Definition: Finance is the systematic and disciplined study of money transactions.
A simple equation summarizing finance:
Finance = Mathematics + Money
In finance, the range of mathematics varies from basic arithmetic to advanced topics like differential equations.
Finance is critical for practical management and decision-making in business.
It is the lingua franca of business.
James Simons - Mathematical professor turned successful hedge fund manager.
Warren Buffett - Renowned investor known for value investing strategies using simple arithmetical calculations.
Jack Welch - Former CEO of General Electric known for his decision-making and ability to enhance corporate value significantly.
All three individuals exemplify a deep understanding of financial language despite their diverse backgrounds.
Components include:
Households
Financial Intermediaries
Non-financial Corporations
Capital Markets
Two primary challenges:
Valuation of assets
Management of assets
Valuation entails determining the worth of an asset, leading to management decisions based on that valuation.
Example of asset value distinction:
Water is essential but inexpensive.
Diamonds are expensive but not necessary for survival.
Core concepts:
Stock: The level of assets (similar to variables).
Flow: The rate of change in assets (akin to derivatives).
Corporate financial decisions revolve around managing cash flows from:
Cash raised from investors
Cash invested in real assets
Cash generated from operations
Household financial management parallels corporate scenarios.
The roles of time and risk are critical in financial analysis; without them, finance becomes basic microeconomics.
Lack of these concepts simplifies financial decisions.
There is no such thing as a free lunch.
Other things equal, individuals prefer more money to less, money now rather than later, and less risk to more risk.
All agents act to further their self-interest.
These notes cover basic financial concepts including present value, net present value (NPV), perpetuity, annuities, and the implications of currency conversion on financial decisions.
The Net Present Value of a project is defined as the difference between the present value of cash inflows and the present value of cash outflows.
Mathematically, it can be expressed as:
$$\text{NPV} = \sum_{t=1}^{N} \frac{CF_t}{(1+r)^t} - C_0$$
where:
CFt = Cash flow at time t
r = Discount rate
N = Total number of periods
C0 = Initial investment (cost)
NPV can vary drastically based on the currency used for calculation.
If cash flows are evaluated in different currencies, exchange rates need to be considered.
In a scenario where exchange rates remain constant:
$$\text{NPV}_\text{local} = \sum_{t=1}^{N} \frac{CF_t}{(1+r)^t}, \text{ assuming } r \text{ is constant over time.}$$
If the exchange rate fluctuates, it could impact the cash flows and NPV depending on the rate of appreciation/depreciation.
The present value (PV) of a future cash flow is calculated as follows:
$$PV = \frac{CF}{(1+r)^t}$$
A firm spends $800,000 annually on electricity. Implementing a specialized lighting system reduces this expense by $90,000 annually for three years at a discount rate of 4%.
Cash flow for cost savings:
CF1 = CF2 = CF3 = 90, 000
Initial cost of systems: C0 = 230, 000
NPV Calculation:
$$NPV = \frac{90,000}{(1+0.04)^1} + \frac{90,000}{(1+0.04)^2} + \frac{90,000}{(1+0.04)^3} - 230,000$$
A perpetuity is an asset that pays a constant cash flow C forever, starting one year from now.
The formula for the present value PV of a perpetuity is:
$$PV = \frac{C}{r}$$
where:
C = Cash flow per period
r = Discount rate (interest rate)
For instance, if a perpetuity pays $100 a year and the interest rate is 10%:
$$PV = \frac{100}{0.10} = 1000$$
If cash flows grow at rate g, the present value is given by:
$$PV = \frac{C}{r - g}$$
with r > g.
An annuity is a financial product that pays a fixed sum C at regular intervals for a fixed number of periods T.
The present value of an annuity can be derived from the present value of a perpetuity:
$$PV = C \times \left( \frac{1 - (1 + r)^{-T}}{r} \right)$$
APR is the nominal interest rate without taking compounding into account.
EAR reflects the actual interest earned or paid after compounding.
FORMULA:
$$EAR = (1 + \frac{r}{n})^{n} - 1$$
where n is the number of compounding periods per year.
If compounding occurs continuously, the formula becomes:
A = Pert
where:
A = Amount of money accumulated after time t
P = Principal amount (the initial amount of money)
e = Base of the natural logarithm
r = Annual interest rate (decimal)
t = Time in years
These concepts are foundational in finance and will aid in understanding more complex financial instruments and investment strategies. Understanding present value, NPV, and compounding effects are vital for assessing investment opportunities.
As of the end of 2007:
Net Revenues: $19 billion
Net Income: $4 billion
Long-term Capital: $145 billion
Assets Under Management: $282 billion
Employees: 28,500
Resulting job dislocation in New York City following Lehman Brothers’ collapse.
The leverage ratio signifies the amount of debt relative to equity. A higher ratio indicates higher risk.
1. Using a 20% Down Payment
Home Price: $500,000
Down Payment: $100,000
Loan: $400,000
Leverage Ratio:
$$\text{Leverage} = \frac{\text{Total Assets}}{\text{Equity}} = \frac{500,000}{100,000} = 5$$
2. Using a 5% Down Payment
Down Payment: $25,000
Loan: $475,000
New Leverage Ratio:
$$\text{Leverage} = \frac{500,000}{25,000} = 20$$
As the example shows, the potential for loss is magnitude greater with higher leverage. A decline in asset value exposes the homeowner under higher leverage to greater loss.
A small percentage change in asset value (e.g., 10%) results in substantial losses for leveraged investors.
When considering market firms like Lehman Brothers:
Effect of a 10%drop on $145 billion: 145 billion × 0.1 = 14.5 billion loss
Nominal Return: Actual dollar growth of wealth without adjusting for inflation.
Real Return: Actual growth of purchasing power, adjusted for inflation.
