I have combined both the slides and the professor's comments during the lecture into one summary. As a result, the summary contains all the necessary information for the exam and is also easy to understand (partly due to the use of graphs). Finally, I have documented weeks separately to keep it org...
All investments are trade-offs between risk and return, with 3 observations:
1. People have money to invest
2. There are several assets to invest in and we have choice between different assets
3. We assume that people have preferences
In other words, there are different possibilities and they are not all equal, some are better than
others (and maybe one is the best). We tend to model that decision as a trade-off between return
(expected gain) and risk (probability and magnitude of loss / shortfall) Assume:
A1: Agent prefer more over less (nonsatiation)
A2: Agents dislike risk (risk-aversion). The definition of risk is an important matter
How should investors, given their preferences, invest their money? (normative)
What can we say about how the market and its participants actually operate and invest? (descriptive)
Both revolve around the risk-return relationship, and both interact: information about how market
work influences investment decisions, which influences the market in its turn. Asset returns also
determine risk, we can separate:
- The chance at a return (high or low)
- The utility of a return – the same return can have different utility in different situations: if
other assets in your portfolio have big negative returns, you like a positive return more. If
your consumption is low, a wipe-out of your investments may hurt more than other ones.
= The amount matters (mean and variance), and the relation with other factors (covariance) matters.
In a formula: Pt = E(Mt+1Xt+1)
P = Price
E = expectation
M = SDF = Marginal utility of future consumption: how much do we value returns (to be used
for consumption) under the conditions prevailing at the time we get those returns.
X = return (distribution); future cash flow (value of investment, not profit)
Today’s price (pt ) is determined by how much I expect to get from it in the future, weighted by how
much I value that payoffs (depends on the conditions boom/recession)
Stochastic = random probability distribution
The SDF can be derived from the utility function, this gives:
'
U (C t +1 )
Mt = ' , denotes time preferences
U ( C1 )
( U(ct, c1+t) = u(ct) + Et[u(ct+1)] , utility = utility now + utility later
U = Utility
C = Consumption
Utility (U) is dependent of the amount consumed, higher consumption now lead to higher utility
Ct = e - pt and Ct+1 = et+1 + xt+1
e = income
c = consumption
x = return
= amount bought of an asset
,SDF, which translates returns to prices by judging its riskiness, depends on marginal utility.
In many cases, m is a linear function of a factor (CAPM).
Mt+1 = a + b’ f
That factor f captures when returns in situation A may be more pleasant than the same returns in
situation B (Boom or recession).
Portfolio theory:
A3 = They maximize utility, and do so for 1 period (maximizing utility = rationality)
A4 = Utility is a function of expected return and variance, and nothing else
Market conditions:
A5 = No distortion from costs, transaction fees, inflation or taxes
If trading has costs, the optimum shifts (another allocation becomes optimal as fees eat part of the
return): investing a small amount in an asset becomes less
attractive, so costs Favor investing in fewer assets (assuming fixed
fees). Inflation and taxes tend to disrupt the choice between
consumption now and investments (for consumption later).
A6 = All information is available at no cost
A7 = All investments are infinitely divisible
If all assumptions hold, it’s giving the exact same set of portfolios
(the efficient set). The degree of risk aversion determines which
efficient portfolio will be chosen.
If we’re optimizing a portfolio given a certain set of assets, and mean-variance preferences, chances
are that the optimum will consist of a lot of relatively small investments. The return of a portfolio is
the return of the assets in the portfolio times the portfolio weights (ω). The variance of the portfolio
is determined by the weights of the assets and the covariance matrix of the assets. If we assume
there is no risk-free asset, portfolio weights sum to 1. We minimize the variance by adjusting the
weights, and subject to the restrictions that the portfolio return equals R p and the weights sum to 1:
Optimal portfolio based on 3 factors:
1. What are the average returns of the assets (the vector r)
2. How are they related (V-1,determines how much you gain through diversification; correlation)
3. How much return do I want / how much risk am I willing to bear (R p)
The exact location of the bullet after a new asset is added, depends on the characteristics of the new
asset, however it will always shift to left (up), which will give the portfolio less volatility.
The first few assets shift the bullet leftward much more than later ones. (or,
within mutual funds, move the mean-variance characteristics closer to the
bullet). Exact shape depends on stock characteristics. However we can assume
that diversification after 60 stocks isn’t as valuable as it is before.
