Intermediate Finance
Mean - variance preferences:
Utility function:
Portfolio theory must assume something about the preferences of the investor
Assuming that the investors can be mapped in terms of utility
Utility = U (Risk , Reward) = U=Er - 1/2Avariance(R)
It is always reasonable to think that a investor will prefer more over less, and hence prefere more reward than less
Investors dislike risk
Investors always prefer less risk over more risk
The utility function is increasing on reward
Utility function is decreasing on risk
What is risk and what is reward?
Reward = Mean (expected return)
Risk = Variance
Concepts of risk tolerence:
Risk averse - wil always pick the least risky option when confronted with two investments with the same E(r)
Risk neutral- the mean/E(r) is the only thing that matters
Risk loving -the riskier the better
For a risk averse investor - A is positive
For risk neutral investor - A is zero
For a risk loving investor - A is negative
Portfolio theory:
Providing optimal ways of investing wealth across financial assets
Portfolio weights: they add up to one
Proportions of wealth assigned to different assets
The return is equal to the sum of the returns of the elements to the porfolio
Statistics: A short review
How do we estimate expected returns in reality? - We dont know the probabilities
Variance of a random variable x:
Var(x) = E((x-E(x))^2)
Often denoted as Var(x) = sigma^2(x)
The variance is the expectation of the squared deviation of a random variable from its mean, and it measures how far a set of (random) numbers are spread of from its mean . The variance itself is diffucult to interpretate
The standard deviaton
Sigma(x) = (var(x))^0,5
Is a measure that is used to quantify the amount of variation or dispertion of a set of data values. A low std indicates that the data points tend to be close to its mean value (expected value). A high std indicated that there is a big spread over a wider range of values. (easy to interpretate)
We use the arithmetic average of past obervations
In reality we dont know probabilities, hence we replace the expected value with the arithmetic average of past observations
Covariance and correlation
To find the risk of a portfolio, one must know the degree to wich stocks move together
Cov(x,y) = E(xy)-E(x)E(y)
The correlation coefficent: p(xy) = Cov(xy)/std.dev(x)std.dev(y)
The correlation always between -1 and 1
The Certainty equivalent:
For any risky portfolio, its certainty equivalent is defined as the risk-free return that provides the same utiity
The utility of a given risky return is just that (risk-free return) - the higher the risk-aversion, the lower certainty equivalent
Indifference curves:
Set of pairs of mean and variance that yield the same utility
Mathematically, we compute that set by finding the pairs of mean and variance that satisfies the equality: U = E® - 1/2Astd.dev® = C
C is a given level of utility
All portfolios that lie on a given indifference curve are equally desirable
Any portfolio lying on a indifference curve (northwest) is more desirable
Different investors have different maps of indifference curves - a more risk-averse investor have more steeply sloped indifference curves
The capital allocation decicion
Risk-free asset versus everything else. The rest: treated as a portfolio whose weights are given
The complete portfolio: Investing a proportion y in the risky portfolio. Rc=(1-y)rf + yrp
If y > 1 -> borrowing position (invest all our money pluss all the money we borrow)
If 0 < Y <1 -> lending position (invest part of our initial wealth in the risky portfolio and part in the T-bill/risk-free asset)
If y < 0 -> short selling the risky portfolio and invest the proceeds plus our initial wealth in the T-bill
Optimal capital allocation
Need to write the utility evaluated at rc as a function of the decicion variable y.
U(rc) = E(rc) - 1/2Avariance(rc) = rf + y(E(rp)-rf) - 1/2Ay^2variance(rp)
We wish to maximize the utility of the investor with respect to the proportion y
Max. for y: rf + y(E(rp)-rf) - 1/2Ay^2*variance(rp)
The firs derivative is set to zero - which gives: y=E(rp)-rf/Avariance(rp)
Interpretation:
-Higher level of risk aversion gives smaller amount invested in risky portfolio.
-Higher level of volatility gives smaller amount invested in rp.
- Larger risk premium gives larger amount invested in rp.