Using the formula:
Real Return ≈ Nominal Return − Inflation Rate
Example: If Nominal Return is 15% and Inflation is 10%, then:
Real Return ≈ 15% − 10% = 5%
Wealth at time t and t+k given as:
Wt = Current Wealth
Wt + k = Wt(1 + r)
With inflation considered for adjustments to purchasing power:
$$\text{Purchasing Power of Wealth} = \frac{W_{t+k}}{I_{t+k}} \quad \text{where } I \text{ is the inflation index.}$$
Fixed income securities are instruments with fixed payoffs, generally involving debt obligations.
Treasury Securities
Corporate Bonds
Municipal Securities
Mortgage-Backed Securities
Coupon Bond Valuation:
Cash Flows: Individual cash coupons received until maturity, plus face value at maturity.
Example: For a $1,000 face value, issuing $50 semi-annually for 3 years.
Bond Price Calculation: Present Value (PV)
$$PV = \sum_{t=1}^{T} \frac{C_t}{(1+r)^t} + \frac{F}{(1+r)^T}$$
Where:
Ct = Cash flow at time t
r = Discount rate
Zero-Coupon Bond Valuation: Present value of a single future cash flow.
$$Price = \frac{F}{(1+r)^n}$$
The behavior of fixed income securities allows for insights into future interest rates and economic conditions.
Understanding financial leverage, the relationship between real and nominal returns, and how fixed income securities are structured provides critical insights into both individual financial decisions and the broader economic landscape. Transitioning from theory to practical application is essential, particularly in understanding current and prospective financial stability.
These notes provide a comprehensive overview of recent trends in financial markets related to interest rates and central banking policies. The focus will be on understanding the implications of these policies, as well as the fundamental concepts related to bond pricing and yield curves.
The Federal Reserve (Fed) chose not to cut interest rates as widely anticipated in the market.
Instead, they extended an $85 billion loan to AIG, indicating a stronger intervention in maintaining liquidity in markets.
In contrast, during the collapse of Lehman Brothers, the Fed did not intervene, which signifies different assessments of systemic risk associated with AIG and Lehman Brothers.
The market initially projected a cut in interest rates based on economic conditions.
The Fed’s choice to hold rates suggests a cautious approach, possibly due to already low rates and concerns about long-term effects of further cuts.
Importance of distinguishing between the cost of borrowing vs. the availability of credit was emphasized.
Spot Rate (Rt): The interest rate applicable for a loan or investment for a specific period beginning today.
Forward Rate (Ft, t + 1): The interest rate agreed upon today for a loan or investment that will begin at a specified future date.
The price of a zero-coupon bond can be written as:
$$P = \frac{F}{(1 + r)^t}$$
where P is the price, F is the face value, and r is the spot interest rate.
Spot rates can differ based on the maturity; hence, the effective annual rate (little r) can be computed as:
$$r_t = \sqrt[t]{\prod_{i=1}^{t}(1 + R_i)} - 1$$
where Ri are the series of spot rates.
Geometric Mean of Spot Rates:
r = (R1R2...Rt)1/t − 1
describes the average effective yield across multiple periods.
A yield curve plots the interest rates of bonds of equal credit quality but differing maturity dates. It can take several forms:
Normal Yield Curve: Upward sloping, indicating higher rates for longer maturities.
Inverted Yield Curve: Downward sloping, often indicating recession.
Flat Yield Curve: Similar rates across maturities; uncertainty in future interest rates.
The yield curve gives insight into future interest rate expectations. For instance:
$$Y_t = \frac{C_1}{(1 + R_1)} + \frac{C_2}{(1 + R_2)^2} + ... + \frac{C_n}{(1 + R_n)^n}$$
where Cn are future coupon payments and Rn are corresponding interest rates.
Several theories explain the shape of the yield curve:
Expectations Hypothesis: Current long-term rates reflect expected future short-term rates. Thus, Ft, t + 1 ≈ E[Rt + 1].
Liquidity Preference Theory: Investors demand a premium for longer-term bonds, causing a typical upward slope in yield curves.
Preferred Habitat Theory: Borrowers have preferred maturities, leading to varying supply and demand conditions across different maturities.
Market Segmentation Model: Market participants prefer specific maturities, impacting supply and demand curves for those maturities.
In conclusion, the understanding of interest rates, yield curves, and the dynamics of monetary policy is fundamental for navigating financial markets. The Fed’s actions can significantly influence these rates, and comprehending the various theories of term structure further enhances this understanding.
These notes cover topics related to the pricing of multiple fixed income securities, the yield curve, arbitrage opportunities, psychological reactions in financial markets, and the law of one price.
Price of Treasury securities has decreased due to:
Decreased market demand
Adjustments in risk perception
Long-term yield changes:
Y30-year increased from 4% to 4.37%
Possible inflationary expectations influencing yields.
Definition: The law of one price states that two identical cash flows must have the same market price.
Arbitrage opportunities arise when identical cash flows differ in price.
Example of arbitrage:
PAsset = PCash flow 1 = PCash flow 2
Where:
$$\begin{aligned}
P_{\text{Asset}} & < P_{\text{Cash flow 1}} \implies \text{Buy Cash Flow 1}\\
P_{\text{Asset}} & > P_{\text{Cash flow 2}} \implies \text{Sell Cash Flow 2}
\end{aligned}$$
A money market fund “breaks the buck” when it cannot return $1 for every $1 invested.
Notable instance: The Reserve Fund providing less than $1 due to market conditions.
Coupon bonds can be priced as a portfolio of pure discount bonds.
Equation for pricing coupon bonds:
$$P = \sum_{t=1}^{T} \frac{C}{(1+y)^t} + \frac{F}{(1+y)^T}$$
Where:
P = Price of the bond
C = Coupon payment
F = Face value
y = Yield to maturity (YTM)
T = Total number of periods
Definition: The weighted average time until cash flows are received.
Useful to measure interest rate risk:
$$D = \frac{\sum_{t=1}^{T} t \cdot \frac{C}{(1+y)^t}}{P}$$
Where D is the duration, and P is the present value of cash flows.
