FM110 Flashcards
1
Q
Discount Rate
A
A discount rate is the reward that investors demand for accepting delayed
rather than immediate gratification.
The discount rate is also called opportunity cost of capital because it is the
return foregone by investing in a capital project rather than investing in freelyavailable securities.
2
Q
Simple Interest
A
Simple interest only pays interest on the original principal (principal is the term
used for the original amount of money invested).
3
Q
Compound Interest
A
Compound interest pays interest not only on the original principal but also on
accumulated interest.
4
Q
Stated Annual Interest Rate
A
The stated annual interest rate indicates the amount of simple interest earned in
a year and does not take into consideration interest earned on interest through
compounding. Therefore we do not discount with stated annual interest rates1
.
When accumulating or discounting cash flows to calculate present values or
future values we always use compound interest. Quoting the annual rate with
simple interest only is simply a convention.
e.g. 0.5% monthly = 0.005(12) = 0.06
5
Q
Effective Annual Rate (EAR)
A
The effective annual rate (EAR) indicates the actual amount of interest that will
be earned at the end of the year after taking into consideration compounding
i.e. interest on interest.
(1+r)^n - 1
6
Q
Converting a Stated Annual Interest Rate to an EAR
A
(1+EAR) = (1 + SAIR/n)^n
7
Q
Future Value
A
If we are given a cash flow of C today it’s future value at time T is given by
FV = C(1+r)^t
where T is the number of time periods and r is the effective rate for the time
period and constant over time.
✓ Note the use of compound interest.
✓ Obviously the future value is larger if r or C or T is larger, ceteris paribus.
8
Q
Present Value
A
Assume that you’re due to receive a payment of C at time T. The current cash
flow today that is equivalent to the future cash flow at time T is given by
PV = C/(1+r)^t
✓ Note the use of compound interest.
✓ Obviously the present value is smaller, if C is smaller, or r or T is larger, ceteris
paribus.
9
Q
NPV Rule
A
Consider an investment project for which you have calculated the NPV.
✓ If the NPV is positive you should invest in the project.
✓ If the NPV is negative, you should turn down the investment opportunity.
The discount rate used in the NPV calculation should reflect the project’s
risk. More risky projects require a greater return and so you should use a
larger discount rate.
If you don’t know the cash flows associated with the project precisely, use the
expected value of each cash flow instead.
10
Q
Why should I trust the NPV rule? Why is it optimal for individuals to invest in
projects with positive NPV and discard projects with negative NPV?
A
It turns out that NPV is optimal (under some assumptions) in the sense that
use of the rule leads to investors maximising their expected wealth.
This is true regardless of how patient or impatient an investor is, and thus
the rule can be used for all investors (they will all agree on which investments
to choose and which to discard).
11
Q
Inflation Rate
A
The rate (usually annual) at which the level of prices in the economy increases.
Denote it by π.
12
Q
Nominal Interest Rate
A
The rate at which the balance of a deposit grows in cash terms. Denote it by r .
13
Q
Real Interest Rate
A
The rate at which the balance of a deposit grows in purchasing power terms.
Denote it by i.
14
Q
The relationship between the real interest rate, the nominal interest rate and
the inflation rate is
A
(1+i) = (1+r)/(1+π)
15
Q
Approximation for Real Interest Rate
A
r = i + π
16
Q
Annuity with Growth
A
- The first cash flow occurs at the end of the 1st period and is denoted by C
- The cash flows must grow at a constant rate each period denoted by g
- The timing of the cash flows occur at constant intervals
- The discount rate is the effective rate for the time period in between cash flows and is constant over time and denoted by r
- There are n time petiods between cash flows where n is finite
- the PV valuation point is one period before the first cash flow
17
Q
Annuity growth formula
A
PV = C((1- (1+g/1+r)^n) / r - g)
18
Q
Annuity
A
- The first cash flow occurs at the end of the 1st period and is denoted by C
- The Cash flow C must be constant each period
- The timing of the cash flows occur at constant intervals
- The discount rate is the effective rate for the time period in between cash flows and is constant over time and is denoted by r
- There are n time periods between cash flows where n is finite
- The PV valuation point is one period before the first cash flow
-> The relevant discount rate to use in the annuity formula is the effective rate in between cash flows i.e. monthly effective rate
-> Used for mortgage problems mainly
19
Q
Annuity Formula
A
PV = C((1 - 1/(1+r)^n )/ r)
20
Q
Perpetuity with Growth
A
- The first cash flow occurs at the end of the 1st period and is denoted by C
- The cash flows must grow at a constant rate each period denoted by g
- The timing of the cash flows occur at constant intervals
- The discount rate is the effective rate for the period between cash flows and is constant over time and is denoted by r
- n the number of periods tends to infinity
- The PV valuation point is one period before the first cash flow
- The discount rate for the period in between cash flows has to be greater than the growth rate between cash flows i.e. r>g
21
Q
Present Value of Perpetuity + Growth
A
C/r-g
22
Q
Perpetuity
A
- The first cash flow occurs at the end of the 1st period and is denoted by C
- The cash flow C must be constant each period
- The timing of the cash flows occur at constant intervals
- The discount rate is the effective rate for the period in between cash flows and is constant over time and is denoted by r
- n the number of periods tends to infinity
- The PV valuation point is one period before the first cash flow
23
Q
Perpetuity Formula
A
PVP = c/r
24
Q
PVP, PVA, Annuity + Growth, Perpetuity + Growth use CFs where they occur at the end of the period, but what if CFs occur at the BEGINNING of the period? What two formulas can we use?
