Lecture 2
Clip 1
Types of a 100-dollar investment:
1. Standard & Poor’s 500: industry leaders and among the largest firms traded on U.S.
Market -> 2nd highest return.
2. Small stocks: securities traded on the NYSE with market capitalization in the Bottom
201 -> Biggest risk and highest return.
3. Corporate bonds: long term, AAA rated US, not very risky, corporate bonds with
maturities of approximately 20 years.
4. Treasury bills: an investment in three-month government debt. There is a short
maturity. -> less return.
The long investment can weight out the risk a bit more than small investment, but you
always have risk. Corporate bonds and treasury bills only have the interest/coupon rate and
the face value at the end.
Realized returns: the return that actually occurs over a particular time period.
𝐷𝑖𝑣𝑡+1 + 𝑃𝑡+1
𝑅𝑡+1 =
𝑃𝑡
Return = dividend yield + capital gain rate.
Asset return as random variables:
- Probability distributions
- Expected returns
- Variance, volatility (=SD) = risk
2
Variance ( R) = 𝑥(𝑅 − 𝐸(𝑅)) = Σ Pr∗ (𝑅 − 𝐸(𝑅))
Standard deviation = √𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒
2
Volatility is mostly measured in % per annum. It is a measure of asset returns.
 𝜎𝑝𝑒𝑟𝑖𝑜𝑑𝑠 = 𝜎𝑝𝑒𝑟𝑖𝑜𝑑 ∗ √𝑇
 Daily to annually = 𝜎𝑎𝑛𝑛𝑢𝑎𝑙 = 𝜎𝑑𝑎𝑖𝑙𝑦 ∗ √252
Clip 2
We can’t predict the annual return of the future, so we look at the past. We look at the
average of the past returns.
̅ = 𝟏 𝚺𝑹𝒕
Average annual return = 𝑹
𝑻
Rt = realized return of a security in year t, for the years 1 through T.
Estimating the variance using the past:
𝑉𝑎𝑟(𝑅) =
1
Σ(𝑅𝑡 − 𝑟̅ )2
𝑇−1
Estimation error: how well is the estimation -> We use a securities historical average return
to estimate its actual expected return. However, the average return is just an estimate of the
expected error.
Standard error = SE =
𝑺𝑫(𝑹)
√𝑻
If you have an error, you can use a confidence interval = historical average return +- 1.96 *
standard error.
Clip 3:
Portfolio weights: a collection of assets is a portfolio. The fraction of the total investment in
the portfolio held in each individual investment in the portfolio. It weights up to 100%.
𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 𝑖
𝑋𝑖 =
𝑡𝑜𝑡𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜
Return of a portfolio: weighted average of the returns on the investments in the portfolio.
𝑅𝑝 = 𝑥1 𝑝1 + 𝑥2 𝑝2 + . . +𝑥𝑛 𝑝𝑛
Expected return on a portfolio: 𝐸 ∗ (Σ𝑥𝑖 𝑟𝑖 )
Clip 4
The volatility of two portfolios is less than the volatility of individual shares. So, investors
advice to diversify:
 John Templeton: “Diversify your investments”
 Warren Buffet: “Wide diversification is only required when investors do not
understand what they are doing”.
Diversification lowers risk in both directions:
- Downward: less weights on individuals
- Upward: there are more individuals
Clip 5
If you buy stocks from companies which move in the opposite direction, you overall risk will
go down. The amount of risk that is eliminated in a portfolio depends on the covariance and
correlation.
Covariance: the expected product of the deviations of two returns from their expected
value. Are they moving together? Yes -> positive.
𝐶𝑜𝑣(𝑅𝑖 , 𝑅𝑗 ) = 𝐸[𝑅𝑖 − 𝐸[𝑅𝑖 ])(𝑅𝑗 − 𝐸(𝑅𝑗 )
Historical covariance: 𝑐𝑜𝑣(𝑅𝑖 , 𝑅𝑗 ) =
1
𝑇−1
̅̅̅
Σ(𝑟𝑖,𝑡 − 𝑟̅̅̅
𝑖 )(𝑟𝑗,𝑡 − 𝑟𝑗 )
Correlation: measure of the common risk shared by stocks that does not depend on their
volatility. It measures a linear relationship between ri and rj. So if ri changes with p%, we
expect Rj changes with Pi,j, * p%.
𝑐𝑜𝑣(𝑟𝑖 , 𝑟𝑗 )
𝑐𝑜𝑟𝑟(𝑟𝑖 , 𝑟𝑗 ) = 𝑝𝑖,𝑗 =
𝑆𝐷(𝑟𝑖 ) ∗ 𝑆𝐷(𝑟𝑗 )
Variance of a portfolio:
𝑉𝑎𝑟(𝑅𝑝) = 𝑥12 ∗ 𝑉𝑎𝑟(𝑟1) + 2 ∗ 𝑥1 ∗ 𝑥2 ∗ 𝑐𝑜𝑣(𝑟1, 𝑟2) + 𝑥22 ∗ 𝑉𝑎𝑟(𝑟2)
Lecture 3
Clip 1
The variance of a portfolio: weighted average covariance of each stock with the portfolio.
