Arhat Virdi
| Week | Lecture | Tutorial |
|---|---|---|
| 1 | The investment problem | Review of first year, e.g. IRR vs NPV, efficient markets hypothesis |
| 2 | Review of EMH, valuation, portfolio choice and CAPM | Evaluation and use of CAPM |
| 3 | More on CAPM - key assumptions, empirical tests. Using CAPM in practical valuation | Other asset pricing models |
| 4 | Other asset pricing models - factor models, consumption-based CAPM, APT. Equity premium and excess volatility puzzles | Behavioural finance |
| 5 | Behavioural finance and the financial crisis. Introduction to capital structure: asymmetric information, financial constraints and agency costs | Capital structure |
| 6 | Tax and trade-off theories of capital structure | Tax and finance |
| 7 | International Finance: diversification, exploitation of legal and fiscal differences by multinationals | International Finance |
| 8 | Corporate governance and CEO compensation | Corporate governance |
| NB | 0th Week TT25 | Collection |
"The special sphere of finance within economics is the study of allocation and deployment of economic resources, both spatially and across time, in an uncertain environment" (Merton 1997)
Today: how do we decide what projects or assets to invest in?
"Tesco has suspended four executives, including its UK managing director, after the supermarket overstated its half-year profit guidance by £250m. That would be almost a quarter of its expected profit for the period. It has launched an investigation headed by Deloitte, and says it is now working to establish the impact of the issue on its full-year results.
"Disappointment would be an understatement," said Tesco chief executive Dave Lewis.
Mr Lewis, who only took the helm on 1 September, said it was "a serious issue", but insisted "it doesn't take away from what I'm able to build at Tesco".
The news prompted a plunge in Tesco's share price, which closed 11.6% lower at 203p.
As a result almost £2.2bn was wiped from Tesco's value on the stock market.
How does information affect prices?
Prices reflect expectations of future value, i.e. future claims to cash flows paid out to shareholders as dividends.
In an efficient market
$$ P_{t}=\frac{E_{t}[D_{t+1}+P_{t+1}]}{1+r} $$
where $P_{t}$ is the share price in period t
$D_{t+1}$ is the dividend paid in period $t+1$
All period $t+1$ terms are expectations
r is the required return on the asset: $r=E_{t}[\frac{D_{t+1}+P_{t+1}}{P_{t}}]-1$
Solving forward, obtain the dividend discount model:
$$ P_{t}=\sum_{j=1}^{\infty}\frac{E[D_{t+j}]}{(1+r)^{j}} $$
(no-arbitrage condition, i.e. there are no opportunities for arbitrage; any deviation from PV pricing leaves a sure way to make money)
Similar approach to valuation of projects
$$ PV=\sum_{j=1}^{T}\frac{E[CF]}{(1+r)^{j}} $$
$$ NPV = \sum_{j=1}^{T}\frac{E[CF]}{(1+r)^{j}} - \text{initial investment} $$
Other criteria also used by practitioners to assess investments
$$ \sum_{j=1}^{T}\frac{E[CF]}{(1+IRR)^{j}}=\text{initial investment} $$
$$ \sum_{j=1}^{T}\frac{E[CF]}{(1+r)^{j}}=\text{initial investment} $$
Graham and Harvey (2002) survey 392 CFOs: IRR and NPV quite popular (~75%)
To evaluate $P_{t}$ we need
Linking preferences to returns and prices
We implicitly assumed that this "price is right"
Formally: Efficient Market Hypothesis
"actual price at every point in time represent very good estimates of intrinsic values" (Fama 1965)
Market is informationally efficient with respect to information set Ω if an investor cannot make economic profits by trading on the information contained in Ω (Jensen 1978)
Market efficiency means that prices are such that investors earn just the required return on average
What information goes into prices, i.e. into $E[D_{t+1}]$?
different levels of information imply different levels of efficiency
What information goes into prices, i.e. into $E[D_{t+1}]$?