A
Annuity Due
Future Value Annuity
25
Annuity Due formula
- We can calculate the PVs of these cash flows using the annuity formula where the number of CFs at the end of each period is (n-1)
PV = C + C(( 1- (1/(1+r)^n-1))/r)
26
Future Value of an annuity formula
FVA = C((1+r)^n -1/r)
- Monthly APR is the SAME as stated annual interest rate with monthly compounding
27
Stockholders
stockholders are the owners of a company. They have voting rights on company decisions (e.g. annual general meetings and extraordinary general meetings)
28
annual general meeting
a yearly meeting where shareholders vote on company policies, elect board members and discuss financial performance
29
Extraordinary general meeting
a meeting held for urgent matters, requiring shareholder votes, outside of the regular AGM
30
Bondholders
lend money to the company through bonds but don't have voting rights. They are paid fixed interest (coupons) but don't recieve any of the company's profit directly
31
Residual cash flows
stockholders are entitled to any leftover cash after the firms expenses and debt obligations are paid
32
common stock
represents ownership in a company and entitles shareholders to a portion of the company's profits. Stockholders may receive dividends, which are payments made from the company's earnings
33
Initial public offering (IPO)
the first sale of a company's stock to the public allowing the firm to raise capital from investors in exchange for ownership shares
34
Shareholders vs Stakeholders
- Shareholders own part of a company through stock ownership. They have financial interest in the companys profit, more ST focused, can make decisions
- Stockholders are affected by the company's decisions, regardless of whether they own stock in the company; employees, customers, supply chain partners, govt; affected by the company's performance.
35
Secondary market
where previously issued shares are traded between investors (e.g. the stock exchange), this market doesn't involve the issuing company but allows investors to buy and sell shares.
(Primary: where securities are created and first issued)
36
Dividends
payments made by companies to shareholders, typically from profits. Unlike bond coupons, dividends aren't guaranteed and can vary depending on the company's performance.
37
Dividend Yield Ratio
compares the annual dividend to the share price, calculated as:
dividend yield = annual dividend/share price
38
Price to Earnings (PE) ratio:
measures the relationship between a company's stock price and its EPS
P/E Ratio: Share price/EPS
38
Perpetuity Formula with dividends as CFs
Po = D/E(r)
38
38
38
3 ways to value a company (to its shareholders)
1. Market Value: the total value of a companys stock as priced by the stock market; mkt value = share price x no. of shares outstanding
2. Book Value: the accounting value of a company's equity as listed on the balance sheet. represents the company's net worth from an accounting perspective
3. Liquidation Value: The amount that would be left for shareholders if the company's assets were sold off and all creditors were paid; worst case scenario of the company's value
39
Expected dividend yield
E0(D1)/P0
39
Expected capital appreciation
% return from the change in stock price
E0(P1-P0)/P0
40
You buy a share today, defined as time t=0, in a corporation that currently has a price of P0. You know that at the end of the year t=1, the firm will pay you a dividend D=! and after the payment the dividend you will be left with a share worth P1. You don't know with certainty the values of D1 and P1 today. How can you estimate the % one year return you will obtain from holding the stock?
E(r) = expected dividend yield + expected capital appreciation
41
Assume for the particular stock we're looking at, investors demand a constant E(r) over time. How will investors value the stock today?
Investors will value the stock today, P0, based on expected dividends and the stock's future price, discounted to the present.
41
If we make the assumption that E0(Pt)/ (1+E(r))^t tends to zero as t tends to infinity then we obtain the stock pricing equation:
P0 = E0(Dt)/ (1+E(r))^t
The stock price is just the Pv of an infinite stream of dividends
Prices will be greater when expected dividends are greater
Prices will be lower when the expected return required by investors E(r) rises
42
Assume that we expect all future dividends to be at a constant level of D. The price of the stock becomes:
P0 = D/(1+E(r))^t
This is just a perpetuity and we know that its PV is equal to
P0 = D/E(r)
Where D is the constant dividend and starts one period after the valuation point
43
Assume we expect dividends to grow at a constant rate in the future. Then the stock price is
P0 = D(1+g)^t-1 / (1+E(r))^t
This is just a perpetuity with growth and we know it can be evaluated as
P0 = D/E(r) - g
44
Gordon Growth Formula
P0 = D/E(r) - g
Shows that the stock price P0 depends on:
- the dividend D: higher dividends make the stock more valuable
- the growth rate g: higher growth rates in dividends increase the stock price because FCFs (dividends) are expected to be larger
- The required return E(r): If investors require a higher return to compensate for risk, the stock price decreases because FCFs are discounted more heavily
45
ROE
ROE is a measure of the amount of earnings that a pound in equity (bookvalue) creates
ROE = EPSt / BV of equity per share t-1
- shows how effectively a company is using its equity to generate profits
46
Plowback Ratio vs Payout Ratio
Plowback ratio is the proportion of earnings retained by the firm and used for investment
Payout Ratio is the ratio of dividends to earnings
47
Earnings growth formula
g = ROE x PlowbackR
48
Deriving earnings growth rate
Change in earnings = new investment x ROE
- We assume that new investment comes from re-investing earnings only and hence depends on earnings and the plowback ratio
new investment = earnings x plowback ratio
earnings growth rate = change in earnings/earnings
= new investment x ROE/earnings
= PlowbackR x ROE
If you plowback earnings into investment projects, this will enable your dividends to grow faster. Growth will be greater when ROE of your investments is larger
49
Dividend growth rate
If ROE and plowback ratio are both constant, then the company reinvests a steady portion of its earnings each year, Because ROE is constant, the return on each pound invested of equity is also steady, leading to constant g of earning which roughly matches growth of dividends.
Dividend growth rate = ROE x PlowbackR
50
PVGO = Present Value of Growth Opportunities
Represents the company's value of future growth opportunities - basically, how much the company's stock price is driven by the potential for future growth rather than its current earnings
When we break down P0, we look at two components:
1) The PV of earnings if all were paid as dividends (i.e. no growth), this is what the stock price would be if the company just paid out all of its current earnings as dividends and didn't re-invest any of it for growth.
2) PVGO: This part represents the additional value investors are willing to pay because they expect the company to grow in the future
P0 = EPS1/E(r) + PVGO
which becomes...
EPS1/P0 = E(r) x (1 - (PVGO/P0))
51
Why should companies not use E/P ratios to understand the required return?