𝑣𝑎𝑟(𝑅𝑝 , 𝑅𝑝 ) = Σ𝑥𝑖 𝑐𝑜𝑣(𝑅𝑖 , 𝑅𝑝 )
If there is a high covariance, everything moves together.
Equally weighted portfolio: a portfolio in which the same amount is invested in each stock.
 Variance of equally weighted: 𝑉𝑎𝑟 (𝑅𝑝 ) = Σ𝑖Σ𝑗𝑥𝑖 𝑥𝑗 𝑐𝑜𝑣(𝑅𝑖 , 𝑅𝑗 )
N = variance terms, 𝑛2 − 𝑛 = covariance terms.
1
1
 𝑉𝑎𝑟 (𝑅𝑝 ) = 𝑛 (𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑖𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙 𝑠𝑡𝑜𝑐𝑘𝑠) + (1 − 𝑛) ∗
(𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑐𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡ℎ𝑒 𝑠𝑡𝑜𝑐𝑘𝑠)
The bigger the N, individual risk become less important, and the covariance gets more
important (=correlated risk).
For a portfolio with arbitrary weights, the SD is calculated: 𝑆𝐷 (𝑅𝑝 ) = Σ𝑖𝑥𝑖 ∗ 𝑆𝐷(𝑅𝑖 ) ∗
𝑐𝑜𝑟𝑟(𝑅𝑖 , 𝑅𝑝 ) -> is always smaller than one.
Unless all the stocks in a portfolio have a perfect positive correlation with one another, the
risk of the portfolio will be lower than the weighted average volatility of the individual
stocks.
 𝑆𝐷(𝑅𝑝 ) = Σ𝑖 𝑥𝑖 𝑆𝐷(𝑅𝑖 )𝑐𝑜𝑟𝑟(𝑅𝑖 , 𝑅𝑝 ) < Σ𝑖 𝑥𝑖 𝑆𝐷(𝑅𝑖 )
Clip 2
Inefficient portfolio: it is possible to find another portfolio that is better in terms of both
expected return and volatility.
Efficient portfolio: there is no way to reduce the volatility of the portfolio without lowering
its expected return.
There are more diversification benefits when your correlation is lower.
Short position: a negative investment in a security. You sell a stock that you do not own and
then buy that stock back in the future. -> advantageous strategy if you expect a stock price
to decline in the future.
Efficient frontier: the line that defines that the highest possible expected return for a given
level of volatility are those on the northwest edge of shaded region. -> no stock on his own
on this line, so we shouldn’t put everything.
Risk can be reduced by investing a portion of a portfolio in a risk-free investment, like T-Bills.
However, doing so will likely reduce the expected return.
On the other hand, an aggressive investor who is seeking high expected returns might
decide to borrow money to invest even more in the stock market.
Arbitrary risk portfolio with risk stocks and risk free treasury bills: 𝐸(𝑅𝑥𝑝 ) = 𝑟𝑓 +
𝑥(𝐸(𝑅𝑝 ) − 𝑟𝑓 )
To earn the highest possible expected return for any level of volatility, we must find the
portfolio that generates the steepest possible line. This combined with the risk-free
investment.
𝑆ℎ𝑎𝑟𝑝𝑒 𝑟𝑎𝑡𝑖𝑜 (𝑅𝑒𝑡𝑢𝑟𝑛 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑜𝑓 𝑣𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦) =
=
𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑒𝑥𝑐𝑒𝑠𝑠 𝑟𝑒𝑡𝑢𝑟𝑛
𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑣𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦
𝐸(𝑅𝑝 ) − 𝑟𝑓
𝑆𝐷(𝑅𝑝 )
The portfolio with the highest Sharpe ratio is the portfolio where the line with the risk-free
investment is tangent to the efficient frontier of risky investments -> Tangent portfolio.
An investor’s preferences will determine how much to invest in the tangent portfolio vs risk
free.
- Conservative: low tangent
- Aggressive: high tangent
- Both types will hold the same portfolio of risky assets -> efficient portfolio.
Clip 3
If we have a risky portfolio P, we add a security if it increases the sharp ratio.
Adding a small amount of x of stock I, changes the weights of the portfolio to: 𝑥𝑖 = 𝑥𝑖 𝑥𝑝 =
1
 Excess return: E(𝑅𝑝 ) − 𝑟𝑓 + (𝑥𝐸(𝑅𝑖 ) − 𝑥𝑟𝑓 )
 𝑆𝐷(𝑅𝑝 ) = 𝑥𝑆𝐷(𝑅𝑖 )𝑐𝑜𝑟𝑟(𝑅𝑖 , 𝑅𝑝 ) + 𝑆𝐷(𝑅𝑝 )𝑐𝑜𝑟𝑟(𝑅𝑝 , 𝑅𝑝 )
We only invest share i, if the investment increases the sharp ratio:
𝐸(𝑅𝑝 ) − 𝑟𝑓 + 𝑥𝐸(𝑅𝑖 ) − 𝑥𝑟𝑓
𝑐𝑜𝑣(𝑅𝑖 , 𝑅𝑒𝑓𝑓 )
≥
𝑆𝐷(𝑅𝑝 )
𝑥𝑆𝐷(𝑅𝑖 )𝑐𝑜𝑟𝑟(𝑅𝑖 , 𝑅𝑝 ) + 𝑆𝐷(𝑅𝑝 )
 𝐸(𝑅𝑖 ) ≥ 𝑟𝑓 + 𝛽𝑖,𝑝 (𝐸(𝑅𝑝 ) − 𝑟𝑓 )
How big should the excess return be of any share to invest in? 𝛽
𝑆𝐷(𝑅𝑖 )𝐶𝑜𝑟𝑟(𝑅𝑖 ,𝑅𝑝 )
𝑆𝐷(𝑅𝑝 )
=
𝐶𝑜𝑣(𝑅𝑖 ,𝑅𝑒𝑓𝑓 )
𝑉𝑎𝑟(𝑅𝑒𝑓𝑓 )
Clip 4
Capital Asset Pricing Model = allows us to identify the efficient portfolio of risky assets
without having any knowledge of the expected return of each security.
