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English written summary for the Investments course of the premaster Finance, Vrije Universiteit Amsterdam.
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Summary Investments – Bodie, Kane & MarcusChapter 1 – The investment environment
Real assets create wealth, financial assets represent claims to parts of all of that wealth and
determine how the ownership of real assets is distributed among investors. Financial assets can be
categorized as fixed income (bond), equity (common stock) or derivative instruments (options and
futures). Top-down portfolio construction techniques start with the asset allocation decision (the
allocation of funds across broad asset classes) and then progress to more specific security-selection
decisions. A portfolio is a collection of investment assets.
Competition in financial markets lead to a risk-return trade-off, in which securities that offer higher
expected rates of return also impose greater risks on investors. The presence of risk however implies
that actual returns can differ considerably from expected returns at the beginning of the investment
period. Competition among security analysts also promotes financial markets where all prices reflect
all available information concerning the value of the security (efficient markets). If new information
becomes available, the price of the security is quickly adjusted. The degree to which investors are
willing to commit funds to stocks depends on their risk aversion and personal preference.
Chapter 2 – Asset classes and Financial instruments
Money market securities are short-term debt obligations, highly marketable, low credit risks and
minimal capital gains or losses. Much of US government borrowing is in the form of Treasury bonds
or notes, which are coupon-paying bonds usually issued at or near par value (with the payment of
the face value at maturity) and reflect the government raising money by selling bills to the public.
Common stock or equities represent ownership shares in a corporation. Each share entitles the
owner to one vote in the annual meeting and a share of dividends paid to the owners (shareholders).
Owners are residual claimants of the income earned by the firm, they are last in line.
Preferred stock usually pays fixed dividends for the life of the firm (perpetuity) and must be
cumulatively paid before the holders of common stock. A call option is a right to purchase an asset at
an exercise price on or before an expiration date. A put option is the right to sell an asset at some
exercise price. Calls increase in value while puts decrease as the price of the underlaying asset
increases. A futures contract is an obligation to buy or sell an asset at a futures price on a maturity
date. The long position (purchasing) gains if the asset value increases, while the short position
(delivering) loses.
Chapter 3 – How securities are traded
Firms issue securities on the primary (new securities) or secondary (existing securities) market, to
raise the capital necessary to finance their investments. Short-selling is the practice of selling
securities that the seller doesn’t own. Securities trading is regulated by government agencies and
self-regulation of the exchanges. Many of the important regulations have to do with full disclosure of
relevant information and prohibit insider trading, which is attempting to profit from inside
information (private information that has not yet made available to the public).
Chapter 4 – Mutual funds and other investment companies
Investment companies are financial intermediaries that collect funds from individual investors and
invest those funds in a potentially wide range of securities or other assets. The value of each share is
called the net asset value and equals the market value of assets held by a fund minus the liabilities of
the fund, divided by the shares outstanding.
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Summary Investments – Annamarie de Ruijter – VU University – 2018/19
,Chapter 5 – Risk, return and the historical record
Interest rates and forecasts of future values are one of the most important inputs into an investment
decision. An interest rate is a promised rate of return denominated in some unit of account (dollars,
euros) over some time period (month, year). The nominal rate is the actual growth of your money
and the real rate the growth rate of your purchasing power, corrected for expected inflation (i). The
real interest rate is determined by supply, demand and government actions (when demand and
supply are equal, an equilibrium rate is reached):
Approximation RR = RN – E(i) and RR = (RN – i) / (1 + i)
The total risk-free return is calculated by: Rf(T) = 100 / P(T) – 1 where T = years
As we compare different time horizons, we use the Effective Annual Rate (EAR), the percentage
increase in funds invested over a 1-year period assuming compounding: 1 + EAR = (1 + Rf(T))^1/T
Annual rates on short-term investments (< 1 year) are often reported using simple interest, called
Annual Percentage Rates (APR). The relationship between the two:
1 + EAR = (1 + APR x T)^1/T and APR = ((1 + EAR)^T – 1) / T
The realized return of a portfolio is called the holding-period return and is defined as:
HPR in % = (Ending price – beginning price + cash dividend) / beginning price
But there is uncertainty about the price and dividend income, so we quantify the rates of return in
different scenarios (R) as a probability (p) weighted average of the rates of return. This is called the
expected return: E(r) = P x R. Then the standard deviation (variance and also called volatility)
captures the risk in squared deviations of the expected return: ^2 = P x (R – E(R))^2. As long as
the distribution is normal/symmetric around the mean , it’s a good measure of risk. The expected
return can also be estimated by the (historic) arithmetic average: E(r) = 1/n x R n = periods
The risk premium (reward) is the expected value of the excess return (difference between the actual
rate of return and the risk-free rate, the rate you can earn by leaving money in risk-free assets). This
trade of between reward and risk measures the attraction of a portfolio:
Sharpe ratio = risk premium / ^2 of excess return
These measures are a good start, but volatility may not cover all risks and what if the distribution
isn’t normal (bell-shaped graph with its peak at 0 and a of 1)? Because normal excess returns
hugely simplify portfolio selection. The measures skewness (which measures the place where most
values are concentrated, thus asymmetry) and kurtosis (which measures the degree of fat tails) are
used instead. Skewed to the left or negatively skewed, has most values concentrated on the left and
underestimates risk. Skewed to the right or positively skewed, has most values on the right and
overestimates risk. With kurtosis the ^2 will underestimate the likelihood of large losses as well as
large gains.