Key Concepts First
E/P Ratio (Earnings Yield): This is just the company’s earnings per share (EPS) divided by the current stock price (P). It’s a way to see what percentage of the stock price is made up by its earnings. A higher E/P ratio might suggest a higher return on investment if we’re only looking at the earnings side of things.
Required Return (E(r)): This is what investors expect to earn based on the risk and growth of a company. It’s not just about the current earnings but also includes the company’s future growth potential.
PVGO (Present Value of Growth Opportunities): This represents the portion of the stock price that investors attribute to future growth opportunities, rather than current earnings. Growth-focused companies have a higher PVGO because investors expect them to grow significantly, and this growth adds extra value beyond just current earnings.
The Equation’s Meaning
The equation:
... shows that the earnings yield (E/P) only reflects part of the required return investors are expecting. The other part comes from the growth potential of the company, which is represented by PVGO. If a company has a lot of growth potential, PVGO is a larger part of the stock price.
Why E/P Alone Doesn’t Work for Growth Companies
When a company has high growth prospects, a large part of its stock price is due to this growth (the PVGO part), not just current earnings. Therefore:
The E/P ratio will only reflect the earnings part and ignore the growth part (PVGO).
This means that for high-growth companies, the E/P ratio will understate the true required return because it misses the value from future growth.
52
Term Structure of Interest Rates
What is the term structure of interest rates?
The term structure describes how interest rates vary by the maturity of cash flows. This variation implies that different cash flows (from bonds or loans) are discounted at different rates based on their time until maturity.
Why does it matter?
Term structure theories attempt to explain these rate differences. This affects pricing of bonds and other financial instruments as each cash flow's maturity affects its discount rate.
53
Spot Rate & PV Calculation
The spot rate (denoted as rt)
is the fixed interest rate for a loan starting today over a specified time t, quoted as an annual rate.
Present Value (PV) Examples:
For £1 received in two years:
1/(1+r2)^2
For £1 received in three years:
1/(1+r3)^3
54
Why are government bonds priced using spot rates
Government bonds are typically default-free and are priced with spot rates to accurately reflect the time value of cash flows across different maturities. This process avoids arbitrage opportunities, ensuring that each cash flow’s present value aligns with market conditions.
Price Formula: The price of a bond with
n years to maturity, annual coupon
C, and face value
F is:
P = C/(1+r1) + C/(1+r2)^2 + C/(1+r3)^2 ...
Why Spot Rates?: Government bonds are priced this way to avoid arbitrage—ensuring consistent pricing across bonds by using current, risk-free spot rates.
55
What is arbitrage in bond pricing?
Arbitrage involves creating a strategy with a positive cash flow today and zero future cash flows. It ensures that any mispricing between a bond and its equivalent cash flows is corrected, as otherwise, investors could profit without risk.
Example: Selling an overpriced bond and buying the equivalent cash flow stream cheaper to lock in a profit.
Bond Example: If Bond A is overpriced relative to its replicating portfolio, one could short-sell Bond A and buy parts of Bonds B and C to create arbitrage profit.
56
What is short selling in bonds?
Short selling a bond involves borrowing and selling a bond you don’t own, with the aim of repurchasing it at a lower price later. This yields profit if the bond price falls.
Process:
1. Borrow and sell the bond at Date 0.
2. Compensate the lender for any coupon payments.
3. Repurchase the bond later at a lower price and return it to the lender.
57
How does a replicating portfolio work?
A replicating portfolio involves combining zero-coupon bonds to match the cash flows of a coupon bond. The combined value must equal the coupon bond's price due to no-arbitrage conditions.
Example:
To replicate a two-year coupon bond (Bond A), buy specific amounts of shorter-maturity zero-coupon bonds (e.g., 0.1 units of a 1-year bond and 1.1 units of a 2-year bond).
58
What is Macaulay Duration?
Macaulay Duration is the weighted average time to receive a bond’s cash flows, measuring interest rate sensitivity.
It reflects the "average" time for cash flows and indicates the bond’s price sensitivity to interest rate changes—the longer the duration, the more sensitive the price is.
59
Modified Duration?
Modified Duration adjusts Macaulay Duration to directly measure price sensitivity to interest rate changes:
D mod = D mac / 1+y
Use: Provides an approximate percentage change in bond price for a given change in yield, making it practical for assessing bond risk.
60
How do bond prices respond to yield changes?
The bond price’s sensitivity to yield changes is measured by the derivative of the price-yield curve.
Formula:
dy/dP ≈ −Dmod × P
Implication: This formula shows that for a given change in yield, the bond price moves inversely, scaled by its modified duration.
61
How are bond prices and interest rates related?
Inverse Relationship: Bond prices decrease when interest rates rise and increase when rates fall.
Sensitivity: Long-term bonds are more sensitive to yield changes than short-term bonds because their cash flows are discounted over a longer period, increasing their duration.
62
What are Spot Rates and Forward Rates in bond markets?
Spot Rate (rₜ): Interest rate on a loan starting today for a specified term.
Forward Rate (f): Agreed-upon rate for a loan starting in the future, determined by spot rates.
Non-Arbitrage Condition:
(1+rt)^t x (1+ft,T) = (1+rt+T)^t+T
Ensures that borrowing at the spot rate and investing at forward rates yields the same as investing at the longer-term spot rate.
63
What is the Expectations Theory of the term structure of interest rates?
Core Idea: The shape of the yield curve reflects investor expectations about future interest rates.
Mechanism: Investors view forward rates as unbiased forecasts of future spot rates. Thus, a bond’s long-term rate represents an average of expected future short-term rates.
Interpretation:
Upward Sloping Curve: Signals that investors expect future interest rates to rise.
Flat Curve: Suggests that future rates are expected to remain similar to current rates.
Downward Sloping Curve: Implies an expectation that future rates will fall.
Implication: An upward sloping term structure can indicate economic expansion, while a downward slope may signal a recession or rate cuts.
64
What is the Liquidity Premium Theory in the term structure of interest rates?