 It uses the optimal choices investors make, to identify the efficient portfolio as the
market portfolio. The portfolio of all stocks and securities in the market.
Three CAPM assumptions:
1. Investors can buy and sell all securities at competitive market prices without
incurring taxes or transactions costs and can borrow and lend at risk free interest
rates.
2. Investors hold efficient portfolios of traded securities- portfolios that yield the
maximum expected return for a given level of volatility.
3. Investors have homogeneous expectations regarding SD, corr and E.
a. Homogeneous: all investors have the same estimates concerning the future.
 All investors will demand the same efficient portfolio.
 Combined portfolio = efficient portfolio.
 Supply securities = market portfolio. Demand for market portfolio = supply of market
portfolio.
S&P 500: is used as proxy for the market portfolio in the US.
AEX: in the Netherlands
MSCI world index: for the whole world. These are value weighted = 𝑥𝑖 =
𝑚𝑎𝑟𝑘𝑒𝑡 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑖
𝑡𝑜𝑡𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑎𝑙𝑙 𝑠𝑒𝑐𝑢𝑟𝑡𝑖𝑒𝑠
Capital market line: when the tangent line goes through the market portfolio. ->
𝑆𝐷(𝑅𝑥 𝐶𝑀𝐿) = 𝑥𝑆𝐷(𝑅𝑚𝑘𝑡 )
This is a combination of risky and risk free.
With the Market Security Line we use the B instead of volatility.
Expected return on a efficient market portfolio (CAPM)
= 𝐸(𝑅𝑖 ) = 𝑟𝑓 + 𝛽𝑖𝑚𝑘𝑡 (𝐸(𝑅𝑚𝑘𝑡 ) − 𝑟𝑓 )
Risk premium = (𝐸(𝑅𝑚𝑘𝑡 ) − 𝑟𝑓 )
Lecture 4
Clip 1
Capital Asset Pricing model: practical way to estimate the equity cost of capital. This is
linked with the Security Market Line equation.
 𝑟𝑖 = 𝑟𝑓 + 𝛽 ∗ (𝑅𝑚𝑘𝑡 − 𝑟𝑓 )
Yield to Maturity: interest rate return an investor will earn from holding the bond to
maturity and receiving its promised payments.
- Little risk: YTM is a reasonable estimate of investors expected rate of return.
- Significant risk of default: YTM will overstate investors’ expected rate of return.
- 𝑟𝑑 = 𝑦 − 𝑝𝐿 = 𝑌𝑇𝑀 − (𝑝𝑟𝑜𝑏(𝑑𝑒𝑓𝑎𝑢𝑙𝑡) ∗ 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑙𝑜𝑠𝑠 𝑟𝑎𝑡𝑒)
All equity comparable:
- Find an all equity financed firm in a single line of business, that is comparable to the
project.
- Use the comparable firm’s equity beta and cost of capital as estimates.
- Assets = debt + equity
𝐸
𝐷
 𝛽𝑢 = 𝐸+𝐷 𝛽𝑒 + 𝐸+𝐷 𝛽𝑑
𝐸
𝐷
 𝑅𝑤𝑎𝑐𝑐 = 𝐸+𝐷 𝑟𝑒 + 𝐸+𝐷 𝑟𝑑 (1 − 𝜏)
These are all in market values, not book values. The E = market capitalization.
𝐷
Target leverage ratio = 𝑟𝑤𝑎𝑐𝑐 = 𝑟𝑢 − 𝐷+𝐸 𝑟𝑒 𝑟𝑑
Clip 2
How can the CAPM be implemented in practice?
1. Market portfolio: the market reports the value of a particular portfolio of securities.
2. Risk free rate: the US treasury bonds, with the same horizon as the firm’s investors. A
lot look at maturity of 10,20,30 years.
3. Market risk premium: estimate by using historical average excess return over the riskfree rate. There are two drawbacks:
a. Standard errors of the estimates are large.
b. Might not represent current expectations.
𝐷𝑖𝑣
You could also do: Dividend yield + expected dividend growth = 𝑝 + 𝑔. It is typical between
4% and 7%
4. Beta estimation: expected change in the excess return of a security for a 1% change
in the excess return of the market portfolio.
