Study Notes on Risk, Return, and Risk Premium

Risk, Return, and Risk Premium

Lecture by Nora Naffa
Corvinus University of Budapest

Course Materials

  • Textbook: Zvi Bodie, Alex Kane, Alan J. Marcus – Investments 12th Edition, McGraw Hill LLC, 2021
    • Relevant Chapters:
    • Chapter 5: Risk, Return, and the Historical Record
    • Chapter 6: Capital Allocation to Risky Assets

Rates of Return in Financial Markets

  • Definition:

    • A rate of return (rr) is defined as the net gain or loss of an investment over a specified time period, expressed as a percentage of the investment’s initial cost.
    • Calculation: The rate of return is calculated by determining the percentage change from the beginning of the period until the end.

  • Example Rates of Return:

    • Netflix Satellite Project: 20%
    • U.S. Stock Market: 10%

Opportunity Cost of Funds

  • The opportunity cost of funds refers to the rate of return on the next-best investment alternative available to the saver, assuming the same level of risk.

Risk and Reward

  • Concept: The idea that risk requires a reward.

    • There is a variability in rates of returns, which affects the investment decisions.

  • Visual Data:

    • Figure 2-3 illustrates rates of return and standard deviations (1926 to 2014), showing relationships between asset types and risk.

Example Rate of Return Graph

  • Graph Axes:
    • Y-axis: Percentage returns
    • X-axis: Standard deviation of returns
  • Trends:
    • Long-term corporate bonds yield lower returns with lower standard deviations compared to common stocks which tend to have higher returns but greater risk.

Interest Rates

  • Nominal Rate of Interest:
    • It is the interest rate paid on debt securities without adjustment for purchasing power loss.
  • Real Rate of Interest:
    • Defined as the nominal rate of interest adjusted for the purchasing power loss due to inflation.

Conversion Between Rates

  • Formulas:

    • Converting nominal rate of interest to real rate:
      ( ext{Nominal or quoted rate of interest}) = ( ext{real rate of interest}) + ( ext{inflation rate})

  • Practice Exercise:

    • A banker offers investment at a quoted rate of 10% and the inflation rate is 6%.
    • Calculate the real rate of interest:
      Let real rate of interest = x
      x = 10 ext{%} - 6 ext{%} = 4 ext{%}

Interest Rate Determinants

  • Components of Interest Rates:
    • Inflation Premium: Compensation for anticipated inflation.
    • Default-Risk Premium: Additional return required for risk of default.
    • Maturity-Risk Premium: Additional return required for longer-term investments.
    • Liquidity-Risk Premium: Extra return needed for investments not easily converted to cash.

Required Rate of Return

  • Real Risk-Free Interest Rate: The minimum return on a fixed-income security in an environment with zero inflation.

  • Formula for Calculating Nominal Interest Rate:
    ext{Nominal interest rate} = ext{real risk-free interest rate} + ext{inflation premium} + ext{default risk premium} + ext{maturity risk premium} + ext{liquidity risk premium}

Historical Rates of Interest

  • Historical data covering the interest rates in Hungary and the USA, with important metrics displayed over years (2000 - 2020).

Comparing Interest Rates

Annual Percentage Rate (APR)

  • Definition: The indicator of interest paid or earned in 1 year, not accounting for compounding.
  • Formula for APR: ext{APR} = ext{interest rate per period} imes ext{number of compounding periods per year}
    • Example: An interest rate of 2% per month compounded monthly results in an APR of 24%.

Effective Annual Rate (EAR)

  • Definition: The annual compound rate that yields the same return as nominal rates when compounded.
  • Formula for EAR:
    ext{EAR} = (1 + ext{APR}/m)^{m} - 1
    where m is the compounding periods per year.

Comparative Rate Examples

  • Loan A: Quoted annual rate of 8.084%, compounded annually
  • Loan B: Quoted annual rate of 7.85%, compounded quarterly
  • Calculation of EAR for both loans to compare their effective returns.

Future Value and Present Value

Formula Involving Nonannual Compounding

FV_n = PV imes (1 + ext{APR}/m)^{mn}

  • Terms:
    • FVn: Future value
    • PV: Present value
    • m: Compounding periods per year
    • n: Number of years

Measures of Return

Holding-Period Return

  • Definition: The rate of return earned on an investment is defined as the dollar gain divided by the invested amount.
  • Example Calculation:
    • Price at beginning: $507.79
    • Price at end: $557.28
    • Dollar return: $49.49
    • Rate of return: rac{49.49}{507.79} = 0.0975 = 9.75 ext{%}

Differences Between Effective Return and Log Return

  • Effective Return:
    r{eff} = rac{P1 + Div1}{P0} - 1
  • Log Return:
    r{cc} = ext{ln} rac{P1 + Div1}{P0}

Expected Rate of Return

  • Definition: The arithmetic mean of all possible outcomes, weighted by the probability of each outcome.
  • Calculation Example: For varying economic states:
    • Economic Recession: 20% chance, cash flow: $1,000
    • Moderate Growth: 30% chance, cash flow: $1,200
    • Strong Growth: 50% chance, cash flow: $1,400
    • Expected cash flow:
      = 20 ext{%} imes 1,000 + 30 ext{%} imes 1,200 + 50 ext{%} imes 1,400 = 1,260
    • Expected rate of return:
      = 20 ext{%} imes 10 ext{%} + 30 ext{%} imes 12 ext{%} + 50 ext{%} imes 14 ext{%} = 12.6 ext{%}

Defining and Measuring Risk

  • Risk Definition:
    "Risk is the potential variability in future cash flows."
  • Quantification of Risk:
    • Use of standard deviation (σ), which is a measure of spread of a probability distribution.
    • Standard Deviation Formula:
      ext{Variance in rate of return: } ext{σ}^2 = ext{ } rac{(r1 - ar{r})^2 imes P{b1} + (r2 - ar{r})^2 imes P{b2} + … + (rn - ar{r})^2 imes P{bn}}{n}

Risk and Diversification

  • Components:
    • Unsystematic Risk: Related to specific investments, diversifiable risk.
    • Systematic Risk: Market-wide factors, non-diversifiable risk.

Summary on Risk Attitudes

  • Risk Averse: A > 0, prefer safer investments
  • Risk Neutral: A = 0, focus on expected return
  • Risk Lover: A < 0, accept lower returns for higher risks

Reward-to-Volatility Ratio (Sharpe Ratio)

  • Sharpe Ratio Formula:
    ext{Sharpe Ratio} = rac{E[rP] - rf}{ ext{σ}_P}
  • Example Calculation:
    ext{Sharpe Ratio} = rac{10 ext{%} - 2 ext{%}}{22 ext{%}}

Historical Rates of Return Data

  • Provides averages, standard deviations, and risk premiums across various asset classes: small-company stocks, large-company stocks, intermediate-term government bonds, corporate bonds, U.S. Treasury bills, and inflation.

Asset Allocation

  • Definition: Process of selecting appropriate asset classes and determining proportions in an investment portfolio aimed at diversification.
  • Key asset classes include stocks, bonds, real estate, and commodities.

Required Rate of Return and the Capital Asset Pricing Model

  • Definition: The minimum rate to attract investment, represented as
    r = rf + β imes (rm - r_f)
  • Where:
    • r: Required return on a security
    • r_f: Risk-free rate of return
    • β: Beta for security measure
    • r_m: Required return on market
  • Security Market Line: Reflects investor attitudes regarding minimum acceptable return for risk levels.

Closing Note

  • Thank you for your attention.
    nora.naffa@uni-corvinus.hu