Financial Modelling and Derivatives - Summary of the lectures.
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Financial Modelling and Derivatives
Summary of the Lectures
VU Bachelor - BK, EBE, IBA
Specialization - Finance / Financial Management
2020 - 2021
,Week 1 (Chapter 10)
The value of $100 invested at the end of 1925
Conclusion from this graph: the higher the risk the higher the potential return.
The value of $100 invested for alternative investment horizons
Above you can see that when the time horizon increases, small stocks will get you less losses.
The value after one year has a lot of losses in small stocks, whereas the value after 20 years
has zero losses in small stocks.
Variables
Discrete Random Variables: only a countable number of distinct values.
Continuous Random Variables: an infinite number of possible values (example: height
measurement).
1
,Example BFI: Expected Return, Standard Deviation and Variance
With this probability distribution we can calculate the following:
Both are measures of the risk of a probability distribution.
In finance, the standard deviation of a return is also referred to as its volatility. The standard
deviation is easier to interpret because it is in the same units as the returns themselves
(example: percent).
Which company has a higher volatility, just from looks? AMC because the returns are more
divergent.
2
,Example BFI: Historical Returns
There are two ways the return of the stock can increase:
1. The company distributes a dividend
2. The price of the stock increases
Realized Return = Dividend Yield + Capital Gain Rate
It can also be that a company distributes dividends multiple times a year, for instance every
quarter. To calculate the historical returns the following assumption has to be made:
All the dividends are reinvested in new stocks of the company.
Example 10.2
3
,What do you observe from these numbers? Typically, Microsoft and S&P 500 move the same
way. In addition, Microsoft is more volatile than the S&P 500.
By counting the number of times a realized return falls within a particular range, we can
estimate the underlying probability distribution.
● Empirical Distribution: when the probability distribution is plotted using historical
data.
In the empirical distribution of annual returns you can see that the treasury bill distribution is
much more narrow (low volatility) and therefore easier to estimate in comparison to small
stocks (high volatility).
4
,Average Annual Return
Variance and Volatility of Returns
Above you can see the following: the higher the return the higher the volatility.
Estimation Error: Using Past Returns to Predict the Future
● We can use a security’s historical average return to estimate its actual expected return.
However, the average return is just an estimate of the expected return.
● Standard Error: a statistical measure of the degree of estimation error.
5
,Example 10.4 (Table 10.2 can be found on page 4)
The historical trade-off between risk and return
Investors are generally risk averse: they will never put together a portfolio that has more
volatility without being rewarded with higher return.
● Excess Returns: the difference between the average return for an investment and the
average return for T-Bills.
● With this excess return investors are being rewarded for the risk that they are taking.
This reward is called the Risk Premium.
● As you can see below: more volatility means more excess return (due to taking more
risk).
Above you can see a linear relationship: investments with a higher volatility have higher risks
and therefore a higher risk premium.
6
,The Returns of Individual Stocks
Is there a positive relationship between volatility and average returns for individual stocks?
● As shown below, there is no precise relationship between volatility and average return
for individual stocks.
● Larger stocks tend to have lower volatility than smaller stocks.
● All stocks tend to have higher risk and lower returns than large portfolios.
No linear relationship for individual stocks because individual stocks carry firm specific and
systematic risk. Portfolios diversified the firm specific risk out, so there is only systematic
risk (explanation below).
Common Versus Independent Risk
● Common Risk: risk that is perfectly correlated, risk that affects all securities
● Independent Risk: risk that is uncorrelated: risk that affects a particular security
Diversification: the averaging out of independent risks in a large portfolio
Diversification in Stock Portfolios: Firm-Specific Risk Versus Systematic Risk
● Firm Specific News: good or bad news about an individual company
● Market-Wide News: news that affects all stocks, such as news about the economy
Consider two types of firms:
● Type S firms are affected only by systematic risk. There is a 50% chance the economy
will be strong and type S stocks will earn a return of 40%; There is a 50% change the
economy will be weak and their return will be –20%. Because all these firms face the
same systematic risk, holding a large portfolio of type S firms will not diversify the
risk.
● Type I firms are affected only by firm-specific risks. Their returns are equally likely to
be 35% or –25%, based on factors specific to each firm’s local market. Because these
risks are firm specific, if we hold a portfolio of the stocks of many type I firms, the
risk is diversified.
7
,Above you can see that if you invest in more than 100 typical firms, the firm specific risk will
be eliminated and all that we are left with is systematic risk. It doesn't matter if you invest in
more than 100 firms, the risk stays the same as there is not any firm specific risk anymore.
No Arbitrage and the Risk Premium
The risk premium for diversifiable risk is zero, so investors are not compensated for holding
firm-specific risk. If this was not the case, investors would buy a lot of firms with firm
specific risk to eliminate firm specific risk through diversification. Because of this the price
of this portfolio would go up. This would set the risk premium to zero.
The risk premium of a security is determined by its systematic risk and does not depend on its
diversifiable risk.
● Is then volatility a useful measure in determining the stock’s risk premium? No
because volatility measures both systematic and firm specific risk for individual
stocks.
Going back to figure 10.7, this is the reason that individual stocks are not on the regression
line. The portfolios are diversified, only holding systematic risk.
Example: Take the most left individual stock in figure 10.7, with a volatility of 22% and
historical return of 9%. When you draw a straight line to the regression line you can see that
the stock has 14% systematic risk. Therefore the firm specific risk is 8%. The reason the
individual stock does not have more return because of the higher volatility is because
investors are not rewarded for firm specific risk, only for systematic risk - the risk premium
for firm specific risk is zero.
Measuring Systematic Risk
To measure the systematic risk of a stock, determine how much of the variability of its return
is due to systematic risk versus unsystematic risk. To determine how sensitive a stock is to
systematic risk, look at the average change in the return for each 1% change in the return of a
portfolio that fluctuates solely due to systematic risk.
8
, ● Efficient Portfolio: portfolio that contains only systematic risk. There is no way to
reduce the volatility of the portfolio without lowering its expected return.
● Market Portfolio: efficient portfolio that contains all shares and securities in the
market. This portfolio can be a good candidate for an efficient portfolio.
Sensitivity to Systematic Risk: Beta (β): the expected percent change in the excess return of a
security for a 1% change in the excess return of the market portfolio.
● How does beta differ from volatility? Beta captures only systematic risk, whereas
volatility captures both systematic and firm specific risk.
Example 10.8
Interpreting Beta (β): a security’s beta is related to how sensitive its underlying revenues and
cash flows are to general economic conditions. Stocks in cyclical industries are likely to be
more sensitive to systematic risk and have higher betas than stocks in less sensitive
industries.
● Which stocks have high betas? To have a high beta, the company needs to have a
large systematic risk. Example: technology stocks. These stocks are highly influenced
by the market. If the market performs well, there is a high demand for tech equipment.
If the market does not perform well there is less demand for tech equipment -> high
beta.
● Which stocks have low betas? To have a low beta, the company needs to have a small
systematic risk. Example: defensive (food) stocks. At all times people need food, so
when bad market news comes it won't affect the food stocks as much as other stocks
-> low beta.
Beta and the Cost of Capital
Estimating the Risk Premium in two steps.
● Market Risk Premium: the reward investors expect to earn for holding a portfolio with
a beta of 1.
9
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