- Linear regression: 𝑎𝑖 + 𝛽𝑖 (𝑅𝑚𝑘𝑡 − 𝑟𝑓 ) + 𝜀𝑖
a=intercept from the regression. 𝛽𝑖 (𝑅𝑚𝑘𝑡 − 𝑟𝑓 )= sensitivity of the stock to market risk. 𝜀𝑖 =
error term, deviation from the best fitting line and is zero on average.
 When E(𝜀𝑖 = 0)= cost of capital = 𝐸(𝑟𝑖 ) = 𝑟𝑓 + 𝛽 ∗ (𝑅𝑚𝑘𝑡 − 𝑟𝑓 ) + 𝛼𝑖
 = risk adjusted performance measure for the historical returns.
-  is positive: stock performed better than CAPM predicted.
-  is negative: below SML
- If the market portfolio was efficient, the stocks would be on SML and a=0.
Considerations when estimating Beta:
1. Time horizon: use at least two years of weekly data. You need enough data.
2. Market proxy: is it correct? A lot use S&P 500.
3. Beta variation and extrapolation: beta varies, so use average industry betas rather
than individual stock betas. When companies mature, they will get closer to 1.
4. Outliers: you can fix this -> winsorize (= set all outliers to a specific percentile of
data), Truncate (= drop all outliers above a percentile) and larger sample.
Clip 3
𝐶1
Spotify worked with direct listing -> growing perpetuity: 𝑉0 = 𝑟 −𝑔
𝑎
Assumptions:
1. Longer horizon of Treasury -> r goes up -> value goes down.
2. Spotify is riskier -> B goes up -> value goes down.
3. Growth rate is higher -> value goes up.
4. Debt is positive -> value goes down.
5. Cash flow has been lower historically -> cash flows go down -> value goes down.
Clip 4
A single factor model uses one portfolio.
Multi factor model uses more than one portfolio, so more betas.
Self-financing portfolio: going long in some stocks and going short in other stocks with equal
market value.
- Go long in high risk and short in small risk.
-
Going short: selling a stock which an investor doesn’t own.
Small-minus-Big portfolio: a trading strategy that each year buys a portfolio of small stocks
and finances this position by short selling a portfolio of big stocks, has historically produced
positive risk-adjusted returns.
High-minus-low portfolio: each year buys an equally weighted portfolio of stocks with a
book to market ratio greater than the 70th percentile of NYSE firms and finances this with
short selling an portfolio with a ratio less than the 30% of NYSE stocks -> positive risk
adjusted returns.
Momentum portfolio: buys the top 30% of stocks and finances this position by short selling
bottom 30% of the stocks -> positive risk adjusted returns -> prior one-year momentum.
Fama-French-Carhart: 𝐸(𝑅𝑠 ) = 𝑟𝑓 + 𝛽𝑠𝑚𝑘𝑡 (𝐸(𝑅𝑚𝑘𝑡 ) − 𝑟𝑓 ) + 𝛽𝑠𝑠𝑚𝑑 (𝐸(𝑅𝑠𝑚𝑑 )) +
+𝛽𝑠ℎ𝑚𝑙 (𝐸(𝑅ℎ𝑚𝑙 )) + 𝛽𝑠𝑚𝑜𝑚 (𝐸(𝑅𝑚𝑜𝑚 ))
There is much debate whether the FFC is a significant improvement over the CAMP. It works
better in the risk of actively managed mutual funds.
Profitability factor (RMW): long firms with robust (high) operating profitability and short
those with weak (low) profitability.
Investing factor (CMA): long firms investing conservatively and short those investing
aggressively -> makes HML factors redundant.
Lecture 5
Clip 1
Efficient markets hypothesis: in an efficient capital market the asset price fully reflects all
available information.
Implications:
1. Because information is reflected in prices immediately, there is no advantage and
investors should expect a normal rate of return.
2. Awareness of information does an investor no good -> no advantage.
3. The price adjusts before the investor has time to trade on it. Because there is a lot of
information.
 Stocks follow the random walk. They are always priced on the level of information
that is available. It moves up and down randomly.
There are 3 types of market efficiency:
1. Weak form efficiency: current prices reflect historical prices. Chartism (technical
analyses: astrology for the finance) = looking for a pattern. This is useless, because
everyone is doing this, you can’t profit.
2. Semi strong form efficiency: prices reflect all public information. Most financial
analyses is useless, because it is hard to perform an financial analyses only on public
information. The annual report is already reflected on the market price.
3. Strong form efficiency: prices reflect all information that is known. Nobody can
consistently make superior profits. There is also private information available.
Eugene Fama: Efficient market Theory. Lars Peter Hansen: meets. Robert Schiller: Behavior
Finance. Richard Taylor: Behavior Economics, Behavior Finance.
Three Pillars of Market Efficiency:
1. Rationality: if everyone who trades is rational. Prices should reflect the information
perfectly.
2. Independent Deviations from Rationality: if people are biased, they should be
unsystematically biased. Then EHM still holds.
3. Unlimited arbitrage: then it still holds.
People can react in different ways:
- Underreaction (momentum): average returns following quarterly announcements ->
slow reaction.
- Overreaction: bubbles.
Biases have the potential to become systematic. It is very unlikely that al people are rational.
Arbitrage involves the simultaneous purchase and sale (or short sale) of securities which are
perfect substitutes to lock in a risk free profit.
