%. The tax rate is t = 30%. The annual rate of return to the investors
historically was about 12% and the annual cash flow was typically 1.7 × 10
6
e.
The asking price that the present owner demands is 8.1 × 10
6
e. Can you expect
to recover your investment over 6 years?
References
1. H. Markowitz, Portfolio Selection. J. Finance 7, 77 (1952)
2. L. Landau, E. Lifschitz, Lehrbuch der theoretischen Physik, Band I: Mechanik (Akademie
Verlag, Berlin, 1979)
3. H. Goldstein, J. Safko, C. Poole, Classical Mechanics (Pearson, Harlow, 2014)
4. J. Tobin, Liquidity preference as behavior towards risk. Rev. Econ. Stud. XXVI, 65 (1958)
5. W. Sharpe, Capital asset prices: a theory of market equilibrium under conditions of risk. J.
Finance 19, 425 (1964)
6. J. Lintner, The valuation of risk assets and the selection of risky investments in stock portfolios
and capital budgets. Rev. Econ. Stat. 47, 13 (1965)
7. E. Fama, K. French, The capital asset pricing model: theory and evidence. J. Econ. Perspect. 18,
25 (2004)