The capital allocation line - CAL
The CAL gives us all possible pairs of expected return and standard deviation that are attainable/feasible
The optimal capital allocation can be seen as a maximization of utility subject to a "budget constrain" (the CAL)
Optimal capital allocation takes place at the point of tangency of an indifference curve and CAL
Different lending and borrowing rate
For y < 1, the lending rate applies, which implies -> E(rc)=rfl + std.dev(rc)*(E(rp)-rfl)/std.dev(rp)
For y>1, the borrowing rate applies ->E(rc)=rfb + std.dev(rc)*(E(rp)-rfb)/std.dev(rp)
Optimal risky portfolios
Now its time to allocate across risky assets. Why not full investment?
By diversifying you might acchieve higher expected return and lower standard deviation. This is because you take advantage of the fact that the assets move in opposite directions (negative covariance/correlation), thus their standard deviations have a cancelling effect
Rabbi Issac bar Aha proposed the following
rule: “One should always divide his wealth
into three parts: a third in land. a third in
merchandise. and a third ready to hand”
Babilonian Talmud (4th Century)
If the correlation coefficient between to assets are 1, then there are no gains from diversification. Hence:
Two sources of risk:
Firm specific risk:
Independent risk sources across firms. Its possible to reduce to negligible levels. Its non systematic and diversifable.
Market risk
Risk that affect all firms, typical macro events. Related to marketwide risk. It is systematic and non diversifiable.
Hence we can write returns as: r = m + e (where cov(m,e)=0)
The opportunity set is the set of all pairs of mean and variance (std.dev) which can be obtained by using all feasible values of portfolio weights
The minimum variance portfolio
The portfolio that gives the lowest variance of all elements of the opportunity set
How to find it?
Minimizing -> var((1-as)rb+asrs)
The efficient set contains all those portfolios in the opportunity set which are above the minimum-variance portfolio
The optimal risky portfolio
Need to find the portfolio with the highes slope (Sharpe-ratio)
Max. as: E(rp)-rf/std.dev(rp)
Se notes for this formula (long and tidious)
Separation property
The choice of the optimal portfolio of risky assets is the same for all (mean-variance) investors. Their specific preferences only determine the fraction to be invested in the risk-free asset. (Just find optimal y, since it incorporates the risk aversion parameter)
Optimal risky portfolio p: point of tangency with CAL with opportunity set of risky assets
To solve:
- Obtain its weights
- Obtain its variance and expected return
Optimal complete portfolio c: point of tangency of indifference curve with the CAL (se illustrasjon)
To solve:
- Obtain fraction y
- Obtain the final weights
Portfolio theory with any number of assets
Matrix algebra
Provides a compact way of expressing systems of equations and multivariable optimization problems - simplifies the derivation of an analytical solution in many instances
A matrix is an array of numbers or variables presented in a table-like manner
A matrix has N rows (rader) and K columns (kolonner), therefor a dimension of N*K
Tyipical a element of a matrix X is usually denoted Xij, where i denotes the row number and j denotes the column
A matrix with N rows and 1 column is known as a (column) vector - it has a dimension of N*1, also called a N-dimensional (column) vector
Matrix addition
Calculated entrywise - (array x + array y)
Both matrices must have the same dimension (N*K)
Matrix multiplication
Scalar product: let c and A be a scalar and a matrix, respectively
The matrix product: X23*Y32
The matrix product: In every matric product we have -> XnkYkm = Znm
Where n and m denotes dimension, and k denotes conformability
Transposition
The transpose of a matrix A, denoted by A´, is the original matrix A with its rows (rader) and columns (kolonner) interchanged (ombyttet)
Transposition wil allow us to write summation like: SumXiYi -> as x´y (important thing to understand)
Remember that in order to perform matrix addition the dimensions need to be the same. That is why we transpose matrices?? (sjekk dette opp)
The expected value of return for a general portofolio in matrix form
Equal to: Er = rf+a´(Er-rf1)
Where rf1 denotes the risk-free rate multiplied by a vector of 1 in order to get into matrices form