Core Idea: Investors prefer liquidity (short-term bonds) over long-term investments due to lower risk and greater flexibility.
Mechanism:
Short-term bonds are less sensitive to interest rate changes, making them more attractive to investors.
To entice investors into holding longer-term bonds, issuers must offer a liquidity premium—extra return on long-term bonds to compensate for added risk and lower liquidity.
Interpretation:
- Natural Shape: The term structure is typically upward sloping, as long-term rates need to be higher to compensate for liquidity preference.
- Exception: Only if interest rates are expected to decrease significantly would the term structure slope downward.
- Implication: This theory explains why long-term rates tend to be higher than short-term rates even when no rate changes are expected.
65
What is the Market Segmentation Theory regarding the term structure of interest rates?
Core Idea: Different investor groups are active in different parts of the yield curve, based on their needs and preferences.
Mechanism:
- Short-Term Investors: Institutions like banks that prefer short-term securities for liquidity and meet near-term liabilities.
- Long-Term Investors: Entities such as pension funds and insurance companies that seek long-term assets to match their long-term liabilities.
Result: Since these groups don’t usually switch between short- and long-term bonds, short-term and long-term rates are determined independently of each other.
Implication: The yield curve's shape can vary widely depending on demand in each segment, potentially appearing upward, downward, flat, or even hump-shaped, without a clear link to expected future interest rates.
66
What is the Preferred Habitat Theory in the term structure of interest rates?
Core Idea: A modification of Market Segmentation Theory, this theory suggests that investors have a preferred range of maturities (habitat) but may deviate if offered a high enough return.
Mechanism:
While investors are typically biased toward a specific maturity range, they may invest in other segments if there is sufficient incentive (e.g., a higher yield).
This movement affects the yield curve, as shifts in demand can steepen or flatten the curve depending on where the incentives are strongest.
Interpretation: Preferred Habitat Theory provides a flexible view, suggesting that changes in the yield curve can result from investor preferences shifting between maturity segments based on available yields.
Implication: This theory allows for mixed influences on the term structure, with both investor preferences and interest rate expectations impacting the yield curve’s shape.
67
What are populations, parameters, samples, and statistics in finance?
Population: The entire set of items or data points from a system under study (often too large to be fully observed).
Parameter: A characteristic or measure of a population (e.g., population mean).
Sample: A manageable subset drawn from the population.
Statistic: A measure derived from a sample, used to estimate the population parameter.
68
How are expected return, variance, and standard deviation of an asset’s returns calculated?
Expected Return: The weighted average of possible returns, where weights are the probabilities of each outcome.
Variance: Measures the risk of an asset as the expected squared deviation of returns from the mean.
Standard Deviation: The square root of variance, providing a measure of risk in the same units as return.
69
What are covariance and correlation, and how do they measure asset return relationships?
Covariance: Measures how returns on two assets move together.
Positive covariance means returns generally move in the same direction; negative covariance means they move in opposite directions.
Correlation: A standardized measure of the linear relationship between two assets, ranging from -1 to +1.
+1: Perfect positive correlation.
0: No correlation.
-1: Perfect negative correlation.
70
How do portfolio weights, expected returns, variances, and covariances contribute to portfolio risk and return?
Portfolio Weights (wi): The proportion of total investment in each asset.
Expected Portfolio Return: Weighted average of individual asset returns.
Portfolio Variance (Two-Asset Case): Considers individual asset variances and covariances
General Portfolio Variance: Sum of variances and covariances across all asset pairs in an N-asset portfolio.
71
What is diversification, and how does it reduce portfolio risk?
Diversification: The process of spreading investments across various assets to reduce unsystematic (firm-specific) risk.
- Empirical Insight: Stocks tend to have correlations less than +1, allowing individual stock risks to partially offset each other in a portfolio.
- Effectiveness: Diversification is most effective when assets have low correlations.
- Limitation: Diversification reduces specific risks but cannot eliminate systematic (market) risk.
72
How does a stock’s beta measure its contribution to portfolio risk, and what does it imply?
Beta (βi): Measures a stock's sensitivity to the overall market movement
βi = Cov(Ri,Rp) = variance x p
Interpretation:
- Positive Beta: Stock generally moves in the same direction as the market.
- Negative Beta: Stock typically moves opposite to the market, acting as a hedge.
- Magnitude: Higher beta means the stock contributes more to portfolio risk.
Examples:
Beta = 0.4: Stock tends to move in the same direction as the market, but with less sensitivity.
Beta = -0.25: Stock moves opposite to market movements, providing a hedging effect.
73
How do you calculate portfolio beta in a multi-asset portfolio?
Portfolio Beta: A weighted average of individual stock betas in a portfolio.
Formula (Two-Asset Portfolio):
𝛽𝑝 = 𝑤1𝛽1 + 𝑤2𝛽2
Implication: The overall portfolio beta shows how sensitive the portfolio is to market movements, guiding investment risk management.
74
What are the key investor preferences assumed in portfolio analysis?
- Prefer portfolios with lower standard deviation/variance for the same expected return.
- Prefer portfolios with higher expected return for the same standard deviation.
- Investors are mean-variance optimizers, valuing reward (mean return) and disliking risk (variance/standard deviation).
75
What is the Portfolio Frontier?
A set of portfolios formed from given securities that minimize variance (standard deviation) for varying levels of expected return.
76
What is the Efficient Frontier?
The portion of the Portfolio Frontier where each portfolio achieves the highest expected return for a given variance.
It lies above and to the right of the
Minimum Variance Portfolio.
77
What is the Minimum Variance Portfolio?
The portfolio with the lowest possible variance (standard deviation).
78
What happens when combining two risky assets with varying correlations?
Corr(R₁, R₂) = 1: Risk increases linearly with expected return. No diversification benefit.
Corr(R₁, R₂) = -1: Maximum diversification benefit. Can even create a risk-free portfolio.
Corr(R₁, R₂) = 0: Diversification reduces risk, but not to the extent of a perfect negative correlation.
79
What are the implications of diversification?