different levels of information imply different levels of efficiency
This has valuable economic intuition
Grossman, Stiglitz (1980):
In Lectures 1-3, assume all investors have full information
in Lecture 4 explore departures from market efficiency
Next:
Suppose 2 possible payoffs from an investment (I)
Expected value of outcomes:
$$ \overline{y}=py_{1}+(1-p)y_{2} $$
Expected utility from the investment:
$$ \overline{u}=pu(y_{1})+(1-p)u(y_{2}) $$
(where $u(x)$ is the utility of receiving a payoff of x)
The certainty equivalent of above lottery is $y_c$, where:
$$ \overline{u}=u(y^{c}) $$
Graph Description: This is a 2D graph illustrating the concept of risk aversion with a concave utility function. The horizontal x-axis represents the payoff, labeled 'y'. The vertical y-axis represents utility, labeled 'u'.
A curve is drawn that starts in the lower left and goes to the upper right, bending downwards. This concave shape shows diminishing marginal utility: each additional unit of payoff provides less additional utility.
Several points are marked:
Risk premium, $r=\overline{y}-y^{c}$
Equivalent definitions of risk averse investor:
Suppose you want to allocate funds to assets X and Y
Asset payoffs:
Consumption from X in good state is then $c_{xg}=wR_{xg}$
generates utility of $u(c_{xg})$, similar for other payoffs
Choose portfolio, i.e. w, to maximise expected utility
$$ p[u(c_{xg})+u(c_{yg})]+(1-p)[u(c_{xb})+u(c_{yb})] $$
First order condition over w yields:
$$ p\frac{\partial u(c_{xg})}{\partial c}R_{xg}+(1-p)\frac{\partial u(c_{xb})}{\partial c}R_{xb} = p\frac{\partial u(c_{yg})}{\partial c}R_{yg}+(1-p)\frac{\partial u(c_{yb})}{\partial c}R_{yb} $$
i.e.: expected marginal utility from investing in X (LHS) = expected marginal utility from investing in Y (RHS)
this approach is not operational even in this simple setting!
(10 minutes)
Alternative: define utility in terms of statistical measures of return and risk
Graph Description: This is a 2D graph representing the Mean-Variance framework. The horizontal x-axis represents risk, measured by standard deviation, and is labeled with the Greek letter sigma ($\sigma$). The vertical y-axis represents the expected return, labeled $E(r)$.
Scattered throughout the graph are numerous 'x' marks, each representing a different possible investment with a specific risk and expected return.
Three curved red lines are drawn across the graph, sweeping up and to the right. These are labeled "Constant utility lines". They are convex, indicating that as risk (sigma) increases, an investor requires a progressively larger increase in expected return to maintain the same level of utility. The utility lines further up and to the left represent higher levels of utility.
Utility of investment with stochastic return r, with variance $\sigma^2$, for investor with risk aversion $\gamma$
$$ U(r,\sigma)=E(r)-\frac{\gamma}{2}\sigma^{2} $$
Is there a positive relationship between volatility and average returns for individual securities?
Graph Description: This is a scatter plot comparing historical risk and return for various assets. The horizontal x-axis is "Historical Volatility (standard deviation)", ranging from 0% to over 50%. The vertical y-axis is "Historical Average Return", ranging from 0% to 25%.
The main feature is a large cloud of data points representing 500 individual stocks. These points are coded by company size: triangles for the 50 largest stocks (stocks 1-50), blue diamonds for mid-size stocks (51-400), and orange circles for the smallest stocks (401-500). The cloud of points shows a weak positive trend, but there is a very wide dispersion, meaning for any given level of volatility, there is a wide range of historical returns.
In contrast, several points representing portfolios are plotted and connected by a dashed line, showing a much clearer, stronger positive relationship between risk and return. These points, from lowest risk/return to highest, are: Treasury Bills, Corporate Bonds, World Stocks, S&P 500, Mid-Cap Stocks, and Small Stocks. These portfolio points lie generally above and to the left of the cloud of individual stocks, indicating they offer better returns for a given level of risk.
The title of the chart is "Historical Volatility and Return for 500 Individual Stocks, by Size, 1926-2004".
