Financial Market Risks

Risk that remains in a fully diversified portfolio. Since portfolio variance converges to the market risk (rather than to 0), it is sufficient to buy a representative slice of the market to get a well-diversified portfolio.

Portfolio whose risk equals the market risk --> \(\beta=1\)

Measure of the sensitivity of a security i to the overall market movements

\(\beta_i=\frac{\sigma_{im}}{\sigma_m^2}=\frac{\text{covariance of i with market}}{\text{variance of market}}\)

\(\beta>1\): asset is more volatile than the market

\(\beta<1\): asset is less volatile than the market

\(\beta=1\): sensitivity of an average stock/well-diversified portfolio

By changing towards higher order of risk measure, the distribution function pays more attention on the outliers --> reflects the risk-return distribution more accurate.

Portfolios with higher order of risk measure have a higher volatitlity, but lower large risks and thus higher expected return.

Portfolio that consists of...

• 25% equities
• 25% gold
• 25% government bonds
• 25% cash

Over the last 40 years, this portfolio showed a very high sharpe ratio.

First risk-return model for valuing a single stock in the portfolio --> mean-beta.

\(r_i=r_f+\beta_i(r_m-r_f)\)

\(r_i\): expected return of stock i; \(r_f\): risk-free return; \(r_m\): expected return of the market portfolio;
\(\alpha=r_f\): profitability; \(\beta_i\): beta of stock i and market portfolio

Assumptions:

• markets are perfectly competitive
• no taxes or transaction costs
• all investors are rational mean-variance optimizers
• all investors have perfect information

Short high-beta assets and long low-beta assets

APT offers an alternative to the CAPM (CAPM is part of APT, but APT goes further).

Assumptions:

• returns depend linearly on multiple arguments (industry factors, interest rates, exchange rates, oil prices etc.)
• number of assets is close to \(\infty\)
• investors have homogenous expectations (as CAPM)
• capital markets are perfect (as CAPM)

Model derived by Fama and French that expands the CAPM by adding size risk and value risk to the market risk factor. It considers the fact that value stocks (high book-value/stock-value) and small-capital firms outperform markets.

Random walk theory states that the next move of a stock price (up or down) is random. Theory was one of the only ones that was first developed in finance and then adopted by physics.

The EMH states that ideally markets should fully reflect all available information instantly, hence, it is impossible to beat the market.

Assumption: humans are fully rational

Conclusion: expected return on bills predict inflation

• Weak form of efficiency: market prices reflect all historical price information
• Semi-strong form of efficiency: market prices reflect all publicly available information
• Strong form of efficiency: market prices reflect all information, both public and private
  --> generally does not hold

• Endogenous shocks are constructuve --> static increase (e.g. word-of-mouth)
• Exogenous shocks are steps --> 0 to 1 instantly (e.g. advert campaign)

• Derivative: any financial instrument that is derived from another --> based on a stock, commodity, exchange rate, asset etc.
  Examples for derivatives: futures, swaps, options
• Options: form of derivative; opportunity to buy (call option) or sell (put option) a stock for a fixed price (exercise/strike price) until a certain expiration date
  Buyer: right to buy (call), sell (put)
  Seller: obligation to sell (call), buy (put)

Combination of a stock and a put option to cancel out risk of loss (by put)

Combination of put and call option to ensure earnings

Options with higher volatility have higher prices (higher propability to earn money).
Value of a call lays between two bounds:
- upper bound = share price
- lower bound = payoff if exercised immediatly

value of call + present value of excercise price = value of put + share price

\(C(t)+K\cdot B(t,T)=P(t)\cdot S(t)+\text{carrying cost}-\text{dividend}\)

\(K\): exercise/strike price

\(B(t,T)=e^{-r(T-t)}\): discounting factor

Theory used to price options. Crucial benefit is that the risk-adjusted return (premium) is not needed, instead the risk-free rate is applied to calculate the option price (risk premium is already included in the stock price).
Option value is based on:

• volatility
• stock price
• exercise/strike price
• risk-free interest rate

\(C=[N(d_1)\cdot S]-[N(d_2)\cdot PV(K)]\)

\(C\): call option price; \(S\): stock price

\(N(d_i)\): cumulative normal density function of d_i; \(PV(K)=\frac{K}{(1+r_0)^n}\): present value of strike price

The right, but not the obligation, to undertake some business decision.
Special characteristic: higher risks --> higher value (from the fact that downside is limited to zero, but upside has no limit)

4 types of real options:

• opportunity to expand
• opportunity to wait
• opportunity to shrink or abandon
• opportunity to vary the ouput or production method mix

Net present value does not take into account changes of the stragety during the lifetime of a project, real options do.

A bond is a securitized loan.

• Principle amount: amount over which interest rates are paid and which has to be repaid at the end
• Coupons: coupon rate - interest rate to be paid
  coupon dates - dates on which coupons are paid
• Maturity: date on which the principle amount has to be repaid

\(PV=\displaystyle\frac{P}{(1+R)^N}+\displaystyle\sum_{i=1}^{N}\frac{C_i}{(1+R)^i}\)

\(C_i:\ \text{coupon rate}\cdot\text{principle amount}\)

\(R\): Yield-to-maturity (YTM)

A 3-year bond return with interest rate \(r_3\) can be seen as \((1+r_3)^3\) or by using different rates for each year \((1+f_1)\cdot (1+f_2)\cdot (1+f_3)\) --> ability to find 2nd year forward rate

\(\text{abnormal return}=\text{r}-(\alpha+\beta\cdot r_m)\)

\(r\): actual return
\(r_m\): market return

• Straddle: simultaneously buy a call and a put with same premium and exercise price
  bet on high volatility --> if price moves heavily in the future (regardless of the direction) one of the options becomes valuable (the higher the price movement the higher the profit)
• Butterfly: simultaneously buy one call with exercise price 100, sell two calls with exercise price 110 and buy one call with exercise price 120
  bet on low volatility --> if price moves only little or is stable, combination of options becomes valuable