The relationship between the covariance (the expected product of the deviations of two returns
from their means) and the correlation (a measure of the common risk shared by stocks that does not
depend on their volatility) can be expressed like this:
When the correlation (or covariance) is positive the stocks tend to move together and when the
correlation (or covariance) is negative the stock tend to move oppositely.
For the computation of a two security portfolio variance we use the following formula:
For the computation of a large portfolio variance (which is equal to the weighted average covariance
of each stock) we use the following formula:
For the case in which we would invest a portion of our money into risk-free assets and a portion into
an arbitrary risky portfolio, we can calculate the volatility of the entire portfolio the following way
(by using the formula for the two security portfolio variance):
In the above computation we can see that the risk-free assets carry no risk and therefore the risk of
the entire portfolio is simply calculated using the risk that the only risky asset in the portfolio
contributes to the overall risk of the portfolio.
We can identify the most efficient (highest amount of return for the risk we take) by using the
formula for the Sharpe-ratio, which is as follows:
,When we have a large portfolio and we want to calculate the contribution of a specific asset (i) to
the portfolio’s variance we can use the following formula:
When we subsequently want to calculate the contribution of that specific asset (i) to the market risk
premium of our portfolio we can use the following formula:
Knowing both of these figures we can calculate a reward-to-risk ratio of that specific asset (i) by
using the following formula:
We can also calculate the reward-to-risk ratio for the market portfolio (also called the market price
of risk):
The CAPM-model is the following model:
Bonds pricing is done using the following formulas:
The yield to maturity (YTM) of a bond is the interest rate that makes the present value of the bond’s
payments equal to its price in the market. So we find the YTM by solving the following bond-
valuation formula for the interest rate r:
,Financial options are contracts that give the owner the right (but not the obligation) to purchase (call
option) or sell (put option) an asset at a fixed price at some future date. The investor buying the
option is the one who can exercise the option and he has a long position in the option. The stock
option quotations can be understood as the following:
• An out-of-the-money option is when the option’s strike price (sell- or buy price agreed upon
in the contract) is lower than the market price of the underlying asset (in the case of a put
option) or higher than the market price of the underlying asset (in the case of a call option).
Thus, if exercised now, the holder of the option loses money on the option.
• An at-the-money option is when the option’s strike price is identical to the price of the
underlying security at the time the option is agreed upon. Thus, if exercised now, the holder
of the option gains and loses nothing on the option.
• An in-the-money option is when the option’s strike price is lower than the market price of
the underlying asset (in the case of a call option) or higher than the market price of the
underlying asset (in the case of a put option). Thus, if exercised now, the holder of the
option gains money on the option.
The payoffs at expiration of an option contract can be computed like so:
• For a long position in a call option, where S is the stock price (at expiration), K is the exercise
price and C is the value of the call option:
• For a long position in a put option, where S is the stock price (at expiration), K is the exercise
price and P is the value of the put option:
A short position in an option contract is taken by the investor that sells the option. That investor
thus has an obligation to provide the cashflow to the buyer of the option (who has a long
position in the option), if the option is exercised with profit by the holder of the option. Thus the
seller’s cash flows are the negative of the buyer’s cash flow.
Put-call-parity is based on the law of one price and it states that positions that provide exactly
the same payoffs (the purchase of the stock and a put vs. the purchase of a bond and a call)
should have the same price. So for a situation in which K is the strike price of the option, C is the
call price, P is the put price and S is the stock price, this gives us the following formula:
When we rearrange the terms of put-call-parity we get an expression for the price of a European
call option for a non-dividend-paying stock:
, Lecture week 1: Risk measurement
Skewness and kurtosis are measures that describe the shape of return distributions. A normal
distribution has a skewness of 0 and a kurtosis of 3. Skewness is a measure of how skewed (tilted)
the distribution of returns is and kurtosis is a measure of how fat the tails of the distribution of
returns is. A distribution of returns with exactly the same mean and exactly the same standard
deviation can look very different when they differ with respect to their skewness or kurtosis.
• A distribution of returns with a negative skewness has a distribution which is slightly tilted to
the left, therefore the probability of observing a negative return is slightly higher than a
normal distribution (with skewness 0). So volatility will understate the total risk of the asset
in this case (seeing as the probability of a negative return is higher than under a normal
distribution).
• A distribution of returns with a positive skewness has a distribution which is slightly tilted to
the right, therefore the probability of observing a positive return is slightly higher than a
normal distribution (with skewness 0). So volatility will overstate the total risk of the asset in
this case (seeing as the probability of a positive return is higher than under a normal
distribution).
This essentially means that while assets might have the same mean and standard deviation, they can
still differ a lot in which asset would be preferred, depending on the skewness of the distribution of
returns (with a positively skewed distribution being preferred).
With respect to kurtosis we can state the following:
• Positive excess kurtosis (which is in excess of the 3 we observe in a normal distribution)
means that the tails of the distribution of returns are fatter, meaning that when things go
bad in the economy the stock has a higher probability of experiencing a negative return. So
for stocks with this positive kurtosis, volatility will understate the total risk of the asset.
• Negative excess kurtosis means that the tails of the distribution are less fat, when compared
to a normal distribution, so when things go bad in the economy, the stock had a lower
probability of experiencing a negative return. Therefore, volatility will overstate the total risk
of the asset.
So we can state that if assets are not normally distributed, skewness and kurtosis can serve as
complementary risk measures to volatility.
Co-movement measures:
• Covariance: quantified degree of co-movement between assets
• Correlation: standardized version of covariance which ranges between -1 and 1.
There are actually also some issues with these measures of co-movement (just as there was with
volatility), seeing as when returns are not normally distributed they are not perfect measures and
during bad times stocks tend to be more correlated with each other. Therefore the covariance and
correlation are dependent on the state of the business cycle.
• Estimating the correlation in good times, can lead to an understatement of risk in bad times,
seeing as correlations generally increase in bad times and to minimize risk you want a
portfolio that is diversified in risk (with negative correlations).
• Estimating the correlation in bad times, can lead to an overstatement of risk in good times.
Since stocks are more correlated in bad times, which will increase risk estimates.
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