Risk and Return: Time-Series Returns

• Lecture VII focuses on Risk and Return

Time-Series Returns

• Time-series returns measured over intervals like monthly, daily, or annually.
• Dollar Return = Divt or Coupont + Change in Market Valuet
• Percentage return is the dollar return divided by the market value from the previous period.
• % returnt = Div Yieldt + Capital Gain Yieldt
• Div Yieldt (Current Yield) is the dividend or coupon payment divided by the previous period's price.
• The formula for Capital Gain Yield is (Pt - Pt-1) / Pt-1.

Stock Investment Return Example

• Investing in 100 shares of Wal-Mart (WMT) at $45 per share, receiving $27 in dividends, and selling the stock at the end of the year for $48 results in a percentage gain of 7.3%.
• Dollar Return = $327
• Percentage Return = 7.3%

Holding Period Return (HPR)

• Defined as the return an investor gets from holding an investment over T years, with R being the return during year i.
• HPR = [(1 + R1) × (1 + R2) ×…× (1 + RT)] - 1

Holding Period Return Example

• Investment returns over four years at 10%, -5%, 20%, and 15% result in a holding period return of 44.21%.

Return Statistics Using Historical Data

• Capital market returns summarized via:
  • Average Return
  • Geometric Return
  • Standard Deviation
• Arithmetic Return is average return.
• Geometric Return shows compound annual growth.
• Geometric Return = [√(1 + R1)(1 + R2) ..... (1 + RT) ] - 1
• The standard deviation measures the dispersion of returns.

Stock Investment Example

• The standard deviation measures risk
• SD indicates an investment's volatility

Risk Aversion and Risk Premium

• Risk aversion assumes investors dislike risk requiring higher returns.
• The risk premium is the extra return to compensate for higher risk.
• Calculated as Risk Premium = Ri - RF (risk return - risk free)

Risk Premium Example

• Example: The Wall Street Journal reports 2% rate for one-year Treasury bills
• Expected return on small-company stocks is 15.3% with a risk-free rate of 2%.

Risk Statistics

• Variance and Standard Deviation are measures of risk analysis.
• Interpretation involves assessing normal distribution.

Normal Distribution

• Adequate size sample, creates bell-shaped curve
• Probability yearly return will be 20% of the mean of 12.3% will be ~ 2/3 (67%).
• Within a normal distribution, specific ranges from the mean relate to the probability.
• +-1 standard deviation is 68.26% chance.
• +-2 standard deviations is 95.44% chance.
• +-3 standard deviations is 99.74% chance.

Standard Deviation Interpretation

• The 20% standard deviation of stock returns from 1926 through 2007:
• Stock returns distribute normally
• Probability of returns within 20% of average of 12.3% occurs at 2/3

Return and Variance Calculations

• Variance = .0045 / (4-1) = .0015
• Standard Deviation = .03873

Average Returns

• Arithmetic average provides expected per period.
• Geometric average measures compound per period.
• Geometric averages are below arithmetic unless all returns equal.
• The arithmetic average is overly optimistic for long horizons.
• Geometric average has over-pessimistic short horizons.

Geometric Return Example

• A geometric average return is an alternative to arithmetic average for investment performance calculations
• Suppose geometric average is 9.58% per year; that provides a 44.21% holding period return.

Not the Same

• Geometric average does not equal the arithmeticaverage

Expected Return vs Required Return

• Expected return forecasts stock gains based on current trends / information.
• Required return is the minimum rate investors will accept for owning a stock given risk.
• People will invest in stock only if minimum return requirements met.

Individual Securities

• Individual securities characteristics evaluated:
  • Expected Return
  • Variance and Standard Deviation
  • Covariance and Correlation

Expected Return, Variance, and Covariance

• The risk and return profile of a dual-asset venture, includes:
• Scenario Analysis involving Recession, Normal, and Boom states
• Probability
• Rate of Return
• Stocks and Bonds
• To illustrate that it is expected return, 11.00% for stock funds and 7.00% for bond funds
• Percentage returns are calculated using this information

Expected Return Formula

• E(R) = P1r1+ P2r2+ ........+ Pnrn.

Standard Deviation Formula

• SD = √Variance
• 14.3% for stocks and 8.2% for bonds.
• Variance = P1(r1 - E(r))^2

Covariance

• Deviation measures how returns change in each state:
  • Weighted takes deviation product times probability.

Covariance and Correlation of Returns

• Covariance measures how returns move together and the extent of the connection.
• Correlation is the measure of returns in relation to each other.
• -1 <= PxY <= 1
• -0.998 translates to "very inversely"

Return and Risk in Stock/Bond Portfolios

• High expected returns and higher risks present from holding more stocks than bonds.
• Portfolios with 50% stocks and 50% bonds are helpful in demonstrating return/risk tradeoffs.

Portfolio Return

• Weighted averages of assets.
• 9% = (0.5)(11%) + (0.5)(7%).

Portfolio Variance / Deviation

• To reduce risk, diversify σp = (WBσB)2 + (WsσS)2.

Efficient Set for Two Assets

• Diversify and include more assets.
• This reduces exposure to risk
• Some portfolios better for higher returns with similar risk

Managing Positions

• Take into account how positions relate in investments

Portfolios Correlations

• -1 <= P <= 1
• Diversify

Diversification

• Can lower risk
  • Does not affect returns

Portfolio Risk with Numbers

• Investing in several stocks could lower systematic variance risk.
• More than 40 stocks are well-diversified.

Risk

• Includes:
  • Systematic
  • Unsystematic

Total Risk

• Combination of :
  • Systematic = Market
  • Unsystematic = asset-specific and reducible

Beta

• In portfolios gauges risk
• It gauges the sensitivity of securities to market shifts.

Calculating Betas from Regression

• Security Returns are graphed vs market returns
• Slope = Change in Y / Change in X

CAPM Model

• Links risk and required returns
• R = RF + Beta*(Rм – RF).
  • Re = expected return
  • Rf = risk free rate

CAPM Model Notes

• Model provides insight into what the fair price of your asset (Investment) should be given the assets beta
• B = 1: Security has the exact same risk as the market
• B > 1: Security is risker then the market
• 0 < B < 1: Security is less risky then the market

Expected vs Required Returns

- Expected Return = what the return is likely to be
- Required Return  = what you ideally demand

Relationship Between Risk & Return Understood Via Graphs

• X axis = Beta (Market Risk)
• Y axis = Expected Return
• With increasing Beta, the steeper the slope, the more reward per unit increase in risk.
• Increase in Betas require an increase in the Expected Return

Calculating Portfolio Beta Examples

• Calculate weights vs holdings