- Law of one price: identical assets have the same price, otherwise there would be an
opportunity for riskless profit (=arbitrage holds the law).
Types of arbitrages:
1. Convertible: currency/derivatives.
2. Perfect substitutes: Shell listed in Amsterdam and London.
3. Statistical arbitrage of close substitutes: not riskless. E.g. buy undervalued BMW,
Short Daimler.
Limits to arbitrages: theoretically arbitrage is riskless but that is not in reality.
1. Implementation costs: liquidity constraints illiquid markets.
2. Noise trader risk: irrational traders can worsen the situation.
3. Fundamental risk: asset value changes, no perfect hedge.
4. Model risk: bad model problem in statistical arbitrage.
5. Synchronization risk: timing game, when do you go short.
Clip 2
2 predictions of the Efficient Market:
1. “Price is right”: prices reflect available information, so accurate signals for resource
allocation.
2. “No free lunch”: market prices are impossible to predict, so hard for investors to beat
the market after taking risk into account.
Twith: online platform for livestreaming. Audience looks, mostly for videogames.
- Chat function = the modern monkeys
- Mike Roberts did it with stocks.
- Popular stocks: DOG, CAKE, MLP, CUK.
- Rule: when the portfolio is less than 25K, Stockstream stops.
Buffet Bed: Vanguard index fund
 Protégé Partners: portfolio of 5 selected hedge funds. Fees are included.
 S&P 500 does it better because of the fees. You cannot beat the market.
Causality is key. When the “price is right”, arbitrage opportunities cannot be exploited -> no
free lunch -> price cannot be right.
 Arbitragers cannot take “arbitrarily large positions”.
Clip 3
How to identify Alpha? -> to improve the performance of their portfolio’s. Investors will
compare the expected return of a security with its required return from the security market
line.
Alpha: the difference between the expected return of a security (=opportunity costs) with
the required return from the security market line.
 𝑎𝑠 = 𝐸[𝑅𝑠 ] − 𝑟𝑠
When the market portfolio is efficient, all stocks are on the SML and have a=0.
Profiting from non-zero aloha stocks: improve performance by buying positive A stocks and
selling negative A stocks.
- Informed investors can profit from uninformed investors, if they don’t hold the
market portfolio.
- The investment strategy does not depend on the quality of an investor’s information
or trading skill.
Market is inefficient when:
1. Investors misinterpret information and think they have a positive alpha when they
have a negative.
2. Care about other aspects than return and volatility, so hold inefficient portfolios.
Rational expectations: everyone is in the market portfolio.
Clip 4
1. Individual investors fail to diversify their portfolio.
a. Familiarity bias: favor investments in companies with which they are familiar.
b. Relative wealth concerns: they want a higher performance than for example
their friends.
2. According to CAPM: hold risk free assets in combination of all risk securities. This is
not in real life.
a. Overconfidence bias: they believe they can pick winners and losers. As a
result, they trade too much.
b. Sensation seeking: individual’s desire for novel and intense risk-trading
experience.
3. Hanging on to losers and disposition effect: an investor holds on to stocks that has
lost their value and sell stocks that has risen in value since the time of purchase.
4. Individuals buy things that have been in the news. Attention, Mood and Experience.
5. Herd behavior: investors make similar trading errors because they are actively
following each other’s behavior.
6. Informational cascade effects: people don’t have much information their selves, so
profit from other information.
Implications of behavioral biases: informed investors can profit from the negative alphas of
other investors.
Stock recommendations -> news? -> stock price increase -> no news? -> more stock increase
-> short this share, it’s hard because of the illiquidity of the unknown companies.
Fund manager value added: most fund managers appear to trade so much that the trading
costs exceed the profits.
Return to investors: actual returns to investors have a negative alpha.
Momentum strategy: buying high returns, selling low returns.
Lecture 6
Clip 1
Derivative: financial instrument whose value depends on or is derived from the value of
another asset.
- Examples: futures, forwards etc
- Used for: hedging, speculation, arbitrage and financial engineering.
They have a key role in transferring risks in the economy. They are traded in:
- Exchange listed: Chicago Board Option Exchange
- OTC: traders working for banks, fund managers and corporate treasures contact each
other directly.
Exchange traded derivatives: contracts are standardized and open outcry system vs
electronic trading.
OTC derivatives:
 Telephone-computer linked network of dealers.
 Contracts negotiated over the phone and be tailored to clients’ need.
 There is a default risk, because of the direct trading.
 Biggest OTC: interest rate, foreign exchange, credit derivatives.
Clip 2
Forward contract: hedging the risk of price fluctuations. It’s the oldest derivative in the
world, from the agricultural.
 Contract: buy and sell an asset at a pre-determined future amount for a certain time
agreed upon today.
 Trades OTC
 Long position: the party that agrees to buy.
 Short position: the party that agrees to sell.
 Payoff from a long position = St-K, Where St = spot rate and K = forward rate.
Clip 3
Future contract is an agreement to buy/sell an asset for a certain price at a certain time.
- Highly standardized in comparing to forward contract.
-
Traded on exchanges. There is no credit risk.