Reduces risk significantly as portfolio returns are less volatile than individual securities.
The benefit is due to non-perfect correlation between securities.
If all securities were perfectly correlated, diversification benefits would disappear.
80
What is the goal of Mean-Variance (MV) Optimization?
Minimize portfolio variance for a given level of expected return.
81
What defines the Portfolio Frontier for N risky assets?
The set of portfolios minimizing variance for varying expected returns.
82
What is the Efficient Frontier?
All portfolios on the frontier with expected returns greater than the minimum variance portfolio
83
How is the Minimum Variance Portfolio identified?
It is the portfolio with the lowest possible risk for N securities.
84
Why isn’t every portfolio on the frontier efficient?
- A portfolio is efficient only if it offers the maximum return for its level of risk.
E.g., Portfolio O dominates M because O offers higher return for the same risk.
85
That determines an investor’s choice on the Efficient Frontier?
Risk aversion level:
- Highly risk-averse investors: Choose portfolios with low risk (low return standard deviation).
- Less risk-averse investors: Opt for portfolios with higher risk and return.
86
What is the role of indifference curves?
Investors choose portfolios where their indifference curves are tangent to the Efficient Frontier, maximizing their utility.
87
What constraints are applied in MV Optimization?
1. Portfolio weights sum to 1
2. Expected Returns matches the Target
88
What drives the benefit of diversification?
- Securities’ returns are not perfectly correlated.
- Diversification minimizes portfolio risk, which is desirable for risk-averse investors.
89
What happens if all securities are perfectly correlated?
The benefits of diversification vanish. The portfolio would behave like a single asset.
90
How does the Portfolio Frontier change with correlation?
Corr = 1: Linear relationship between risk and return.
Corr = -1: Significant risk reduction; potential for risk-free portfolio.
Corr = 0: Typical diversification curve, less extreme than -1.
91
What happens when combining a risk-free asset and a risky asset/portfolio?
The risk-return opportunity set is linear in expected return-standard deviation space.
92
What is the Capital Allocation Line (CAL)?
A linear line representing the risk-return tradeoff between a risk-free asset and a risky asset/portfolio.
To work out the gradient, use the Sharpe Ratio formula
93
What does a point on the CAL represent?
Between risk-free asset and portfolio A: Partly invested in the risk-free asset and partly in portfolio A.
To the right of portfolio A: Borrowing at the risk-free rate to invest more than 100% in portfolio A.
94
How does the inclusion of a risk-free asset change the efficient frontier?
The new efficient frontier is a straight line (CAL) joining the risk-free rate to the tangency portfolio on the efficient frontier of risky assets.
95
What is the tangency portfolio?
The portfolio of risky assets where the CAL is tangent to the efficient frontier of risky assets only. Maximises the Sharpe Ratio.
96
Why is the tangency portfolio important?
It is the optimal risky portfolio for all investors.
All investors combine the risk-free asset with the tangency portfolio, varying proportions based on their risk aversion.
97
How do risk preferences affect portfolio choice?
Highly risk-averse investors: Invest more in the risk-free asset and less in the tangency portfolio.
Less risk-averse investors: Borrow at the risk-free rate and invest heavily in the tangency portfolio.
98
What does the CAL illustrate about risk-return tradeoff?
Combining a risk-free asset with a risky portfolio allows investors to:
Achieve lower risk than the risky portfolio with partial allocation to the risk-free asset.
Achieve higher returns by leveraging and borrowing at the risk-free rate.
99
What are the key assumptions of CAPM?
1. N risky assets and one risk-free asset:
2. Borrowing and lending occur at the same risk-free rate.
3. Costless trading: No transaction costs or taxes.
4. Mean-variance optimization: Investors care only about expected return and variance.
5. Homogeneous expectations: All investors have the same information and expectations about returns, variances, and covariances.
6. Single-period investment horizon: All investors make decisions for the same time frame.
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Why do all investors hold the same tangency portfolio?
With homogeneous expectations and mean-variance optimization, all investors derive the same efficient frontier and select the same tangency portfolio.
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How do investors differ in their portfolios?
They combine the risk-free asset and tangency portfolio in different proportions based on risk aversion.
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What is the market portfolio?
If all investors hold the same tangency portfolio, this portfolio represents the market portfolio.
The weight of each asset in the market portfolio equals its proportion in the overall market.
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What are the key assumptions of the CAPM?
1. N risky assets and one risk-free asset:
- Investors can borrow/lend at the same risk-free rate.
2. Costless trading: No transaction costs or taxes.
3. Mean-variance optimization:
- Investors care only about mean return and variance.
4. Homogeneous expectations:
- All investors have the same information and beliefs.
5. Single-period time horizon:
- Investors optimize over the same time frame.
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Why do all investors hold the same tangency portfolio in CAPM?
- Homogeneous expectations mean all investors use the same inputs (expected returns, variances, and covariances) for optimization.
- The tangency portfolio maximizes the Sharpe ratio and is optimal for all investors.
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What does the efficient frontier look like for all investors under CAPM?
The efficient frontier, derived from risky assets only, is identical for all investors.
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Why do investors combine the tangency portfolio with the risk-free asset?
Borrowing/lending at the risk-free rate allows investors to adjust their risk exposure while maintaining efficiency.
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What is the market portfolio under CAPM?
The market portfolio is the tangency portfolio in equilibrium.
It represents the value-weighted portfolio of all risky assets.
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Why is the market portfolio optimal in equilibrium?
- Demand: All investors hold risky assets in proportions dictated by the tangency portfolio.
- Supply: The market portfolio summarizes the supply of risky securities.
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How are individual asset weights in the market portfolio determined?
Weight = Asset’s market capitalization / Total market capitalization.
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How is the risk of a stock measured in CAPM?
Risk is measured by its beta, which represents its contribution to the market portfolio’s risk.
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What is the Capital Market Line (CML)?
The CML is the efficient frontier in expected return-standard deviation space when a risk-free asset is included.
It represents combinations of the market portfolio and the risk-free asset.
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What is the slope of the CML?
