Investments
Summary of the Lectures
VU Bachelor - BK, EBE, IBA
Specialization - Finance
2020 - 2021
,Week 1 (Chapter 5-8)
Examples
Cumulative return of Heineken and Royal Dutch Shell between 1996 and 2015.
● Both Heineken and Shell appear to be volatile.
● There seems to be a co-movement between the companies -> there seems to be a
systematic risk component of returns.
● Also, the co-movement between the companies appears to be stronger in bad times.
QUANTIFYING RISK
Volatility – a good start for measuring risk. What are the limitations of volatility?
● Volatility is very helpful when returns are normally distributed because you only need
average return and volatility to measure risk.
SKEWNESS AND KURTOSIS
New measures to describe the shape of return distributions:
Know that a normal distribution has a skewness of 0 and a kurtosis of 3. This is why we
express kurtosis normally as excess kurtosis.
Skewness
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,Above: normal, positively skewed and negatively skewed.
● What does it mean when a distribution is negatively skewed? It means that a
distribution is tilted to the left. So the probability of observing a negative return is
slightly higher. The opposite is the case for positively skewed distributions.
● So with a negatively skewed distribution the volatility is going to understate the risk
of an asset and a positively skewed distribution volatility is going to overstate the risk
of an asset.
● This means that you would rather hold an asset with the highest possible skewness.
(Excess) Kurtosis
Above: normal and fat-tailed (more spreaded)
● What does it mean for a distribution to have positive (excess) kurtosis? The kurtosis
here is 0.35. If you look at the end of the left tail of the distribution, there is more mass
for the asset that has positive kurtosis. So when the times are bad, there is a bigger
chance of negative returns.
● So if an asset distribution has positive kurtosis the volatility is going to understate the
risk and an asset distribution with a negative kurtosis is going to overstate the risk.
This is why skewness and kurtosis are important measures for calculating risk. If the
distribution is not normally distributed, volatility is no longer a perfect measure.
COVARIANCE AND CORRELATION
Comovement measures. both work well with normally distributed assets:
● Covariance: quantifies the degree of co-movement between assets.
● Correlation: the standard diversion that moves between -1 and 1.
What are the limitations of covariance/correlation?
● When the distribution is nor normally distributed the measures are not perfect
anymore.
● Also, in bad times correlations tend to increase. So this means that the covariance and
the correlations are dependent on the state of the world. If you estimate
covariance/correlation in good times, you are going to understate the risk in bad times.
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,Above you can see the skewness and kurtosis of both stocks.
● Both have a positive skewness, which indicates that the probability of a positive return
is higher than with a normal distribution.
● Both have excess kurtosis, so both stocks have very fat tails. This indicates that the
probability of observing a very low return is much higher than under a normal
distribution.
THE REWARD-TO-VOLATILITY RATIO
The risk premium: expected excess return over the risk free rate
The reward-to-volatility (Sharpe) ratio:
EX: Sharpe Ratio of a portfolio with equal weights in Heineken and Shell in 2006 = 0.23
STYLIZED FACTS OF ASSET RETURNS
● Serial correlation of daily returns is close to zero. So returns are therefore very hard to
predict from the past.
● On the other hand, variance has positive autocorrelation for small horizons. When
there is a high daily volatility, it tends to stay that way for small horizons.
● The distribution of daily returns has fatter tails than the normal. Higher probability of
large losses (look at kurtosis of Heineken and Shell!). Increasing the holding horizon
(i.e. month, year) brings the distribution closer to normal.
● The distribution of returns is asymmetric (typically negatively skewed). There are
more large drops in returns than upward moves.
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,Above you see the return distribution of the two stocks.
● Note that the return autocorrelation is very close to 0, as stated in the first stylized fact.
There is very little predictability.
● Also the variance is positive, as stated in the second stylized fact.
● When you zoom in in the return distribution of Heineken you can see that the stock
has a positive kurtosis. This is consistent with the third stylized fact. Typically, stocks
have a higher kurtosis than the normal distribution would have.
RISK MEASUREMENT: VAR
Value-at-Risk (VaR): a measure of downside risk.
● It measures the potential loss over a specified horizon such that there is a (low)
probability 𝞪 that the actual loss will be larger.
● VaR as a quantile of the projected distribution of returns.
● The 5% VaR, commonly estimated in practice, is the return at the 5th percentile when
returns are sorted from high to low.
● So if you compute the 5% VaR, you are 95% confident your returns are not going to
be lower than the threshold value.
● VaR is the highest return of the worst cases!
VAR USING NORMAL DISTRIBUTION (First Approach)
If we assume that returns are normally distributed with mean μ and standard deviation 𝝈:
VAR USING NORMAL DISTRIBUTION: EXAMPLE
For an investment of $100 in a stock with annual mean of 10% and STDev of 15%, the 5%
1-year VaR (i.e. 95% confidence level) is:
$14,68 means that you are 95% certain that your losses are not going to be lower than $14,68.