Specifications future contract:
 Asset
 Size
 Delivery location
 Delivery time
 Price quotes, limits and position limits
 Don’t put the entire value of the account, but a sufficient fud in your margin account.
o Margin account: amount of money deposited by an investor with the broker,
to minimize the possibility of a loss through a default of the contract.
Characteristics of futures:
- They are settled daily.
- Settlement price: the price just before the final bell each day and used for the daily
settlement process.
- Closing out a futures position involves entering an offsetting trade. Most contracts
are closed out before maturity.
- Open interest: number of contracts outstanding equal to the number of long
positions or number of short positions.
Clip 4
Forward long
𝑆𝑡 − 𝐹𝑡𝑇
Unlimited
Limited (max: 𝐹𝑡𝑇 )
Pay off
Possible gain
Possible loss
Forward short
𝐹𝑡𝑇 − 𝑆𝑡
Limited (max: 𝐹𝑡𝑇 )
Unlimited
Forward prices assumptions:
1. No transaction costs.
2. All trading profits are taxed at the same time.
3. Risk free rate of interest is the same for borrowing and lending.
4. Market participants will take advantage or arbitrage opportunities as they occur.
𝑅 𝑚
Compounding: 𝐴 (1 + 𝑀)


𝑐
𝐷𝑖𝑠𝑐𝑜𝑢𝑛𝑡 = 𝑒 𝑟𝑡
This is continuously.
S0 = spot price today.
F0 = future or forward price today.
T = time until delivery date.
R = rf interest rate for maturity T.
 𝐹0 = 𝑆0 ∗ 𝑒 𝑟𝑡
𝑇
 = (1 + 𝑟𝑓 ) ∗ 𝑆0
 𝐹0 = 𝑒 𝑟𝑓 ∗ 𝑆0
When a stock has dividend:
 𝐹0 = (𝑆0 − 𝐼) ∗ 𝑒 𝑟𝑇 , where I = PV(Div)
 𝑃𝑣 𝑜𝑓 𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑 = 𝐴 ∗ 𝑒 −𝑟𝑓∗𝑡
When an investment provides a yield (used for stock index futures):
 𝐹0 = 𝑆0 ∗ 𝑒 (𝑟−𝑞)∗𝑇
 𝑞 = 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑦𝑖𝑒𝑙𝑑.
When investing in foreign currencies:
 𝐹0 = 𝑆0 ∗ 𝑒 (𝑟−𝑟𝑓)𝑇
 𝑟 = 𝑑𝑜𝑚𝑒𝑠𝑡𝑖𝑐 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒
 Rf = foreign interest
 S0 kan ook de exchange rate zijn.
When having negative income, like storage costs:
 𝐹0 = 𝑆0 ∗ 𝑒 (𝑟+𝑢)𝑇
 𝑢 = 𝑠𝑡𝑜𝑟𝑎𝑔𝑒 𝑐𝑜𝑠𝑡 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑎𝑠 𝑎 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑠𝑠𝑒𝑡 𝑣𝑎𝑙𝑢𝑒
 𝐹0 = (𝑆0 + 𝑈) ∗ 𝑒 𝑟𝑇
𝑐
 𝑈 = 𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑠𝑡𝑜𝑟𝑎𝑔𝑒 𝑐𝑜𝑠𝑡𝑠 = 𝑒 𝑟𝑓
Investment assets: held by significant numbers of people purely for investment purposes.
Consumption assets: held primarily for consumption.
Forward contract is worth 0 (except bid offer spread). When it is first negotiated. Later it will
be positive or negative.
Lecture 7
Clip 1
Option: it gives the holder the right to buy or sell at a certain price -> OTC and on exchange.
A future/forward contract gives the holder the obligation to buy or sell at a certain price,
because you said you will do it.
2 types of option:
1. Call option: buyer of the contract (holder) has the right to buy the asset by a certain
time for a certain price.
2. Put option: the holder has the right to sell the asset by a certain time for a certain
price.
Terminology:
 Exercise price/strike price: price in the contract.
 Expiration date/maturity: date in contract.
 There are two options:
o European: the right to buy or sell can be done only on the expiration date.
o American: can be done anytime up to the expiration date.
Moneyness of an option: where the price in the contract (the exercise) is relative to the
current stock price.
- Higher/positive: holder will exercise the price -> in the money
- Lower: out the money
- Equal: at the money
Intrinsic value: value option is exercised immediately.
Clip 2
Long option in an option contract: the value at expiration:
 C= max(S-K,0)
 S = share price, K = exercise price, C = value of the call, max = max of the two
quantities in the parentheses.
It can never go below zero, because you have the right. Otherwise, you won’t buy it.
Put option is the opposite of a call option:
 P = (K-S, 0)
Long option = positive correlated and short option is negative correlated. That’s why they
ask for a fee.
The higher the exercise price, the lower the price of the option. The lower the call option
price, the higher is the chance that the share price will be above the exercise price.
Option pays off profiles:
1. Buying a call: going long in a call -> right to buy.
2. Selling a call: writing a call/going short in a call -> obligation to sell.
3. Buying a put: going long in a put -> right to sell.
4. Selling a put: writing a put, going short in a put -> obligation to buy.
Options vs forward/futures:
- Forward/futures: are obligations and options have on party with an option and one
with an obligation.