a´is the transposed version of matrix a
Variance of a portfolio with N assets: a´Sum*a (see notes)
The utility of a general portfolio in matrix form
U = rf + a´(Er-rf1)-1/2AaSum*a
The inverse
Multiplication by the inverse of a matrix
The inverse of a square matrix A is denoted by A^-1. It is equal to I. Which has the quality of being equal to:AA^-1= I = 1 (see notes)
Division
Solving systems of linear equations (see notes)
The expected beta-return representation
It is possible to write the expected return of any return r: Er-rf = Beta(r,a)(Er-rf)
The CAPM
Its the cornerstone of modern financial theory: it is an equilibrium model which gives an prediction of the risk-expected return relationship of any financial asset
Assumptions:
- Many individual investors who are price takers
- Two-period model
- Investment are limited to traded financial assets which are perfectly diversifiable
- No short-sale restrictions
- A single risk free asset at which any investor can borrow or lend
- No taxes and no government
- No transaction costs
- Investors are rational mean-variance optimizers
- Homogenous belifs
Equilibrium when:
Investors maximize their utility: optimizes as in Portfolio theory
Market clears: the aggregation of the portfolio of all individual investors must equal the total supply of assets (the market portfolio)
Equilibrium gives us the CAPM allocation prediction and CAPM pricing prediction
The market portfolio
M = the aggregation of all financial wealth in the economy
The return has weights (determined in equilibrium) on each security which are equal to their relative market value
The relative market value is the market value of the security divided by the sum of the aggregated market values of all securities
Allocation: The separation property - all investors choose the same portfolio of risky assets p (the tangency portfolio)
The CML is the CAL of the market portfolio (see formula)
Pricing:: Mean variance return, therefor the expected beta-return representation holds
Er-rf = Beta(r,A)(E(rA)-rf)
Where beta(r,A)=Cov(r,A)/var(A)
The expected value of any return is given by:
Er -rf = Beta(r,rm)(E(rm)-rf)
Where Beta(r,rm)=cov(r,rm)/var(rm)
Interpretation
Beta: Measures quantity of market risk.
Correlation: Establishes if b contains "good" or "bad" risk through its sign (-+)
CAPM pricing prediction: The risk premium of any asset is equal to its amount of risk (measured by its beta with the market portfolio) times the price of that risk (measured by the risk premium of the market portfolio)
The security market line (SML): Gives the risk premium for individual assets or portfolio
The relevant measure of risk is the beta from the market portfolio, due to the fact that investors dont get any reward for bearing firm-specific risk
Benchmark for asset management: "good" or "bad" stocks? - find assets alpha. It is the difference between actual expected return and its associated CAPM prediction
Index models
The single-index model
Firm specific vs. systematic risk: returns move together only because of systematic risk, any firm specific risk is uncorrelated across assets. All sources of risk grouped into one index
The systematic components are grouped into one single random variable with different sensitivities across different stocks: mi=Beta(i)(rm)
Excess returns: Ri = alfa(i)+beta(i)Rm+ei
We can estimate the single-index model: The SCL. Here we can find intercept and slope for excess returns. We can also decompose the variance of an asset into firm specific and market risk. We can also find R^2 of the regression by dividing the market risk with the variance of the stock
The single-index and CAPM:
Cov(Rhp,Rs&p)=BetaHP*variance(s&p)
BetaHP=cov(Rhp,Rs&p)/variance(s&p)
Then the relationship of any return is: E(Rhp)=alfa(hp)+beta(hp)*E(Rs&p)
Active management and the single-index model:
The idea is to find an optimal active portfolio with two concerns in mind:
- Attention can only be devoted to a limited numer of securities
- Balance between agressive strategies and diversification
A model for security selection
Passive strategy = rm (marked/index)
Active strategy: a portfolio A consisting of limited set of securities with return rA
The return of the portfolio: rp = (1-aA)rm + aArA
The weight on A is the one that provides the highest slope of the CAL of P (see notes)