Image Description: A portrait photograph of Harry Markowitz. He is an older man with glasses and thinning white hair. He is wearing a suit and tie and is smiling gently at the camera. The image has a green border.
Individual stocks are rarely held in isolation; rather they are held as elements of a portfolio.
How should we combine assets in our portfolios?
DIVERSIFICATION: The risk of an asset's poor return can be offset by another asset, if returns are not too correlated.
Don't put all your eggs in one basket!
Let $r_{A}$ and $r_{B}$ be the returns on asset A and B
These are stochastic variables, with:
Weights: w = fraction of wealth in A, 1 - w in B
Portfolio returns
$$ r_{p}=wr_{A}+(1-w)r_{B} $$
Expected return for the portfolio is:
$$ E(r_{p})=wE(r_{A})+(1-w)E(r_{B}) $$
And the portfolio variance is:
$$ \sigma_{p}^{2}=E(r_{p}-E(r_{p}))^{2} = w^{2}\sigma_{A}^{2}+(1-w)^{2}\sigma_{B}^{2}+2w(1-w)\sigma_{AB} $$
where $\rho_{AB}=\frac{\sigma_{AB}}{\sigma_{A}\sigma_{B}}\in[-1,1]$ is the correlation coefficient between $r_{A}$ and $r_{B}$
Assume means, variances and correlations are constant
Consider the polar cases:
What does the variance of the portfolio look like?
When $\rho_{AB}=+1$ we get:
$$ var(r_{p})=[w\sigma_{A}+(1-w)\sigma_{B}]^{2} $$
Or: $std(r_{p})=w\sigma_{A}+(1-w)\sigma_{B}$
Can write $w=\frac{std(r_{p})-\sigma_{B}}{\sigma_{A}-\sigma_{B}}$
Thus, can write $E(r_{p})$ as a function of $std(r_{p})$
$std(r_{p})=w\sigma_{A}+(1-w)\sigma_{B}$ and $E(r_{p})=wE(r_{A})+(1-w)E(r_{B})$
$$ \Rightarrow E(r_{p})=std(r_{p})\frac{E(r_{A})-E(r_{B})}{\sigma_{A}-\sigma_{B}}+c $$
Graph Description: This graph shows the risk-return tradeoff for a portfolio of two assets, A and B, that have perfect positive correlation. The horizontal axis is risk ($\sigma_p$) and the vertical axis is expected return ($E(r_p)$). Two points, labeled 'A' and 'B', represent the individual assets. A straight line connects point A and point B. This line represents all possible portfolios that can be created by combining assets A and B. The straight line indicates that there is no reduction in risk from diversification; the portfolio's risk is simply a weighted average of the individual assets' risks.
Now $std(r_{p})=|w\sigma_{A}-(1-w)\sigma_{B}|$
there exists a combination with zero risk
Graph Description: This graph shows the risk-return tradeoff for a portfolio of two assets, A and B, with perfect negative correlation. The horizontal axis is risk ($\sigma_p$) and the vertical axis is expected return ($E(r_p)$). Points 'A' and 'B' represent the individual assets. All possible portfolio combinations of A and B are represented by two straight line segments that meet at a point on the vertical axis. This creates a V-shape pointing to the left. The point on the vertical axis represents a portfolio with zero risk, which is possible to construct because the assets are perfectly negatively correlated.
$$ std(r_{p})=\sqrt{w^{2}\sigma_{A}^{2}+(1-w)^{2}\sigma_{B}^{2}+2w(1-w)\sigma_{AB}} $$
again $std(r_{p})$ can be lower than min{$\sigma_A$, $\sigma_B$}
diversification: low $r_{A}$ compensated by higher $r_{B}$
Graph Description: This graph shows the risk-return tradeoff for a portfolio of two assets, A and B, in the general case where correlation is between -1 and +1. The horizontal axis represents risk and the vertical axis represents expected return. Points 'A' and 'B' represent the individual assets. The line connecting them, which represents all possible portfolios, is a curve that bows out to the left. This curvature demonstrates the benefit of diversification: by combining the assets, it is possible to create a portfolio with a lower level of risk than either of the individual assets.