TRANSFORMING THE TIME HORIZON
● If we increase the time horizon, VaR increases as well.
● The variance of a T-day return is T times the variance of a 1-day return.
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,If returns are independent and identically distributed:
VAR USING HISTORICAL RETURNS (Second Approach)
Assumption: the distribution of tomorrow’s returns r t+1 is well approximated by the
ₘ
empirical distribution of past observations.
Below: assumption returns are normally distributed -> estimating VaR at a given level 𝞪.
Expected shortfall is another measure for downside risk.
EXPECTED SHORTFALL
Also called conditional tail expectation (CTE)
● VaR measures number of losses, CTE measures magnitude of losses
○ VaR takes the highest return from the worst cases.
○ CTE takes an average return of the worst cases.
● Because of this, CTE More conservative measure of downside risk than VaR
CTE computes the expected value of losses that exceed the VaR:
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,CTE VS VAR: |CTE|>|VaR|
Remember: VaR is just a threshold level and CTE is an average.
● Note that when confidence interval increases, the VaR and CTE also increase because
we move more left of the distribution.
● Also, note that when the confidence interval increases the difference between VaR and
CTE also increases. This happens because the stocks have fat tails.
● How do we interpret the 0.022 normal distribution VaR of Heineken? We are 95%
confident that our daily losses are not going to be higher than 0.022.
● How do we interpret the 0.033 historical distribution VaR of Heineken? We are 99%
confident, using a historical returns approach, that over the next 1 day our losses are
not going to be higher than 0.033.
● At a 95% interval the average of the worst observations (CTE) of Heineken is going to
be 0.030.
RISK AND RISK AVERSION
How to rank portfolios based on investor’s preferences over their risk-return trade-off?
Assign a utility (welfare) score. If you face the following set of portfolios, what would you
choose?
We need some sort of formula to determine that.
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,MEAN VARIANCE UTILITY
Measures the utility/welfare/happiness achieved with a portfolio.
“A” is risk-aversion coefficient. Investors are:
● Risk averse if A>0
● Risk-neutral investor if A=0 -> Decisions are only based on E(r)
● Risk loving if A<0
In general investors are risk averse.
So the choice of portfolio depends on the risk aversion:
CAPITAL ALLOCATION ACROSS RISKY AND RISK-FREE ASSETS
Consider the case when the investor has to allocate her portfolio between risk-free
money-market securities (with return rf ) and a risky one (with return rE ).
The return of the portfolio is:
● w = the weight on the risky asset
PORTFOLIO OF ONE RISKY AND A RISK-FREE ASSET
The expected rate of return of the portfolio:
The standard deviation of the portfolio:
● 𝝈 E = the standard deviation of the risky asset
So with all formulas combined the max Utility with respect to w is:
Optimal weight in the risky asset:
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,When you look at the derivation of w we can see the following:
● The higher expected return the higher the weight
● the higher the risk averseness the lower the weight
● the higher the volatility the lower the weight
OPTIMAL PORTFOLIO ALLOCATION WITH ONE RISKY ASSET (EXAMPLE)
● When you have a w of 0 you are only holding Rf and with a w of 1 you are only
holding the risky asset
● You can vary the amount of weights
● The optimal portfolio is w 0.5 or 0.6 as the U is the highest there. See also below.
● From 0w to 0.5w the U is increasing, we are willing to take the risk in moderate levels
● After 0.6w you are taking too much risk and therefore there is a ‘penalty’ which
decreases U.
OPTIMAL PORTFOLIO ALLOCATION WITH ONE RISKY ASSET
What are the required portfolio returns for varying σP levels, such that utility remains
constant, i.e. the investor is indifferent between the risk-return combinations?
Thus, we can find the required expected portfolio return for different levels of risk (through
σP ), and for a given utility score.
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, OPTIMAL PORTFOLIO ALLOCATION WITH ONE RISKY ASSET (EXAMPLE)
Indifference curves:
● Why is it a curve? For higher levels of volatility we want an increasingly higher
expected return.
● The higher A, the curvier the curve gets because we want increasingly extra return for
higher volatility.
● For higher levels of utility, the curve simply shifts upward.
FINDING THE OPTIMAL PORTFOLIO USING INDIFFERENCE CURVES
● The black line represents all possible combinations between Rf and the risky asset.
● This is because with an increase in w of the risky asset the E(r) and volatility
increases.
● The lower indifference curve is not a good one because we can draw another one
which generates higher levels of utility
● However the higher one is not attainable because it does not intersect with the black
line
● So the middle one is the best one because it generates higher levels of utility than the
lower one and intersects with the black line, unlike the higher one.
● The place where the utility curve and the black line intersects is the optimal portfolio.
PORTFOLIOS OF TWO RISKY ASSETS
Consider an investor who allocates their portfolio between two risky assets:
● a stock fund (rE , 𝝈E)
● a bond fund (rD , 𝝈D)
The first step: determining the variance.
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