The Sharpe Ratio of the market portfolio
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What is the Security Market Line (SML)?
The SML plots the E(r) against beta (B)
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What does the slope of the SML represent?
The Market Risk Premium
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What if a stock lies above the SML?
The stock has a positive alpha and is underpriced.
Trading implication: Overweight the stock in your portfolio.
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What if a stock lies below the SML?
The stock has a negative alpha and is overpriced.
Trading implication: Underweight the stock in your portfolio.
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How is CAPM used for stock valuation?
1. Discount rate: CAPM provides a risk-adjusted discount rate for estimating the present value of future dividends.
2. Portfolio selection:
- Combine the risk-free asset and market portfolio to maximize utility.
- Adjust weights for stocks with positive/negative alphas.
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How is CAPM used for project valuation?
Use the CAPM expected return as the discount rate for NPV calculations of projects.
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What are the empirical implications of CAPM?
1. Estimated alpha for all stocks should be zero.
2. Expected returns should be linear in beta.
3. Slope of the SML equals the market risk premium.
4. Expected returns depend only on beta.
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What empirical issues challenge CAPM?
- SML often has a slope lower than the MRP.
- Other factors like size and value affect returns:
Small-cap stocks: Tend to have higher returns than large-cap stocks.
High book-to-market stocks: Tend to have higher returns than low book-to-market stocks.
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What are some alternatives to CAPM?
1. Fama-French Three-Factor Model:
Adds size and value factors.
2. Arbitrage Pricing Theory (APT):
Multi-factor model without specific assumptions like CAPM.
3. Inter-temporal CAPM:
Extends CAPM for multiple periods.
4. Consumption-based Asset Pricing Models:
Focus on consumption patterns to explain returns.
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What are the key takeaways of CAPM?
1. Beta is the only measure of risk affecting expected returns.
2. Expected returns are linear in beta.
3. Stocks with non-zero alphas suggest mispricing and trading opportunities.
4. CAPM provides a benchmark for risk-adjusted performance.
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What is "profit" in the context of market efficiency?
Profit refers to earning returns above the required return for a security’s risk (abnormal returns), not just positive returns.
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What is the basic idea of efficient markets?
In an efficient market:
- All relevant information is instantly and accurately reflected in prices.
- No abnormal returns can be earned through trading.
- Securities are fairly valued at all times.
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What are abnormal returns?
Returns that are not explained by a security’s systemic risk factors (e.g., CAPM beta).
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How are abnormal returns calculated using CAPM?
AR = Actual Return - CAPM Expected Return
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What is the Efficient Market Hypothesis (EMH)?
Formulated by Eugene Fama (1970), EMH states:
- Asset prices reflect all available information.
- No return predictability exists in efficient markets.
- Sophisticated investors quickly eliminate return predictability.
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What are the three forms of market efficiency?
1. Weak Form:
- Prices reflect all past trading data (e.g., prices, volumes).
- No patterns in past data can be used for abnormal returns.
2. Semi-Strong Form:
- Prices reflect all past and publicly available information (e.g., earnings, news).
- Fundamental analysis cannot yield abnormal returns.
3. Strong Form:
- Prices reflect all past, public, and private information.
- Even insider trading cannot generate abnormal returns
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What evidence supports the Weak Form?
Tests of serial correlations in returns:
- Insignificant autocorrelations in market indices (e.g., S&P 500).
- Stock prices follow a random walk.
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What evidence supports the Semi-Strong Form?
Event studies (e.g., stock price responses to earnings or takeovers):
- Prices adjust quickly to corporate announcements.
- Mutual funds fail to generate abnormal returns (Carhart, 1997).
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Is the Strong Form of market efficiency realistic?
Unlikely to hold:
- Insiders sometimes profit using private information.
- Insider trading is illegal, making direct testing difficult
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How do prices adjust in an efficient market?
1. Prices react to the unexpected element in new information.
2. Trades reflect updated expectations, balancing market views.
- Overvalued assets are shorted.
- Undervalued assets are bought.
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What assumptions underlie market efficiency?
1. Rationality: All investors are rational and process information instantly.
2. Independent deviations: Optimists and pessimists cancel out irrational behavior.
3. Dominance of rational investors: Professionals with capital correct mispricings.
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What is the Joint Hypothesis Problem?
Tests of market efficiency require an asset pricing model.
If abnormal returns are observed:
- It may reflect market inefficiency.
- Or, the asset pricing model used (e.g., CAPM) may be incorrect.
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What anomalies challenge the EMH?
1. Momentum and Reversal: Stocks continue trends or revert unexpectedly.
2. Post-Earnings Announcement Drift: Prices adjust slowly after earnings.
3. Twin-Stock Puzzles: Identical stocks trade at different prices.
4. Asset Price Bubbles: Prices deviate from fundamentals.
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What factors limit arbitrage?
1. Implementation Costs:
Commissions, bid-ask spreads, short sales constraints.
2. Noise Trader Risk:
Mispricings may worsen in the short term.
Arbitrageurs with capital constraints may avoid correcting these mispricings.
Why does this matter?
Arbitrage constraints can allow mispricings to persist, contradicting the EMH.
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What are the practical implications of EMH?
1. Passive vs. Active Investing:
- Passive strategies (e.g., index funds) are preferred due to lower costs.
- Active strategies rarely outperform after accounting for costs.
2. Investment Strategies:
- Weak and semi-strong efficiency suggest active management has limited value.
- Strong form suggests even insider knowledge offers no advantage.
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Summary of EMH Evidence
1. Weak Form: Supported by empirical data (e.g., random walk in prices).
2. Semi-Strong Form: Supported by event studies and mutual fund performance.
3. Strong Form: Rarely holds due to insider trading evidence and regulatory issues.
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What is a derivative security?
A financial security whose value is derived from other underlying variables such as:
- Stocks
- Currencies
- Interest rates
- Indexes
- Commodities
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What are the primary types of plain vanilla derivatives?
Forwards and Futures: Obligation to buy/sell an asset at a pre-specified price on a future date.