- Forward/future: set strike price, no money exchanges after the contract closed.
- Option: set cost of opinion at time 0.
Trading strategy: you can combine a few:
 Covered call, protective put: position in one option.
 Bull/bear/box/butterfly: spread in two or more option of the same type.
 Straddle/strip/strap: combination of call and puts.
Straddle: long a call and put option on the same stock, exercise date, stike price. When
investors expect the stock to be volatile and move up or down a large amount but no idea
which direction -> more money when strike and stock price are far apart.
Strangle: you do not receive money if the stock price is between the two stock prices.
Butterfly spread: go long in two call options with different strike prices and short in two call
options with a strike price equal to the average strike price of the first two calls. This makes
more money when stock and strike prices are close.
Clip 3
European options are non lineair:
- Call: max{St – K, 0}
- Put: max{K-St, 0}
𝑆0 = 𝑠𝑡𝑜𝑐𝑘 𝑝𝑟𝑖𝑐𝑒 𝑡𝑜𝑑𝑎𝑦
𝐾 = 𝑠𝑡𝑟𝑖𝑘𝑒 𝑝𝑟𝑖𝑐𝑒
𝑇 = 𝑙𝑖𝑓𝑒 𝑜𝑓 𝑎𝑛 𝑜𝑝𝑡𝑖𝑜𝑛
𝜎 = 𝑣𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑠𝑡𝑜𝑐𝑘 𝑝𝑟𝑖𝑐𝑒
𝐷 = 𝑃𝑉 𝑜𝑓 𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑𝑠 𝑝𝑎𝑖𝑑 𝑑𝑢𝑟𝑖𝑛𝑔 𝑙𝑖𝑓𝑒 𝑜𝑓 𝑎𝑛 𝑜𝑝𝑡𝑖𝑜𝑛
𝑟 = 𝑟𝑖𝑠𝑘 𝑓𝑟𝑒𝑒 𝑟𝑎𝑡𝑒 𝑓𝑜𝑟 𝑚𝑎𝑡𝑢𝑟𝑖𝑡𝑦 𝑇 𝑤𝑖𝑡ℎ 𝑐𝑜𝑛𝑡. 𝑐𝑜𝑚𝑝
All of these factors affect the option prices.
Upper bound call: it can never be worth more than the stock price -> 𝐶 ≤ 𝑆𝑡
𝐾
Lower bound call: can never be less than its intrinsic value -> 𝐶 ≥ max(𝑆0 − (1+𝑟)𝑡 , 0)
Call option is worth more than the stock price, then short the call and make riskless profit.
European call is worth less than it’s intrinsic value, then the investor buys the call, short the
stock and invest k/(1+r)^t in a riskless bond. This is riskless profit.
American call is worth less than it’s intrinsic value, then investor buys the call, exercise it and
sell the share in the market.
Upper/lower bounds put
 The most a put can be worth is when the value of the stock is zero
o American: = 𝑃 ≤ 𝐾
o European: P < of gelijk aan k/(1+r)^T
 Least a put can be worth is the instrinsic value.
o American: P  max(K-S0, 0)
𝐾
o European: 𝑃 ≥ max((1+𝑟)𝑡 − 𝑆, 0)
Upper bound put: when a European put option is worth more than k/(1+r)^T, investors will
short the put and invest it in a riskless bond that will be worth K, which is riskless profit.
Lower bound put: when a European put is worth less than it’s intrinsic value, an investor will
buy the put, borrow k/(1+r)^T and buy and hold the share.
For American puts, the time value can be negative when the stock price is at it’s minimum.
Clip 4
Two possibilities for protecting against price drop:
 Protective put (stock + put)
 Riskless bond + call
 Same payoffs, same price.
 𝑆 + 𝑃 = 𝑃𝑉(𝐾) + 𝐶, is the put, call parity.
Assumptions:
- Perfectly comparable put option in the market.
- Put option is prices correctly in the market.
Lecture 8
Clip 1
Binominal option pricing model: a technique for pricing options based on the assumption
that each period, the stock’s return can take only two values. It can go up or down.
Binominal tree: a timeline with two branches at every date representing the possible events
that could happen at those times.
Replicating portfolio: portfolio consisting of a stock and a risk free bond that has the same
value and pay offs in one period as an option written on the same stock.
- Law of one price: current value of the call and the replicating portfolio must be equal.
What is it worth today?
 ∆ ∗ 𝑆0 + 𝐵 = 𝐶
𝐶 −𝐶
𝐶𝑑 −∆𝑆𝑑
∆ = 𝑆𝑢−𝑆 𝑑 , 𝐵 = (1+𝑟)
𝑇
𝑢
𝑑
Clip 2
There are no expectations:
- No need for the probabilities. They are implicit in the share price.
- One with the price of 20, won’t have a big chance to go to 90.
Risk neutral probabilities: because of everyone is risk neutral, the financial assets have the
same cost capital, risk free rate of interest.
 𝑝𝑆𝑢 + (1 − 𝑝)𝑆𝑑 = 𝑠(1 + 𝑟)𝑇
𝑃 ∗𝐶 +(1−𝑝)∗𝐶𝑑
 𝐶 = 𝑢 𝑢 1+𝑟
𝑆(1 + 𝑟)𝑇 − 𝑆𝑑
𝑃=
𝑆𝑢 − 𝑆𝑑
No assumption on the risk preferences of investors is necessary to calculate the option price
using the Binominal model. It works for any set of preferences, including risk-neutral
investors.