Measures sensitivity of the bond price to changes in yield:
$$D^* = \frac{D}{1+y}$$
Where D* is the modified duration.
Measures the curvature of the bond price yield relationship.
Allows for approximating changes in bond prices with changing yields.
$$\begin{aligned}
P_{\text{new}} & = P_{\text{original}} \left( 1 - D^*\Delta y + \frac{1}{2}C\Delta y^2 \right)\end{aligned}$$
Risky debt and credit ratings
Case studies on subprime securities
The yield curve is a graphical representation of interest rates on debt for a range of maturities. It reflects financial market sentiments and can indicate the health of the economy.
The yield on a three-month Treasury bill is measured at
71 − 72 basis points
indicating improving market conditions as compared to the previous week (previously about 30-40 basis points).
Concerns about inflation affect the long end of the yield curve. The yield on a 30-year bond is now
4.22%
which is lower than before.
Market prices reflect aggregate sentiment and may not be "correct" but instead represent current expectations of economic conditions.
Duration measures the sensitivity of a bond’s price to changes in interest rates. It reflects the average time it takes to receive cash flows from the bond.
Convexity describes the curvature in the relationship between bond prices and interest rates, illustrating how duration changes as interest rates change.
The bond price P(y′) at a new yield y′ can be approximated as:
$$P(y') \approx P(y) + P'(y) \cdot (y' - y) + \frac{1}{2} P''(y) \cdot (y' - y)^2$$
where P′(y) is the first derivative (sensitivity to yield changes) and P″(y) is the second derivative (reflecting convexity).
Credit ratings serve as a measure of default risk for corporate bonds. Major agencies include:
Moody’s
Standard & Poor’s (S&P)
Fitch
Investment Grade:
AAA: Highest quality
AA: High quality
A: Upper-medium quality
Non-Investment Grade:
Anything below BAA is considered more speculative with higher default risks.
The default premium varies by bond rating. For example, BAA bonds experience higher spreads compared to Treasury bonds due to perceived risk:
Spread = Yield (BAA) − Yield (Treasury)
Historically, Moody’s BAA bonds typically show lower default rates than non-investment grade over a longer horizon.
Securitization transforms illiquid assets into liquid securities by pooling them and issuing new securities based on the pooled assets, which could offer varying levels of risk and return.
Senior Tranche: Less risky, paid before junior tranche.
Junior Tranche: Riskier, receives payouts after senior tranche.
Consider a simple example where we pool two risky bonds:
Each bond has a face value of $1,000 with a default risk of 10%.
Creating a tranching structure changes the risk profile, and respective defaults are modeled as:
Probability that both bonds fail: pfail = (0.1)(0.1) = 0.01
Probability that at least one succeeds: 1 − pfail = 0.99
The correlation of underlying assets significantly affects risk assessments:
Perfectly correlated assets will lead to systemic risk as the probability of simultaneous defaults increases.
If traditional models assume independence but reality shows correlation:
Both senior and junior tranches can see valuations plunge as defaults spike beyond historical expectations.
Potential pitfalls include over-reliance on credit ratings and assumptions about asset independence. Continuous monitoring and adjustment of risk models are crucial for preventing financial dislocations and ensuring stability in markets.
The evolution of financial products such as bonds, along with their ratings and securitization methods, is essential for understanding investment dynamics and managing associated risks.
Equity represents ownership in a corporation.
When you own equity, you own a sequence of cash flows, primarily in the form of dividends or capital gains.
Dividends (D): Cash payments made to shareholders.
Capital Gains: Appreciation in the value of the equity.
Companies, particularly early-stage growth companies, may not pay dividends to reinvest in operations.
Residual Claimant: Shareholders are entitled to assets after bondholders are paid.
Limited Liability: Maximum loss is limited to the amount invested; personal assets are protected.
Voting Rights: Shareholders can influence company decisions through voting.
Primary Market: Initial issuance of securities.
Secondary Market: Trading of existing securities.
The price of a stock today (Pt) is determined by the present value of expected future dividends:
$$P_t = \sum_{t=1}^{\infty} \frac{D_t}{(1+r_t)^t}$$
where Dt is the dividend at time t and rt is the risk-adjusted return.
Dividends Dt can fluctuate and are based on company performance and market conditions.
Risk-Adjusted Return (r): Reflects the riskiness of the dividends.
If dividends are constant:
$$P = \frac{D}{r}$$
If dividends grow at a constant rate g:
$$P = \frac{D_0(1+g)}{r-g}$$
where D0 is the most recent dividend.
Price is an increasing function of expected cash flows and inversely proportional to the discount rate:
If r increases, P decreases.
Evaluating growth opportunities and their impact on stock prices can lead to rapid price fluctuations.
The relationship between the cost of equity capital (r), dividend yield (D/P), and growth rate (g):
$$r = \frac{D}{P} + g$$
This formula captures the total return required by investors.
Understanding equity pricing is complex, as it incorporates various risk factors and growth assumptions. Investors and financial managers must continuously evaluate market conditions, company performance, and the broader economic environment to make informed decisions regarding equity investments.
The current state of the stock market indicates a decline.
The three-month T-bill rate has increased by 60-70 basis points.
The Federal Reserve (Fed) is focused on maintaining liquidity in markets.
Despite market fears, overall interest rates remain relatively low.
Investors appear to be hesitant and are waiting for clearer signals regarding economic policies and the market.
Historical perspectives show that crises can create opportunities for investors who are willing to take risks.
Key insights from psychology and behavioral finance suggest that fear can cloud rational thinking.
A forward contract is a commitment to buy or sell a commodity or asset at a predetermined price at a future date.
Features include:
Customization: Agreements tailored specifically between two parties.
Non-Standardization: Terms can vary, with no fixed quantity or price.
Over-the-Counter Trading: Contracts are usually traded directly between parties and not on exchanges.
The forward price (F) is determined in such a way that the present value of the contract is zero at the initiation:
PV(F) = F ⋅ e − rt = 0
where r is the risk-free interest rate, and t is the time until settlement.