Assume we have N risky assets:
Therefore, the portfolio return is:
$$ r_{p}=w_{1}r_{1}+\cdot\cdot\cdot+w_{N}r_{N}=\sum_{i=1}^{N}w_{i}r_{i} $$
Therefore, the portfolio expected return is:
$$ E(r_{p})=\sum_{i=1}^{N}w_{i}E(r_{i}) $$
Portfolio variance depends on
$$ \sigma_{p}^{2}=\sum_{i=1}^{N}w_{i}^{2}\sigma_{i}^{2}+\sum_{i}\sum_{j\ne i}w_{i}w_{j}\rho_{ij}\sigma_{i}\sigma_{j} $$
For any target expected return $\overline{r}$:
$$ \min_{\{w_{1},\cdot\cdot\cdot,w_{N}\}}(\sum_{i=1}^{N}w_{i}^{2}\sigma_{i}^{2}+\sum_{i}\sum_{j\ne i}w_{i}w_{j}\rho_{ij}\sigma_{i}\sigma_{j}) $$
Subject to:
$$ \sum_{i=1}^{N}w_{i}=1, \sum_{i=1}^{N}w_{i}E(r_{i})=\overline{r} $$
Graph Description: This is a Mean-Variance diagram. The horizontal axis represents risk ($\sigma$) and the vertical axis represents expected return ($E(r)$). Many 'x' marks are scattered on the graph, each representing a risky asset or portfolio.
A C-shaped curve opening to the right envelops all the possible risky portfolios from the left. This curve is labeled the "Minimum Variance Frontier". For any given level of expected return, the point on this frontier represents the portfolio with the least possible risk.
The top half of this curve is highlighted in red. This upper portion is known as the "Efficient Frontier". Any portfolio on the efficient frontier offers the highest possible expected return for its level of risk. Rational investors would only choose portfolios on this efficient frontier.
Efficient frontier: maximize $E(r)$ for given $std(r)$
cannot improve one dimension without hurting the other
Graph Description: This is another Mean-Variance diagram comparing the benefits of diversification with more assets. The horizontal axis is "Volatility (standard deviation)" from 0% to 40%, and the vertical axis is "Expected Return" from 0% to 15%.
Individual points for 10 stocks (IBM, GM, Disney, GE, McDonald's, Merck, Campbell Soup, Exxon Mobil, Edison International, Anheuser-Busch) are scattered in the middle and right of the graph.
Two efficient frontiers are drawn. The first, a dashed curve labeled "Efficient Frontier with Exxon Mobil, GE, and IBM", is further to the right. The second, a solid curve labeled "Efficient Frontier with all 10 Stocks", is positioned to the left of the first one. This illustrates that by increasing the number of stocks in the portfolio from 3 to 10, the efficient frontier shifts to the left, meaning it is possible to achieve better risk-return combinations (either higher return for the same risk, or lower risk for the same return).
In general, with $w_{i}=(1/N)$:
$$ \sigma_{p}^{2}=\frac{1}{N^{2}}\sum_{i=1}^{N}\sigma_{i}^{2}+\frac{1}{N^{2}}\sum_{i=1}^{N}\sum_{j\ne i}^{N}cov(r_{i},r_{j}) $$
$$ \sigma_{p}^{2}=\frac{1}{N}[\frac{1}{N}\sum_{i=1}^{N}\sigma_{i}^{2}]+[1-\frac{1}{N}][\frac{1}{N(N-1)}\sum_{i=1}^{N}\sum_{j\ne i}^{N}cov(r_{i},r_{j})] $$
$$ \sigma_{p}^{2} = \frac{1}{N}\text{average variance} + (1-\frac{1}{N})\text{average covariance} $$
What happens when N goes to infinity?
Hence, when held in a portfolio some but not all -- of the risk of a stock disappears
Contribution of i to portfolio risk is smaller than $w_{i}^{2}\sigma_{i}^{2}$
$$ \text{total risk of the stock} = \text{risk that cannot be diversified away} + \text{risk that can be diversified away} $$
Graph Description: This graph illustrates how portfolio risk changes as the number of assets in the portfolio increases. The horizontal x-axis is labeled "# of assets". The vertical y-axis represents "Portfolio variance".
