L2 BA2

Measuring Financial Market Performance

• average annual returns big factor
  • S&P 500 (some risk): 6.66%
  • savings account (no risk): 4.70%
• variation (risk) is also big factor
• risk/return relationship is deciding factor
  • performance evaluation should consider both

Estimate Future performance

• most assets future performance is not known
  • bank will tell you, S&P 500 will not
• assumption: future returns will be similar to the past
  • we can analyse historical data and extrapolate into the future

Historical Returns

• non-dividend paying stock: simple return
• with paying dividends:
  • simple return: $r_t=\frac{p_t-p_{t-1}}{p_{t-1}}=\frac{p_t}{p_{t-1}}-1$todo add dividend
• simple vs log-return
  • the larger the difference in price the larger the log/simple return difference is
  • depends on application, more convenient in certain situations
    • e.g. aggregation

Aggregate Returns

• simple returns: compounding
• log returns can be simply added up
• per annuum (p.a.)
  • “stock return in May was 12.68% p.a.”
  • extrapolate the e.g. monthly return for a whole year
  • $r_{May}=1.1268^{\frac{1}{12}}=0.01=1\%$

Cross-Section Portfolio

• simple returns: weighted sum
• log returns: impossible
  • convert to simple returns for weights, log the weights, then multiply everythingtodo check that

Excess Returns

• investment increases in value additional to the cash flow it provides
• extra “benefit” of taking on risk
• expressed relative to EURIBOR or FED fund rate
  • almost 0-risk, the risk of the USA defaulting is very low
  • portfolio manager are payed relative to an index
    • e.g. S&P 500

Measure Risk

• volatility/variance
  • QM2 math
  • sample estimate of volatility
    • todo ba2 2 48
  • measures dispersion in returns
• covariance
• correlation
  • if correlation or covariance is positive, then 2 stocks tend to improve together
• $\mathbb{V}$ is not a lineartodo ba2 2 50

Examples

• todo type some math
• no correlation
  • is just weighted sum
• negative correlation
  • increases risk of portfolio

Value at Risk

• “what is the worst that could happen?”
• $\alpha$ … chance of the worst case happening $\text{VaR}=M\ast \alpha \ast \sigma \ast \sqrt{T}$

Expected Shortall

• “when the worst is happening, how bad is it?”
• if the value falls below x% then the average return will be y%

Skewness

• are there more extreme values at the left or the right
  • positive … right tail
  • negative … left tail

Sharpe-Ratio

• combine Risk and Return
• $r_f$ … risk free rate
• result: amount of excess return per unit of risk