The buyer is referred to as "long" and the seller as "short".
There is significant counterparty risk; the risk that one party will default on the contract.
Futures contracts are similar to forward contracts but differ in several key aspects:
Standardization: Contracts for standardized quantities and qualities of commodities.
Daily Settlement: Marking to market every day where gains and losses are settled daily.
Clearinghouse Participation: Involves a clearing corporation that serves as an intermediary, reducing counterparty risk.
Futures prices are derived from the expectations about future spot prices.
Pfutures = Pspot + CostofCarry − ConvenienceYield
Most futures contracts are cash-settled meaning no actual delivery of the commodity occurs.
However, some contracts require physical delivery, which can lead to issues if not properly managed.
NYMEX crude oil futures may have contract prices such as $76.06 per barrel.
Contract margins are set to manage daily fluctuations and risk, typically requiring only around 5% of the total contract value.
In times of uncertainty, understanding financial instruments such as forwards and futures can help manage risk and maintain operational focus.
Companies must evaluate their exposure to commodity price fluctuations and determine the appropriateness of using these financial tools.
Futures and forward contracts are agreements to buy or sell an asset at a future date for a specified price. They serve as risk management tools.
Futures Price (Ft, T): The agreed price for the future transaction at time t for delivery at time T.
Spot Price (St): The current market price of the asset.
Contract Size: The quantity of the asset covered by the contract (e.g., in barrels for oil).
Consider an oil futures contract issued on July 27, 2007, for oil delivered in December for a price of $75.06 per barrel. Each contract is for 1,000 barrels.
Futures Price: F = 75.06
Contract Size: 1, 000 barrels
Total Contract Value: 1, 000 × 75.06 = 75, 060 dollars
No cash changes hands at the time of contract execution. The value of the contract adjusts daily based on price changes, leading to potential gains or losses.
The initial margin is the upfront cash deposited to open a position (e.g., $4,050).
The maintenance margin is the minimum account balance required to maintain the position (e.g., $3,000). Falling below this triggers a margin call.
If the futures price moves, the difference affects your margin account:
Profit/Loss = (Current Futures Price − Initial Futures Price) × Contract Size
Initial futures price: \$0.7455 per pound (for cattle). After a decrease to $0.7435:
$$\begin{aligned}
\text{Contract Size} &= 40,000 \text{ pounds} \\
\text{Value of Position} &= 40,000 \times 0.7435 = 297,400 \\
\text{Loss} &= 298,200 - 297,400 = 800 \text{ dollars}\end{aligned}$$
If the futures price rises above the initial price, the holder profits.
If the futures price falls below the initial price, the holder incurs losses.
To determine the prices of futures and forwards, we equate the present value of payoffs. The relationship can be derived as:
F0, T = S0(1 + r)T
Where r is the risk-free interest rate.
Cost of carry: storage costs or convenience yield may affect the pricing.
Standard relationship (net costs):
Ft, T = St(1 + r)T − t + Storage costs − Convenience yield
The market operates under conditions of supply and demand, and arbitrage opportunities can arise if futures prices deviate from expected spot prices.
Futures and forward contracts can serve not only as tools for hedging risk but also as methods for speculation. Understanding their pricing is crucial in capital markets.
Option pricing is a fundamental concept in financial markets, especially in the context of derivatives. Understanding the basic principles will provide insights into more complex financial instruments.
Options are contracts that give the holder the right, but not the obligation, to buy or sell an underlying asset at a predetermined price within a specified time period.
Call Options: Give the holder the right to buy the underlying asset at a specific strike price.
Put Options: Give the holder the right to sell the underlying asset at a specific strike price.
Understanding payoff diagrams is key to visualizing the value of options at expiration.
The payoff of a call option at expiration is given by:
Payoff = max (0, ST − K)
where ST is the stock price at expiration and K is the strike price.
The payoff of a put option at expiration is given by:
Payoff = max (0, K − ST)
The net payoff after paying the option premium P is given by:
Net Payoff = Payoff − P
This changes the shape of the payoff diagram, shifting it downwards by the premium amount.
If the premium paid for the call option is P, then the net payoff diagram will be:
Net Payoff = max (0, ST − K) − P
If the premium paid for the put option is P, then the net payoff diagram will be:
Net Payoff = max (0, K − ST) − P
In a futures contract, there is no initial premium to be paid. Thus, payoffs are symmetric. In contrast, options provide asymmetric payoffs.
Implied volatility is a crucial concept for option pricing. It represents the market’s expectation of future volatility and can be inferred from option prices. The Chicago Board Options Exchange (CBOE) publishes the VIX index, representing the market’s expectation of volatility.
The foundational work in option pricing theory emerged in the early to mid-20th century, particularly through the contributions of:
Louis Bachelier in 1900, introducing the concept of price random walk.
Fischer Black and Myron Scholes in the 1970s, formulating the Black-Scholes model.
The Black-Scholes formula is a key result that gives the theoretical price of European call and put options. It can be represented as follows:
Call = S0N(d1) − Ke − rTN(d2)
where:
$$d_1 = \frac{\ln(S_0 / K) + (r + \sigma^2 / 2) T}{\sigma \sqrt{T}}, \quad d_2 = d_1 - \sigma \sqrt{T}$$
with N(d) being the cumulative distribution function of the standard normal distribution.
Understanding options and their pricing is crucial for navigating financial markets. The asymmetry of payoffs, the impact of volatility, and the historical development of pricing theory provide a comprehensive background for approaching advanced topics in finance.
This document covers key points from lectures on option pricing and the binomial option pricing model. This method serves as a foundational tool for understanding the valuation of options and other derivative securities.
The binomial model allows for a straightforward approach to derive theoretical pricing formulas for options. It is simpler than the Black-Scholes formula and can be calibrated using historical stock price behaviors.
Let S0 be the current stock price.
A call option with strike price K expires in one period.