Optimal A is the one with highest information ratio
Bond prices and yields
Definition
Bond are debt (obligasjon)
Issuers are borrowers
Holders are creditors
Can be governments (state or local) or companys
Elements
- Maturity date
- Face or par value
- Coupon rate
- Coupon frequency
- Redemtion value
Special bonds:
Callable bonds
Puttable bonds
Convertible bonds
Floating-rate bonds
Even more special
Indexed bonds
Catastrophe bonds
Asset-backed bonds
Inverse floaters
Bond price over time
Price goes up/down (and the yield down/up) as market rates fluctuate. There is a built-in capital gain or loss. The bond price approaches "par value/face value" as maturity approaches
The built in capital gain/loss will offset a below/above-market coupon rate so that the holding-period return and yield are equal (as long as the yield stay constant during the period)
If the cupon rate is higher than the yield to maturity, then the price of the bond is lower 1 year forward (IMPORTANT)
If the yield to maturity is higher than the coupon rate, then 1 year forward the price wil be higher (IMPORTANT)
Bond pricing in between coupon dates:
Invoice price = the present value of all future payoffs
The coupon of this period does not entirely belong to the new holder: flat price:
Flat price = invoice price - accrued interest
Default
Default risk
Bond issuers may not satisfy their obligations to repay their debt: bond default risk = credit risk
Yield to maturity has to be interpreted as the maximum possible yield to maturity (IMPORTANT)
Default premium
Is the yield spread between the bond and a riskless asset (US og Norwegian bonds for instance)
Derivatives markets
Futures and forwards
Forward contracts
A forward contract is a agreement between two counterparties - a buyer and a seller. The buyer agrees to buy an underlying asset from the seller
Bond price converges to face value!
The delivery of the asset occurs at a later time, but the price is determined at the time of purchase
Futures
Futures are similar to forward, but feature formalized and standarized characteristics
An agreement to buy an underlying asset at some specified time in the future at a predetermined price
Key differences:
Forwards
- Trade OTC (known counterparty)
- Customized
- Counterparty risk
Futures
- Exchange traded (unknown counterparty)
- No counterparty risk
- Clearing houses and margin deposits
- Standardized
- Secondary trading - liquidity
Key terms:
Futures price: agreed-upon price at maturity
Long position: agree to purchase
Short position: agree to sell
Profit = spot price minus original futures price
Profit = original futures price minus spot
Types of contracts:
Agricultural commodities, metals and minerals (including energy contracts), forreign currencies, fiancial futures (interest rates or stock indexes) etc..
Trading places
Clearinghouse: acts as a party to all buyers and sellers - obligated to deliver or supply delivery
Closing out positions: reversing the trade, take or make delivery.
Most trades are reversed and do not involve actual delivery
Margin and trading arrangements:
Initial margin: Capital or "safe" securities deposited to absorb losses
Market to market: Each day the profits or losses from the futures price are reflected in the account
Maintainence or variation margin: An established value below which a traders margin may not fall
Margin call: When the maintainence margin is reached, broker will ask for additonal margin funds
Convergence of price: Actual commodity of a certain grade with a delivery location or some contracts cash settlements
Trading strategies
Speculation
Short - positioned for fall in price
Long - positioned for increased price
Hedging
Long hedge - protecting against a rise in price
Short hedge - protecting against a fall in price
Basis and basis risk (hedging)
Basis: the difference between the futures price and the spot price. Over time the basis will likely change and eventually converge
Basis risk: the variability in the basis that will affect profits and/or the hedging performance
Spot - futures parity: There are two ways of acquire an asset for some specified date in the future.
- Purchase it now and store it
- Take a long position in futures
The two strategies must have the same financial cost - therefor it is a parity
With a perfect hedge the futures payoff is certain - there is no risk!