Measures that indicate vulnerability to extreme negative returns are the value at risk (VaR) and
expected shortfall or conditional tail expectation (CTE). The VaR measures potential loss that will
exceed a specific probability (assumes normality). Given a confidence level of 95% you sort the
observations from high to low and cut off the highest loss at the 5 th percentile. You can scale VaR to a
different time horizon by multiplying with T. CTE focusses on the average expected returns in the
5% left tail, thus the VaR will always be higher.
The four facts of asset returns that will remain the same in repeated experiments:
1. The correlation (degree of linearity between two variables) of daily returns is close to zero
and hard to predict.
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Summary Investments – Annamarie de Ruijter – VU University – 2018/19
, 2. The variance/volatility has positive autocorrelation (the degree of similarity between a graph
and the same in a different time horizon) for small horizons, because uncertainty isn’t solved
in one day and the random walk hypothesis (efficient markets).
3. The distribution of daily returns has fatter tails than normal, a greater chance of large losses.
4. The distribution of returns is asymmetric (negatively skewed), there are more large drops
than upward movements so the probability of low returns is higher.
Chapter 6 – Capital allocation to risky assets
Speculation is undertaking a risky investment for its risk premium, where the risk must be sufficient
to affect the decision and the expected profit greater than the risk-free alternative. A gamble is to
bet on an uncertain outcome for enjoyment of the risk itself. A fair game is a risky investment with a
risk premium of zero (gamble), so a risk-averse investor will reject it. Risk averse (A > 0) investors will
go for a low-risk portfolio, risk tolerant (A < 0) ones for a high-risk portfolio and risk-neutral ones will
solely judge on expected returns. These preferences and the tradeoff between expected return and
volatility of a portfolio are expressed as a utility function, with a penalty if the investor is risk averse.
Higher utility values are portfolios with more attractive risk-return profiles:
U = E(r) – ½ x A x ^2 where A is the risk-aversion coefficient
This tradeoff is graphically represented by indifference curves, where a favorable portfolio has a
higher expected return and lower variance (quadrant I). Every point representing a portfolio on the
line of the curve, are portfolios with the same utility value and are equally attractive to investors but
have different A’s. Higher indifference curves correspond to higher levels of utility. A portfolio may
be constructed with risky (P) and risk-free (F) assets to reduce overall risk. If you divide the total
amount of risky securities by the complete portfolio, you’ll get the relative proportions of these
assets within the portfolio (these aren’t changed). After this, the investor must choose the proportion
of the investment budget to these two categories (capital allocation). Suppose y is invested in P and
(1 – y) is left to invest in F, the return on the complete portfolio is:
Rc = y x r(p) + (1 – y) x rf E(rc) = y x E(rp) + (1 – y) x rf c = y x p
Firstly, we can solve the maximalization problem by replacing E(r) and , taking the derivative
(setting this to zero) and solving for y:
Max U = Rf + y x (E(rc) – rf) – ½A x y^2 x p y* = (E(rp) – rf) / A^2p
Otherwise you can trial and error with different values of y from 0 (only risk-free assets) to 1 (only
risky assets). The third method is to plot the indifferent curves, with different levels of volatility with
a fixed utility. The standard deviation of the portfolio is the standard deviations of the risky and risk-
free assets multiplied by the weight y and (1 – y). After this step, all the possible portfolios and risk-
return combinations are plotted in a plane as a straight line called the capital allocation line (CAL).
The slope of the line equals the increase in expected return of the complete portfolio per unit of
additional standard deviation (reward-to-volatility ratio). This expected return of the complete
portfolio is the same as the risky portfolio alone, because the
risk-free rate has zero : S = (E(rp) – rf) / p
The CAL is then compared with the indifferent curve in the
same graph. Now the investor must choose one optimal
portfolio with the highest utility, at the point on the utility
curve where taking more risk doesn’t reflect the additional
expected return (C). The optimal one (y*) in the risky asset is
proportional to the risk premium and inversely proportional
to the variance and degree of risk aversion. This happens
when the highest indifference curve is tangent to the CAL.
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Summary Investments – Annamarie de Ruijter – VU University – 2018/19
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