Tomorrow’s stock price can either go up or down:
$$\begin{aligned}
S_1 =
\begin{cases}
u S_0 & \text{(if up)} \\
d S_0 & \text{(if down)}
\end{cases}
\end{aligned}$$
where u > 1 (up factor) and d < 1 (down factor).
Probabilities of the respective outcomes:
$$\begin{aligned}
P(\text{up}) = p \quad \text{and} \quad P(\text{down}) = 1 - p
\end{aligned}$$
The payoffs at expiration of the call option are defined as:
$$\begin{aligned}
C_1 = \max(S_1 - K, 0) =
\begin{cases}
u S_0 - K & \text{(if up)} \\
0 & \text{(if down)}
\end{cases}\end{aligned}$$
To find the price C0 today, we use the notion of no-arbitrage, where we construct a riskless portfolio.
The expected option price can be expressed as:
$$\begin{aligned}
C_0 = \frac{1}{r} \left( p \cdot C_u + (1-p) \cdot C_d \right)\end{aligned}$$
Where:
$$\begin{aligned}
C_u &= \max(u S_0 - K, 0) \\
C_d &= \max(d S_0 - K, 0) \\
r &= 1 + \text{risk-free rate}\end{aligned}$$
The risk-neutral probabilities facilitate pricing without explicitly needing the probability p of outcomes.
The parameters C0, S0, K, u, d, and r determine the option’s price.
The derived price is a weighted average of potential option payoffs.
The binomial model can be extended for multiple periods through a tree structure, allowing for complex option pricing scenarios.
As the number of time periods approaches infinity and the intervals shrink, the binomial model converges toward the Black-Scholes formula, which is mathematically derived from continuous-time models.
The relationship between risk and expected return is foundational in finance, influencing portfolio management and asset pricing.
Expected Return: μ = 𝔼[R]
Variance of Returns: σ2 = 𝔼[(R − μ)2]
Standard Deviation: $\sigma = \sqrt{\sigma^2}$
Correlation: Measures how securities move relative to one another, crucial for diversification. Defined as:
$$\begin{aligned}
\text{Corr}(X, Y) = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y}
\end{aligned}$$
Understanding option pricing through the binomial model provides foundational knowledge for derivatives valuation. This approach not only covers policy decisions but also sets the stage for more complex models, including the celebrated Black-Scholes framework.
Cox, J.C., Ross, S.A., and Rubinstein, M. "Option Pricing: A Simplified Approach." Journal of Financial Economics, 1979.
Black, F. and Scholes, M. "The Pricing of Options and Corporate Liabilities." The Journal of Political Economy, 1973.
Understanding market dynamics and characteristics is crucial for financial decision-making. Key elements include:
Market prices are generally random and unpredictable.
Prices should react quickly to new information.
Investors should not be able to earn abnormal returns after risk adjustment.
A well-functioning market is one that does not exhibit predictable patterns in stock prices. If stock prices were predictable, investors could exploit these patterns, leading to volatility reduction.
Financial markets trend toward randomness; prices merely drift over time.
The more investors attempt to forecast prices, the more random prices become.
1. Real Interest Rates: The average real interest rate has been slightly positive:
$$\begin{aligned}
r_{real} = r_{nominal} - r_{inflation}\end{aligned}$$
For example:
$$\begin{aligned}
r_{1\text{yr}} &= 0.38\% \text{ (monthly)}, \\
r_{inflation} &= 0.32\% \text{ (monthly)}, \\
r_{real} &\approx 0.06\% \text{ (monthly)}.\end{aligned}$$
2. Average Returns:
Value Weighted Index (VW): 1% per month.
Equal Weighted Index (EW): 1.18% per month.
Example: Motorola had an expected rate of return of approximately 1.66% per month.
3. Risk-Reward Trade-off:
Higher risk associated with higher average returns.
T-bills exhibit low risk (σT − bills ≈ 5% monthly) compared to stocks like Motorola (σMotorola ≈ 10% monthly).
Expectations of return must include both price fluctuations and any coupon payments:
$$\begin{aligned}
R_{total} = \text{Price Fluctuation} + \text{Coupon Payment}\end{aligned}$$
Volatility can be quantified through standard deviation. A key measure of risk is:
$$\begin{aligned}
\sigma = \sqrt{\frac{1}{N-1} \sum_{i=1}^{N} (R_i - \bar{R})^2}\end{aligned}$$
where Ri represents individual returns and R̄ is the average return.
Investigating the daily return of a market index versus specific stocks shows little predictability:
$$\begin{aligned}
R_{today} \text{ vs } R_{tomorrow}\end{aligned}$$
A portfolio is a combination of securities, denoted by weights that sum to one:
$$\begin{aligned}
\omega = \left[ \omega_1, \omega_2, \ldots, \omega_n \right]\end{aligned}$$
where $\sum_{i=1}^{n} \omega_i = 1$.
Using weights greater than one indicates that the investor is utilizing leverage:
$$\begin{aligned}
\text{Long Asset} + \text{Short Asset} \to \text{Net Investment}\end{aligned}$$
Expected returns of the portfolio can be computed as:
$$\begin{aligned}
E(R_p) = \sum_{i=1}^{n} \omega_i E(R_i)\end{aligned}$$
Risk (standard deviation) can be defined for a two-asset portfolio as:
$$\begin{aligned}
\sigma_p = \sqrt{\omega_1^2 \sigma_1^2 + \omega_2^2 \sigma_2^2 + 2\omega_1 \omega_2 \text{Cov}(R_1, R_2)}\end{aligned}$$
The goal of an investor is to choose a portfolio that maximizes expected return for a given level of risk:
$$\begin{aligned}
\max_{E(R_p)} \text{ subject to } \sigma_p < \text{target risk}\end{aligned}$$
Empirical anomalies challenge traditional efficiency theory:
Size Effect: Smaller firms often provide a higher return than larger firms.
Value Premium: Value stocks outperform growth stocks.