A perfect hedge should return the riskless rate of return (no arbitrage opportunity)
This relationship can be used to develop futures pricing relationship
If the spot-futures parity is violated, arbitrage is possible (not realistic): If the futures price is too high, short the futures and acquire the stock by borrowing at rf. If the futures price is too low, go long futures, short the stock and invest the rest in rf
Futures price versus expected spot price:
Expectations
Normal backwardation
Contango
Options
What is it?
A call option gives the holder the right to purchase an asset for a specified price, called exercise or strike price, on or before some specified expiration date
A put option gives the holder the right to sell an asset for a specified price, called the exercise or strike price, on or before som specified expiration date
The purchase price of the option is called the premium
Option terminology:
Buy - long
Sell - short
Call
Put
Market and exercise price relationships:
In the money: exercise of the option is profitable if
- Call: market price > exercise price
- Put: exercise price > market price
Out of the money: Exercise not profitable if
- Call: market price < exercise price
- Put: exercise price < market price
At the money: exercise price and asset price are equal
American options
European options
The option can be exercised at any time before the specified expiration date
The option can only be exercised on the specified expiration date
Different types of options
- Stock options
- Index options
- Futures options
- Forreign currency (FX) options
- Interest rate options
Option strategy:
An unlimited variety of payoff patterns can be achieved by combining puts and calls with different maturities (its like like Lego)
Protective put
Limit loss - position: Long the stock and long the put
Covered call
Some downside protection at the expense of giving up gain potential. Position: Own the stock and write a call
Other strategies
Straddle: Long call and long put
Spreads: A combination of two or more call options or put options on the same asset with different exercise prices or time to expiration
Vertical or money spread: Same maturity, different exercise price
Horizontal or time spread: Different maturity dates
Put-call parity:(only valid for European options)
Since the payoff on a combination of a long call and a short put are equivalent to leveraged equity, the prices must be equal. If prices are not equal there wil be arbitrage possibilities
C + X/(1+rf)^T = S0 + P
Optionlike securities
Callable bonds
Convertible securities
Warrants
Collateralized loans
Exotic options
Asian options
Barrier options
Lookback options
Currency translated options
Binary options
Option valuation:
Intrisic value:: Profit that could be made if the option was exercised immediately
Time value: The difference between the option price and the intrisic value
Factors influencing option values: CALL
Stock price (increases)
Exercise price (decreases)
Volatility of stock price (increases)
Time to expiration (increases)
Interest rate (decreases)
Dividend rate (decreases)
Restrictions on option value: CALL
- Value cannot be negative
- Value cannot exceed the stock value
- Value of the call must be greater than the value of levered equity
Early exercise
It never pays to exercise a Call option before expiration
Always worth more than its intrisic value, but for American options early exercise is sometimes optimal.
Replication arguments in Finance
The ability to create a perfect hedge is key to replication arguments
Since we have locked in the year-end we can discount using the risk-free rate
Hence, we can express the opions value in terms of the current stock price, the stock price in year end is redudant
Dont need to know the options beta or expected return
Dynamic hedging
Set up perfectly hedged portfolio on each node, and work backwards through the tree. The portfolios are perfectly hedged over a tiny period. By continously revising the hedge ratio, the portfolio can remain hedged and earn the risk-free rate over each tiny interval
As the sub period get infinetessmall we approach what is called continously time, and with that assumption we can derive closed form formula for option valuation
Black-Scholes Merton model
N(d) can (loosely) be viewed as risk-adjusted probability that the call option will expire in-the-money
If both N(d) terms is close to 1, then c=0 (the value of the call is about zero)
For midrange values of N(d), c can be viewed as the present value of the calls potensial payoff, adjusted for the probability of expire in-the-money
Assumptions for BSM:
- The stock will pay no dividends during the lifetime of the option
- Both r and std.dev are constant through the lifetime of the option
- Stock prices are continous (no jumps)
Using the BSM: Hedge ratio - the number of stocks required to hedge against the price risk of holding one option
Portfolio insurance:
Buy put: results in downside protection with unlimited upside potential
Limitations:
- Tracking errors (if indexes are used)
- Maturity of puts may be too short
- Hedge ratios/delta change as stock values change