A curve starts high on the left side of the graph and slopes steeply downwards as it moves to the right. It then gradually flattens out, approaching a horizontal dashed line. This shows that adding assets to a portfolio rapidly reduces its variance at first, but the benefit diminishes as more assets are added.
The total risk is broken down into two components. An arrow points from the horizontal dashed line up to the curve in the early, steep part of the graph; this area is labeled "unique risk". This is the risk that can be diversified away. Another arrow points from the x-axis up to the horizontal dashed line; this area is labeled "systematic risk". This is the market risk that remains even in a well-diversified portfolio.
Break down the set of all possible causes of risk an investor can face into two groups:
Total risk = Systematic Risk + Idiosyncratic Risk
Graph Description: This graph begins to introduce a risk-free asset into the Mean-Variance framework. The horizontal axis is risk ($\sigma_p$) and the vertical axis is expected return ($E(r_p)$). The familiar C-shaped efficient frontier of risky portfolios is shown, along with scattered 'x' marks representing individual assets. A new point is added, labeled '$r_f$', located on the vertical axis. Since it is on the vertical axis, its risk ($\sigma$) is zero. This point represents the risk-free asset.
Combine $r_{f}$ with some $r_{R}$, what do you get?
$E(r_{p})=E(r_{f})+w(E(r_{R})-E(r_{f}))$
$\sigma(r_{p})=w\sigma(r_{R})$
Graph Description: This graph builds on the previous one. It shows the efficient frontier of risky assets and the risk-free asset point, $r_f$. A point labeled 'R' is selected on the efficient frontier. A straight line is drawn starting from $r_f$ and passing through point R. This line represents all possible portfolios that can be formed by combining the risk-free asset with the risky portfolio R. Any point between $r_f$ and R is a combination of lending (investing in the risk-free asset) and investing in R. Any point beyond R on the line is a combination of borrowing at the risk-free rate to invest more in R.
Combine $r_{f}$ with some $r_{T}$, what do you get?
$E(r_{p})=E(r_{f})+w(E(r_{T})-E(r_{f}))$
$\sigma(r_{p})=w\sigma(r_{T})$
Graph Description: This graph shows the optimal combination of the risk-free asset and a risky portfolio. A straight line is drawn from the risk-free rate, $r_f$, so that it is tangent to the efficient frontier of risky assets. The point of tangency is labeled 'T' and called the "Tangency portfolio". This line, known as the Capital Allocation Line (CAL), represents the new efficient frontier. It lies above the old risky-asset-only frontier, indicating that for any level of risk greater than zero, an investor can achieve a higher expected return by combining the tangency portfolio with the risk-free asset.
Tangency portfolio: maximizes slope
$$ \frac{E(r_{T})-r_{f}}{\sigma(r_{T})} $$
(Sharpe ratio)
Efficient frontier is now a line: Capital Allocation Line ($CAL_{T}$)
under certain conditions this becomes the Capital Market Line (CML)
Graph Description: This graph is the same as the previous one, showing the Capital Allocation Line (CAL) tangent to the efficient frontier at point T. It is further annotated to explain the investment strategies along the line. The segment of the CAL between the risk-free asset ($r_f$) and the tangency portfolio (T) is labeled "Risk averse, Lending, w<1". This indicates that investors with lower risk tolerance will hold a portfolio combining the risk-free asset and portfolio T. The portion of the CAL extending beyond point T is labeled "Risk seeking, Borrowing, w>1". This indicates that investors with higher risk tolerance will borrow money at the risk-free rate to invest more than 100% of their capital in the tangency portfolio T. This illustrates the two-fund separation theorem: all optimal portfolios are combinations of just two "funds"—the risk-free asset and the tangency portfolio of risky assets.
All efficient portfolios combine risk-free asset and the tangency portfolio
Risk aversion pins down location on CAL