Momentum: Past winning stocks tend to continue winning.
These empirical observations and principles of portfolio theory form the backbone of modern finance. Future discussions will elaborate on risk measurement and portfolio construction strategies to optimize investment decisions.
We left off discussing risk and return.
Assumption: Investors favor higher expected return and dislike risk (measured by volatility).
Graph representation:
X-axis: Standard deviation of the portfolio (σ)
Y-axis: Expected return of the portfolio (E[R])
Aim: Maximize happiness, depicted as moving northwest (higher returns, lower risk).
Construct a portfolio through a collection of securities.
Goal: Achieve the best possible position on the mean-standard deviation graph.
We need to understand mean and variance of a portfolio.
Expected return of a portfolio (E[Rp]):
$$E[R_p] = \sum_{i=1}^{n} \omega_i E[R_i]$$
Where:
ωi = weight of asset i
E[Ri] = expected return of asset i
Variance calculation:
σp2 = E[(Rp−E[Rp])2]
Expanding variance:
$$\sigma^2_p = \sum_{i=1}^{n} \omega_i^2 \sigma_i^2 + \sum_{i=1}^{n} \sum_{j \neq i} \omega_i \omega_j Cov(R_i, R_j)$$
Where:
σi2 = variance of asset i
Cov(Ri, Rj) = covariance between assets i and j
Covariance can be written in terms of correlation:
Cov(Ri, Rj) = ρijσiσj
Notice that variance is not a simple average of individual variances due to cross-products (covariances).
For two assets A and B:
E[Rp] = ωAE[RA] + ωBE[RB]
Variance becomes:
σp2 = ωA2σA2 + ωB2σB2 + 2ωAωBCov(RA, RB)
Plotting different combinations of assets leads to a curved bullet shape in the mean-standard deviation space.
Indicates that risk does not increase linearly with expected return.
Diversifications benefit from low or negative correlations; it can reduce overall portfolio risk.
If perfect negative correlation (ρ = − 1):
E[Rp] remains positive while ensuring σp = 0
Introduce a risk-free asset (e.g., T-bills):
E[Rp] = ωrisk − freeRrisk − free + ωstocksE[Rstocks]
If σrisk − free = 0, variance simplifies to:
σp2 = σstocks2
Investment decisions must be made to achieve a portfolio along the efficient frontier.
Key Insight: There are portfolios that allow better risk-return trade-offs than any individual asset.
The trade-off between expected return and risk is a central theme in finance. This lecture provides insights into portfolio construction, risk management, and the implications of diversification.
The expected return of a portfolio is calculated as a weighted average of the expected returns of its components:
$$E(R_p) = \sum_{i=1}^{n} w_i E(R_i)$$
where wi represents the weight of asset i in the portfolio, and E(Ri) is the expected return of asset i.
The variance of a portfolio (risk) is given by:
$$\sigma_p^2 = \sum_{i=1}^{n} w_i^2 \sigma_i^2 + \sum_{i \neq j} w_i w_j \sigma_{ij}$$
where σi2 is the variance of asset i, and σij is the covariance between assets i and j.
The efficient frontier is the set of portfolios that provides the highest expected return for a given level of risk. This can be visualized as a curve (bullet-shaped) in a risk-return graph.
Rational investors prefer to be on the upper branch of the efficient frontier.
Portfolios below the frontier can be improved upon through diversification.
An investor will never want to hold 100% of a single security when better trade-offs are available on the frontier.
The tangency portfolio is of particular importance:
It represents the highest Sharpe ratio (risk-adjusted return).
When mixed with a risk-free asset (T-Bills), it delineates the optimal risk-return trade-off line.
A critical aspect of portfolio construction involves understanding covariance among assets:
Adding assets that are not perfectly correlated decreases overall portfolio risk.
The more assets included, the closer the portfolio can get to the leftmost point of the bullet shape (lower risk).
The Sharpe Ratio is defined as:
$$\text{Sharpe Ratio} = \frac{E(R_p) - R_f}{\sigma_p}$$
where Rf is the risk-free rate, and σp is the standard deviation of the portfolio returns.
For any efficient portfolio:
$$E(R) = R_f + \frac{\sigma_p}{\sigma_{M}} (E(R_M) - R_f)$$
where RM is the return of the market portfolio, and σM is the volatility of the market portfolio.
The CAPM describes the linear relationship between the expected return of an asset and its risk, measured in terms of beta:
E(Ri) = Rf + βi(E(RM) − Rf)
where $\beta_i = \frac{\text{Cov}(R_i, R_M)}{\sigma_M^2}$ is a measure of the sensitivity of the asset’s returns to market returns.
Diversification reduces risk, making efficient use of the covariance matrix.
The tangency portfolio forms the basis for maximizing return per unit of risk.
Understanding beta allows for determinations of expected returns, impacting investment and capital budgeting decisions.
Investors must consider changes in parameters over time, market inefficiencies, and the limitations of this framework in real-world applications. Future discussions will delve deeper into project financing and corporate investment decisions based on these principles.
The risk-reward trade-off is represented by the Capital Market Line (CML), which describes the expected return of portfolios based on their risk.
Efficient Portfolio: A portfolio that offers the highest return for a given level of risk or the lowest risk for a given level of return.
Capital Market Line (CML): Represents portfolios that optimally combine risk and return.
Security Market Line (SML): Represents the relationship between systematic risk (beta) and expected return for all securities.
The equations for CML and SML are as follows:
Capital Market Line (CML):
$$E(R_p) = R_f + \frac{\sigma_p}{\sigma_m}(E(R_m) - R_f)$$
Where:
E(Rp) = Expected return of the portfolio
Rf = Risk-free rate
σp = Standard deviation of the portfolio’s returns
σm = Standard deviation of the market’s returns
E(Rm) = Expected market return
Security Market Line (SML):
E(Ri) = Rf + βi(E(Rm) − Rf)
Where:
E(Ri) = Expected return of asset i
$\beta_i = \frac{\text{Cov}(R_i, R_m)}{\text{Var}(R_m)}$ is a measure of the sensitivity of asset returns to market returns.