 Gives the same option price no matter what the actual risk preferences and expected
stock returns are.
In the real world: risk averse and positive risk premium.
Hypothetical risk neutral world: no compensation for risk.
The risk neutral probabilities overweight the bad states and underweight the good states.
Monte carlo simulation: simulate a lot of random paths. Take the average of all these
different values. That should be equal to the price of your call.
Clip 3:
Backward induction: key of the tree. Extending the construction portfolio back one tick at
the time.
Moving up or down are not the way stock prices behave annually or daily. But by increasing
the length and increasing the number of periods, there is a realistic model for stock prices.
Lecture 9
Clip 1
Black scholes option pricing model: a technique for pricing European-style options when
the stock can be traded continuously. It can be derived from the Binominal option pricing
model by
- Length of each period be zero
- Number of periods can grow infinitely large
When there is no dividend
 𝐶 = 𝑆 ∗ 𝑁(𝑑1) − 𝑃𝑉(𝐾) ∗ 𝑁(𝑑2)
N(d) = cumulative normal distribution = p that an outcome from a standard normal
distribution will be below a certain value.
𝑆
]
𝑃𝑉(𝐾)
𝑑1 =
, 𝑑2 = 𝑑1 − 𝜎√𝑇
𝜎 ∗ √𝑇
For the N(d1), look at the distribution table.
European put option: 𝑝 = 𝑃𝑉(𝐾) ∗ (1 − 𝑁(𝑑2 )) − 𝑆(1 − 𝑁(𝑑1 ))
𝐿𝑛 [
Dividend paying stocks:
 PV(DIV) = PV of any dividend paid prior to the expiration of the option.
 𝑆 𝑥 = 𝑆 − 𝑃𝑉(𝐷𝑖𝑣)
Only SD is not observable, so you can estimate it by historical data or use the implied
volatility by using the prices and the formula.
Clip 2
𝜕𝐶
= 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑜𝑝𝑡𝑖𝑜𝑛 𝑝𝑟𝑖𝑐𝑒 𝑎𝑠 𝑎 𝑟𝑒𝑠𝑢𝑙𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑡ℎ𝑒 𝑠ℎ𝑎𝑟𝑒 𝑝𝑟𝑖𝑐𝑒.
𝜕𝑆
It is always smaller than 1, change in Call price is always smaller than change in stock price.
∆=
Replicating portfolio of a call option, has always a long position in stock and a short position
in bond -> leveraged position in the stock -> positive beta.
Delta hedging: options showing a delta which is the opposite to that of the current options
holding to maintain a delta neutral position.
Pro’s delta hedging:
- Hedge the risk of adverse price changes in the underlying asset.
- Protect profits from an option or stock position in the short term without scarifying
the long term holding.
Cons:
- Constant rebalancing -> high trading costs.
- Rebalancing is expensive, as options can lose time value.
Clip 3
What do you need for option Greek?
 Stock price = Delta/Gamma




Strike price
Exercise date = Theta
Rf = Rho
Sd = vega
Gamma: rate of change in the delta with respect to the price change in the underlying asset.
Theta: rate of change of the value of the portfolio with respect to the passage of time ->
time decay.
- Usually negative: time passes, option less valuable.
- Not same parameter as delta, because there is uncertainty about the stock price but
not about the passage of time.
Vega: change in option value with respect to the asset volatility.
- Very high are low -> sensitive to small changes.
Rho: rate of change of the value of the portfolio with respect to the interest rate.
 Options with different underlying assets, can have the same rho.
In an ideal world, traders can do hedging for all these. That’s not in the real world, because:
1. Managing a larger portfolio dependent on a single asset. They make delta zero at
least once a day.
2. Zero gamma or vega is harder, because it is hard to find options that can be traded in
the volume required at competitive prices.
Lecture 10
Clip 1
Real option: the right to make a particular business decision, such as a capital investment.
- Difference with financial option, is that real options and the underlying assets are not
traded in competitive markets.
Decision nodes: node on a decision tree at which a decision is made -> ∎ -> vb. I am going
or not.
Information nodes: type of node indicating uncertainty that is out of control of the decision
maker -> ⊙ -> is the weather good?
The value of the real option can be computed by comparing her expected profit without the
real option to wait until the weather is revealed, to the value with the option to wait.
Options to:
- Delay an investment opportunity.
- Grow
- Abandon an investment opportunity.
 In real world, there is a cost in delaying a decision.
Clip 2
If you wait to make a decision, it might generate in the interim and the competitor could
develop a competing product.
Clip 3
Other factors affecting the decision to wait:
- Volatility: waiting is more valuable when there is more uncertainty.
- Dividends: absent dividends (=not optimal to exercise a call option early) and in real
option context (=better to wait unless there is a cost).
Clip 4
Growth option: real option to invest in the future.
Abandonment option: the option to disinvest.
Clip 5
Option to abandon: many times, it is more profitable to abandon instead of starting a new
one. But managers often don’t want to focus on this alternative.