The bullet: what are the combinations of variance and required return that
will be optimal. The optimal portfolio will contain a lot of assets (in reality
encountered by transactions costs). The bullet can be replicated with 2 assets that are on the efficient
frontier (portfolio separation). Main influences on the bullets are men returns and correlations. To
,arrive at the optimal portfolio requires you the know the preference structure (Utility
curves/functions).
Formula of the week and extra explanations:
The optimal portfolio will be determined by the average return of the assets you can invest in (r), the
relation between the assets (V-1 ; determines how much you gain through diversification), and how
much return you want / risk you’re willing to bear (Rp )
Covariance = measure the relationship between two variables
Variance refers to the distribution of one variable
Other things equal, an asset which does badly in states of nature like a recession, in which the
investor feels poor and is consuming little, is less desirable than an asset that does badly in states of
nature like a boom in which the investor feels wealthy and is consuming a great deal.
The former asset will sell for a lower price; its price will reflect a discount for its ‘‘riskiness,’’ and this
riskiness depends on a co-variance, not a variance.
Alternatives to investment C change, alternatives are going to be more or less attractive to invest. If A
and B are also more riskier now, the portfolio weights will change also for C.
When everyone is trading ETF’s, buying and selling the same stock, it would not affect other stocks
when everyone buys the same stock; When selling a single apple stock, it would automatically include
the whole ETF.
Extra explanation about Bullet and Efficient frontier:
The bullet summarizes the investment possibilities: combination of
stocks that are efficient; highest possible return for a given amount of
risk. Where the efficient frontier is the above side of the bullet with
only efficient portfolios. When adding a risk free object, there appears a
tangency portfolio and another Efficient frontier (Capital market Line in
the CAPM).
So, only when investing fully tangency point (intersection between the
bullet and the Efficient Frontier), you don’t make use of the Rf. Otherwise you always will go long or
go short on the Rf.
, Lecture 2, Portfolio theory, CAPM, theory and testing
Portfolio theory: the risk free object
The existence of a risk-free object will generally result in portfolios with higher utility, especially for
risk averse agents. Incorporating the risk free object mathematically means we drop the weight
restriction, instead of adding the risk free object (interest rate or bonds) to the
formula.
The straight line created in the figure is the capital Market Line in the CAPM.
Portfolio separation still holds but now between 2 special assets, the risk-free object
and the tangency portfolio. Each investor can (and will) maximize his utility by
combining risky and risk-free assets. If all agents have the same expectations (they
agree on the parameters: expected return and covariance matrix) and A5-A7 hold,
they will all have the same tangency portfolio.
The next logical step is the equilibrium model: If everyone needs the same portfolio for their
optimum and if all assets are to be held (returns get adjusted if they are so low no-one wants the
assets), then the tangency portfolio must contain all assets - it must in fact be the market portfolio.
Restrictions
In reality investors face problems, since there are limits on shortselling and it is not that easy to get a
bank loan for it. The restriction will limit the possibilities in portfolio optimization which depends on
long and short positions. If there is no risk free object, one can only get returns between the
minimum and the maximum of the risky assets. If there is a risk free object,
- Between Rf and Tangency portfolio = long Rf
- Beyond Tangency portfolio = short Rf
Restrictions on portfolio weights are used to limit exposure to specific sectors, but they limit
diversification and lead to an increase in variance instead of decreasing the risk as it was supposed to
do. Furthermore, optimization might result for example in a weight of 0.734, this has to be 0 or 1 with
no infinite divisibility, which results in both cases in a worse risk/return trade-off.
CAPM
The CAPM builds on portfolio theory but goes much further. It is an equilibrium model; describing the
entire market. It is more descriptive in nature, as it works not well enough to warrant using it the
normative way. CAPM requires more assumptions:
A8. All investors have the same expectation regarding returns and covariances (Needed to
arrive at the same bullet for everyone, and thus the same tangency portfolio for everyone).
A9. All investors can lend and borrow at the risk-free rate.
A10. Asset markets are characterized by perfect competition: no-one is big enough to have an
influence on asset prices (and hence returns), everyone is a price taker (if not, everyone’s
decisions will influence prices and hence everyone else’s decisions; game theory).
The CAPM does not assume all investors have the same (or similar) preferences; they differ in their
utility curves. It just needs them to follow the MV-criterion.
The relation of the return of any asset with the tangency portfolio (and in equilibrium, the market
portfolio) makes the importance of diversification even more clear: diversifiable risk is not priced (you
can get rid of it, so no one is willing to pay for it).
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