When β = 1:
E(Ri) = E(Rm)
This implies the asset has the same risk as the market.
When β = 0:
E(Ri) = Rf
This indicates a risk-free asset, which may still be volatile.
When β < 0:
E(Ri) < Rf
A rare situation where investors expect lower returns than the risk-free rate.
Beta serves as a measure of systematic risk. Unlike the total volatility (sigma), beta captures the risk inherent to a security relative to the market portfolio.
The beta of a portfolio can be computed as a weighted average of the individual betas of its components:
$$\beta_p = \sum_{i=1}^{n} w_i \beta_i$$
Where wi is the weight of asset i in the portfolio.
Performance can be measured through alpha, which indicates whether a security or portfolio outperformed the expected return predicted by the SML:
α = E(Ri) − (Rf+βi(E(Rm)−Rf))
CAPM has practical applications in finance, allowing investors to estimate the expected return on assets, and assess the attractiveness of investing based on the risk the asset adds to a well-diversified portfolio.
Given historical data:
For Microsoft, β = 1.49
For Gillette, β = 0.81
Assume Rf = 5% and E(Rm) − Rf = 6%
The required rates of return can be calculated using:
Microsoft:
E(RMSFT) = 5% + 1.49 × 6% = 13.94%
Gillette:
E(RGIL) = 5% + 0.81 × 6% = 9.86%
Understanding the relationship between risk (beta) and expected return (SML) is crucial for making informed investment decisions. The CAPM framework provides a systematic approach to evaluate whether an investment is attractive based on its risk profile compared to the market.
This lecture continues the discussion on the Capital Asset Pricing Model (CAPM) and transitions into the topic of capital budgeting. The CAPM is a critical framework used in finance to understand the relationship between risk and expected return.
The CAPM formula is given by:
E(Ri) = Rf + βi(E(Rm) − Rf)
where:
E(Ri) is the expected return on the asset
Rf is the risk-free rate
βi is the beta of the asset
E(Rm) is the expected return of the market
The CAPM asserts that the expected return of an asset is proportional to its systematic risk, as measured by beta.
In practice, empirical tests of CAPM can yield various results. For instance, when analyzing Biogen and Motorola, significant deviations from the expected alphas were noted, indicating possible undervaluation or inadequacies in the CAPM.
The goodness of fit of CAPM can be assessed using the R-squared statistic, which indicates how well the model explains the variability of asset returns. For Biogen and Motorola, the R-squared values were reported as:
Biogen: R2 = 0.175
Motorola: R2 = 0.33
These values suggest a reasonable, but not perfect, fit in financial data due to the inherent noise in the returns.
Empirical evidence shows varying performances of CAPM across different types of portfolios. Notably, the comparison between small-cap and large-cap portfolios reveals:
Small stocks: Average monthly return of 1.33%, β = 1.4
Large stocks: Average monthly return of 0.9%, β = 0.94
Using the CAPM framework, these returns can be expected based on the calculated risk-free rate and market premium. The analysis provides an approximation of expected returns versus actual performance.
While CAPM is widely used, it does have its limitations:
Does not capture all risks that influence asset prices.
Recent literature suggests additional factors (e.g., value, size anomalies) may explain variations in expected returns.
In practice, these factors lead to the development of multi-factor models—beyond the traditional CAPM.
The next section focuses on capital budgeting, which involves planning and evaluating potential large expenditures or investments.
The NPV is calculated as:
$$NPV = \sum_{t=0}^{T} \frac{C_t}{(1 + r)^t}$$
where:
Ct = cash flow at time t
r = discount rate
T = total time period
The NPV should be positive to consider an investment viable. Key considerations in capital budgeting include cash flow estimation and the appropriate discount rate.
Important points in cash flow analysis include:
Use cash flows rather than accounting earnings for decision-making.
Compute after-tax cash flows.
Focus on project-specific cash flows.
The discount rate for cash flows should reflect the project’s risk profile. For varying project phases, different discount rates might apply—especially if risks change over time.
The CAPM serves as a foundational model in finance but must be understood within the context of its limitations. Capital budgeting principles, particularly the NPV rule, provide a structured approach to evaluating investments, urging a focus on cash flows rather than accounting measures.
The lecture focuses on capital budgeting and project financing.
Alternatives to Net Present Value (NPV) will be discussed, specifically:
Payback Period
Internal Rate of Return (IRR)
Profitability Index
NPV is recommended as the primary criterion for capital budgeting decisions.
APV is an extension of NPV that accounts for:
Taxes
Project interactions
Strategic alternatives
Optionality
Recommended for advanced courses on capital budgeting.
Definition: The payback period (k) is the time required for an investment to generate an amount of income sufficient to recover the initial investment.
Cash flows are represented as cf0, cf1, …, cfk.
Criterion:
Accept the project if k ≤ t*
where t* is a predetermined threshold.
Issues with Payback Period:
Ignoring cash flows after the payback period.
Not considering the scale of investment and risk.
May lead to short-sighted decisions focused on career risk.
Incorporates discounting of cash flows to address the time value of money.
Still ignores cash flows beyond the payback period.
Useful for addressing liquidity concerns.
Definition: The IRR is the discount rate (r) that makes the net present value (NPV) equal to zero.
$$\text{NPV} = 0 \Rightarrow \sum_{t=1}^T \frac{\text{cf}_t}{(1 + r)^t} - \text{cf}_0 = 0$$
Acceptance Rule:
Accept if IRR ≥ r* (hurdle rate)
Select project with the highest IRR for mutually exclusive projects.
Shortcomings of IRR:
May not exist or might yield multiple IRRs depending on cash flow patterns.
Can lead to incorrect project rankings, particularly with non-conventional cash flows.
Ignores scale of project, making it less suitable for heterogeneous investments.