Options:
- Call Option: Right to buy an asset at a pre-specified price before or on a specified date.
- Put Option: Right to sell an asset at a pre-specified price before or on a specified date.
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What are the two types of derivative markets?
1. Over-the-counter (OTC) markets:
- Direct trades between banks, fund managers, and corporate treasurers.
- Often customized but less regulated.
2. Exchange-traded markets:
- Standardized contracts traded on exchanges (e.g., Chicago Board Options Exchange).
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What are the common uses of derivatives?
1. Hedging:
To reduce risks associated with economic activities or investments (e.g., oil companies hedging price risks).
2. Speculation:
To gain exposure to certain risks deliberately (e.g., betting on FTSE100 movements using futures).
3. Arbitrage:
To exploit price differences between derivatives and their underlying assets for risk-free profits.
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What is a forward contract?
A bilateral agreement to buy/sell an asset at a pre-specified price (forward price) on a future date (maturity).
144
Key characteristics of forward contracts?
OTC contracts, not traded on exchanges.
The contract value is zero at inception (no money changes hands initially).
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What is a futures contract?
Similar to forwards but:
- Standardized and exchange-traded.
- Gains/losses are settled daily (marked to market).
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What are long and short positions in a forward contract?
1. Long position:
Obligation to buy the asset at the forward price.
Payoff at maturity: S(T) - F(0,T)
where:
S(T) is the market price at maturity
and F(0,T) is the forward price
2. Short position:
Obligation to sell the asset at the forward price.
Payoff at maturity: F(0,T) - S(T)
.
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How can forwards be used for hedging?
Example: A wheat farmer hedges against price fluctuations:
Contract: Sell 5000 bushels of wheat in 6 months at £4.50/bushels (£22,500 total revenue).
Outcomes:
If market price =
£3.50: Farmer gains £1.00/bushel (£5000 total gain).
If market price = £4.75: Farmer loses
£0.25/bushel (£1250 total loss).
Regardless of market prices, the farmer secures revenue of
£22,500.
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How do forwards differ from futures?
Feature
Forwards
Futures
Trading Venue
OTC
Exchange-traded
Customization
Fully customizable
Standardized contracts
Settlement
At maturity
Daily (marked to market)
Counterparty Risk
High
Low (cleared by exchange)
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What are the payoff formulas for basic derivatives?
1. FORWARDS
- Long: S(T) - F(0,T)
- Short: F(0,T) - S(T)
2. OPTIONS
- Call Option Payoff: max(S(T) - K,0) where K = strike price
- Put Option Payoff: max(K - S(T), 0)
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How are futures and forwards similar?
Both have the same payoff profiles:
- Long position payoff: S(T) - F(0,T)
- Short position payoff: F(0,T) - S(T)
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How are futures and forwards different?
1. Trading Location:
- Forwards: OTC (customised bilateral agreements)
- Futures: Exchange-traded (standardised)
2. Settlement:
- Forwards: Settled at maturity
- Futures: Settled daily (marked to market)
3. Counterparty Risk:
- Forwards: Higher risk (no clearinghouse)
- Futures: Lower risk (clearinghouse ensures settlement)
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What is marking to market?
- A process where gains/losses are settled daily based on the market price of the futures contract
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What tools are used in marking to market?
1. Margin Account:
- Holds funds to cover potential losses
2. Initial Margin:
- Minimum deposit required to open a contract
3. Maintenance Margin:
- Minimum balance required in the margin account. If the balance falls below this, the account must then be topped up to the initial margin level.
EXAMPLE
- Contract size: 100 oz, Futures price: £300/oz, Initial margin: £1500, Maintainance margin: £1000
WEEK 1: Futures price drops to £298.40 (Loss = 100 x (300 - 298.40) = 160)
WEEK 2: Futures price drops to £294.60 (Additional Loss = 100 x (298.40 - 294.60) = 380)
Margin falls below maintenance level, requiring a top up to £1500
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What principle is used to price forwards and futures?
1. Absence of Arbitrage:
- Investors exploit price differences until no arbitrage opportunities remain.
- Ensures fair pricing of derivatives.
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What are the laws governing arbitrage pricing?
1. Law of One Price:
- Identical payoffs in all states = identical prices
2. Law of Payoff Dominance:
- If Portfolio A's payoff is always greater than or equal to Portfolio B's payoff, then Price of A >= Price of B
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How is the forward price derived for an investment asset with no income?
1. Portfolio 1:
- Enter a long forward contract
- Pay F(0,T) at maturity to receive the asset.
2. Portfolio 2:
- Buy the asset at S0 today.
- Borrow S0 at the risk-free rate r until maturity
Arbitrage-Free Condition:
F(0,T) = S0(1+r)^T
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How is the forward price adjusted for fixed cash income?
1. Portfolio 2 (modified):
- Buy the asset at S0 today.
- Borrow the present value of F(0,T) and the present value of income (I).
Forward Price:
F(0,T) = (S0 - PV(I))(1+r)^T
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What if an asset pays a continuous yield y?
The forward price becomes:
F(0,T) = S0e^(r-y)T
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What are two special features of commodities?
1. Storage Costs:
- Commodities require storage, increasing the forward price.
- Adjusted Forward Price:
F(0,T) = (S0 + PV(U))(1+r)^T
where U is storage cost
2. Convenience Yield:
- Benefits from holding the physical commodity (e.g, during seasonal demand)
- Net convenience yield NCY = Convenience Yield - Storage Cost.
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How are forward exchange rates calculated?
1. Spot exchange rate (S0): Home currency per foreign currency
2. Risk-free rates:
rH: Home currency rate.
rF: Foreign currency rate.
3. Forward Exchange Rate:
F(0,T) = S0 x (1+rH)^T / (1+rF)^T
Example:
Spot rate:
S0 = 0.75 GBP/USD, rH = 2%, rF = 1%, T=1
F(0,T) = 0.75 x (1.02/1.01) = 0.7574 GBP/USB
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What is an option?