Definition:
$$\text{Profitability Index} = \frac{\text{Present Value of Cash Flows}}{\text{Initial Investment}}$$
Criteria:
Accept if the profitability index ≥ 1.
Maximize the profitability index when projects are mutually exclusive.
Shortcomings:
Does not consider the scale of investment, leading to potentially poor decisions when comparing different-sized projects.
NPV remains the best method for capital budgeting.
Understanding alternatives like Payback Period, IRR, and Profitability Index helps in comprehensively evaluating projects.
Importance of considering career risk and cultural inertia in investment decisions.
The following notes summarize key concepts covered in a course on efficient markets and behavioral finance, providing insights on market behavior, decision-making, and the underlying psychology that drives financial choices.
The Efficient Market Hypothesis asserts that asset prices reflect all available information. This implies that:
It is impossible to consistently achieve higher returns than the overall market.
Any price changes in stocks occur as new information becomes available.
Following the space shuttle disaster, it was observed that Morton Thiokol’s stock price dropped significantly in a short time:
"Stock prices reflected the enormous amounts of gathered information about the company’s practices."
The market’s reaction was swift, implying that the information pertaining to the company was already discounted in the stock price.
Over the last two decades, criticisms of the EMH have emerged, highlighting various market inefficiencies. A humorous illustration is the following quip regarding economists and their rational behavior:
"If that were a real, genuine $100 bill, someone would have already taken it, so it must be a counterfeit."
This satirically underscores the idea that if profitable opportunities exist, they should be recognized and acted upon swiftly by market participants.
Traditional finance relies on the assumption that market participants are rational. However, behavioral finance challenges this by showcasing human biases, including:
Loss Aversion
Overconfidence
Mental Accounting
Anchoring and Framing
The theory of behavioral finance posits that investors frequently make irrational decisions due to cognitive biases. One prominent example is loss aversion, described by Kahneman and Tversky:
"People prefer to avoid losses rather than acquiring equivalent gains."
Most individuals tend to choose a certain smaller gain over a risky option that offers greater potential returns. Conversely, they often gamble to avoid a certain loss despite the risk of greater losses occurring.
Consider two urns containing red and black balls. The first urn has an equal number of red and black balls. The second urn has an undisclosed distribution, yet individuals prefer the second urn which introduces uncertainty:
"Even with the same expected value, the lack of information leads to different preferences."
The relationship between rationality and emotion has been clarified by recent neurological research. In particular, the work of Damasio illustrates that emotions are crucial for rational decision-making.
Elliot lost part of his brain due to surgery and exhibited rational behavior yet became unable to make effective decisions. Damasio concludes that:
"Without emotion, rational decision-making becomes impaired."
According to Paul MacLean’s triune model:
The Reptilian Brain: Controls basic survival functions.
The Mammalian Brain: Manages emotions and social behavior.
The Hominid Brain: Responsible for higher cognitive functions, including logic and decision-making.
The Adaptive Markets Hypothesis combines elements of both rational finance and behavioral finance:
"Market participants are influenced by psychological biases, yet they learn and adapt through experience."
When making financial decisions, individuals should be mindful of the emotional and cognitive biases that may influence their judgment. A heuristic approach that incorporates emotional regulation can lead to better results.
The interplay between market efficiency and behavioral finance sheds light on the complexities of financial decision-making. By understanding these dynamics, investors can refine their strategies and better navigate the markets.
The adaptive markets hypothesis suggests that markets satisfy the following six properties:
Individuals act in their own self-interest, but they also make mistakes.
Individuals learn and adapt over time.
Competition drives adaptation and innovation.
Natural selection shapes the survival of various heuristics used by individuals.
The only thing that truly matters is survival.
An illustrative example is the problem of getting dressed in the morning, showcasing how personal heuristics have evolved over time based on life experiences.
The adaptive markets hypothesis leads to various implications that contrast with traditional efficient market theory:
The relationship between risk and expected return (e.g., illustrated by the Sharpe Ratio) is not constant over time. This is due to the variability in individual preferences and experiences, which can shape attitudes towards risk.
$$\text{Sharpe Ratio} = \frac{E[R] - R_f}{\sigma}$$
where E[R] is the expected return, Rf is the risk-free rate, and σ is the standard deviation of the return.
Individuals may be indelibly altered by significant historical events (e.g., the Great Depression), leading to long-lasting impacts on their financial behaviors and preferences.
Market dynamics fluctuate between periods of efficiency and inefficiency based on investor behavior. The adaptive markets hypothesis suggests that these behaviors can shift due to prevailing social or economic conditions.
Limited arbitrage exists, meaning opportunities for profit (or "free lunches") do arise but may not be consistently available due to competitive pressures.
The relationship between profits (pastures) and investors (sheep) is illustrated. As profits diminish due to increased competition, the investor population may decline, followed by a resurgence of profit opportunities.
Market efficiency is analyzed through the random walk hypothesis, which asserts that past prices do not predict future prices. An empirical examination reveals that autocorrelation in stock returns varies over time:
$$\rho_{t} = \frac{\text{Cov}(X_t, X_{t-1})}{\sigma^2}$$
where Xt is the return at time t and σ2 is the variance.
A rolling analysis of the first-order autocorrelation of the S&P 500 from 1871 to 2003 shows fluctuations in market efficiency, suggesting that market participants change through different eras.
The role of pain in risk management was examined. Past financial gains can anesthetize individuals to potential future risks, leading to overextension and reckless decision-making.
Research shows that financial rewards stimulate the same brain pathways as addictive substances. This can lead to poor decision-making when individuals are experiencing prolonged periods of profitability without accompanying pain.
The adaptive markets theory posits that financial systems share more characteristics with evolutionary biology than physics. Thus, understanding market dynamics requires a focus on behavioral aspects of investors and the ecological context in which markets operate.
Discussions about improving financial regulations and adapting strategies based on historical and behavioral insights point toward the need for ongoing development of theories that can encompass market behavior across different states (normal versus distress).