- A contract giving the right, but not the obligation, to buy or sell a specific quantity of an underlying asset at a pre-determined price (strike price) on or before a specific maturity date.
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What are the five key terms in an options contract?
1. Underlying asset (e.g. stock, index, commodity)
2. Quantity of the asset
3. Maturity date (expiration date)
4. Strike Price (exercise price)
5. Right to buy or sell (call or put)
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What are the types of options?
1. Call Option: right to buy the underlying asset.
2. Put Option: right to sell the underlying asset
164
How do European and American options differ?
- European options: can only be exercised at maturity
- American options: can be exercised any time before or on maturity.
165
What is the pay off for a call option?
CT = max (0, ST - K)
Exercise (in the money) if ST > K
Do nothing (out of the money) if ST < K
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What is the payoff for a put option?
PT = max (0, K-ST)
Exercise (in the money) if ST < K
Do nothing (out of the money) if ST > K
167
What are payoff and profit diagrams?
- Payoff Diagrams: Show potential payoffs at maturity, excluding initial option costs
- Profit Diagrams: include the cost of the option premium
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How do payoffs differ for long and short positions?
1. Long Call Option:
Payoff: max(0, ST-K)
Profit: max(0, ST-K) - C (includes premium C).
2. Short Call Option:
Payoff: -max(0, ST-K)
Profit: -max(0,ST-K) + C
3. Long Put Option:
Payoff: max(0, K-ST)
Profit: max(0, K-ST)-P (includes premium P)
4. Short Put Option:
Payoff: -max(0, K-ST)
Profit: -max(0,K-ST) + P
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Option Strategies- Straddle
STRADDLE
SetUp: Long a call and a put with the same strike price K and maturity.
PayOff:
- Gains if stock price moves significantly in either direction.
- Losses if stock price remains close to K
Use Case: Expect significant price movement but unsure of the direction.
170
Reverse Straddle
SetUp: Short a call and a put with the same strike price K and maturity.
PayOff:
Gains if the stock price remains close to K
Losses if the stock price moves significantly
Use Case: Expect low volatility and price stability around K
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Bull Spread (Using Calls)
SetUp: Long a call with strike price K1 and short a call with a higher strike price K2 (same maturity).
PayOff
Gains if the stock price rises moderately.
Loss limited to the difference K2 - K1
Losses if the stock price drops below K1
Use Case: Mildly bullish outlook with protection against extreme price movements.
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What does it mean to "buy" or "write" an option?
1. Buy an option (long position)
- Pay the premium for the right to exercise
- Profit potential is unlimited for calls, capped for puts
- Loss limited to the premium paid
2. Write an option (short position)
- Receive the premium upfront
- Obligated to fulfill the contract if exercised
- Loss potential is unlimited for calls, capped for puts
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What is the Binomial Option Pricing model? Why is it useful?
- A model used to calculate the fair value of an option.
- Assumes that the underlying stock price can only move to two possible levels in a single period
Up: Su = uS(0)
Down: Sd = dS(0) where u>1 and 0
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Key Formulas in the Binomial Model
1. Stock price at maturity
2. Call option payoff
3. Put option payoff
4. Delta: Shares in the replicating portfolio
5. Borrowing or investing amount
6. Value of the Option
1. S(u) = uS(0), S(d) = dS(0)
2. C(u) = max(0, S(u) - K) ; C(d) = max(0, S(d) - K)
3. P(u) = max(0, K - S(u); P(d) = max(0, K - S(d)
4. Δ = C(u) - C(d) / S(u) - S(d)
6. C(0) = ΔS(0) + PV(B)
Where PV(B) = B/(1+r)^T
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Why is the Binomial Model important? Does it depend on probabilities?
- it aligns with no-arbitrage principles
- provides a framework to calculate option prices based on underlying stock movements
It doesn't depend on probabilities. The replicating portfolio matches payoffs, ensuring prices are independent of actual probabilities.
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What is the Risk-Neutral Method in Binomial Option Pricing?
- A method to value options by assuming a risk-neutral world.
- In a risk-neutral world:
1. The expected return of all securities is the risk-free rate.
2. The probabilities used for up(q) and down (1-q) movements are not actual probabilities but risk-neutral probabilities.
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What's the formula for Risk-Neutral Probabilities (q):
q = (1+r) - d/ u-d
where
r: risk-free rate for the period
u: upward price factor
d: downward price factor
1-q: risk-neutral probability of a down move.
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How do you calculate the option price (C0) using the risk-neutral method?
C(0) = 1/1+r (q . C(u) + (1-q) . C(d))
where
C(u) = call option value if the stock moves up
C(d) = call option value if the stock moves down
1+r = discount factor to present value
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Why use the risk neutral method? Does the risk neutral method rely on real-world probabilities? What makes this method efficient?
It simplifies pricing by ignoring actual probabilities and focusing on risk-neutral probabilities. It applies to pricing any derivative, not just options.
No, it assumes a hypothetical world where all securities earn the risk-free rate
It calculates the expected payoff under risk-neutral probabilities and discounts it at the risk-free rate.
180
What does the Black-Scholes Option Pricing Model do?
It provides a formula to calculate the fair price of a European call and put options in a continuous time framework.
181
How does the Black-Scholes Option Pricing Model differ from the Binomial Option Pricing Model?
The Black-Scholes model assumes continuous price evolution, while the binomial model assumes discrete steps.
182
What are the key assumptions of the Black-Scholes model?
1. The stock price follows a geometric Brownian motion with constant volatility (σ).
2. The stock's percentage returns are normally distributed over time.
3. The risk-free interest rate (𝑟) is constant.
4. There are no transaction costs, dividends or taxes
5. Options are European-style (exercised only at maturity)
6. Markets are efficient, and arbitrage opportunities don't exist.
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How does volatility and time to maturity impact option prices?
Higher volatility increases both call and put option prices because it raises the likelihood of extreme stock price movements
Longer time to maturity generally increases the value of options because there is more time for the stock price to move favorably.
