Averages: Arithmetic, Geometric, and Dollar-Weighted

Averages

  • Using historical data.

  • Different types of averages:

    • Arithmetic (simple) average

    • Geometric (time-weighted) average

    • Dollar-weighted average

1. Arithmetic (Simple) Average

  • Arithmetic (simple) average: Return earned in an average period over multiple periods. It is an additive average.

  • Example:

    • Consider returns of 10%, 25%, -20%, and 25% over four periods.

    • Arithmetic average = 10+25−20+254=10\frac{10 + 25 - 20 + 25}{4} = 10%

  • In general:

    • Arithmetic Average = r<em>1+r</em>2+r<em>3+…+r</em>nn\frac{r<em>1 + r</em>2 + r<em>3 + … + r</em>n}{n}

  • Example to illustrate the difference between arithmetic and geometric average:

    • If you invest $1 at time 0 and the arithmetic average return is 10%, after 4 years, the accumulated value would be (1+0.10)4=1.4641(1 + 0.10)^4 = 1.4641

    • However, this may not be the actual compounded growth due to the nature of averaging.

2. Geometric (Time-Weighted) Average

  • Geometric average

    • Assume an investment of $1 at time 0 with returns of 10%, 25%, -20%, and 25% over four periods.

    • Future Value at the end of year 4: FV4=(1+0.10)(1+0.25)(1−0.20)(1+0.25)=1.375\text{FV}_4 = (1 + 0.10)(1 + 0.25)(1 - 0.20)(1 + 0.25) = 1.375

    • Starting with $1, the investment grows to $1.375, which is different from the $1.464 obtained using the simple arithmetic average.

    • The annually compounded rate of return is calculated using the TVM equation, solving for I%.

    • N = 4, PV = -1, PMT = 0, FV = 1.375 gives I% = 8.29% (geometric mean).

  • Formula:

    • r<em>GA=(1+r</em>1)(1+r<em>2)(1+r</em>3)…(1+rn)n−1=8.29%r<em>{GA} = \sqrt[n]{(1 + r</em>1)(1 + r<em>2)(1 + r</em>3)…(1 + r_n)} - 1 = 8.29\%%

  • Geometric Average: Average compound return per period over multiple periods. It is a multiplicative average.

  • The geometric average reflects the compounding effect, while the arithmetic average ignores it.

  • Relationship between Geometric and Arithmetic Average:

    • r<em>GA≤r</em>AAr<em>{GA} \le r</em>{AA}

    • r<em>GA≈r</em>AA−12Variancer<em>{GA} ≈ r</em>{AA} - \frac{1}{2} \text{Variance}

  • $r{GA} = r{AA}$ only when all the returns are equal (e.g., 5% in every period).

  • For returns over nn periods:

  • The correct compound return for an investment is given by the geometric average, not the arithmetic average, of historical returns.
    *Example using historical returns:

Year 1: −50%, Year 2: +50%
Calculate the arithmetic average.
Calculate the geometric average.
Explain the results.

  • Example: Consider the following historical returns.

Investment Year 1 Year 2 Year 3 Simple Ave Geometric Ave
A 5% 5% 5% 5% 5%
B 6% 5% 4% 5% 4.99%
C 9% 5% 1% 5% 4.94%
a) Calculate the arithmetic average.
b) Calculate the geometric average.
c) Explain the results.
Volatility kills you. Do you agree with this statement?

3. Dollar-Weighted Average

  • Dollar-weighted average

    • Consider an investment with inflows and outflows over time.

    • Time 0: -$1m, Time 1: -$0.1m, Time 2: -$0.5m, Time 3: $0.8m, Time 4: $1m

    • The internal rate of return (IRR) for this investment is calculated using a financial calculator.

    • irr(−1,−0.1,−0.5,0.8,1.0)=4.17%irr(-1, {-0.1, -0.5, 0.8, 1.0}) = 4.17\%%

    • 4.17% is the dollar-weighted return for this investment.

  • The dollar-weighted return considers the cash flows of the investment, i.e., how many dollars flow in and out.

Problems

  • Geometric Average Example:

    Year

    Return

    One Plus Return

    Compounded Return:

    1926

    11.14

    1.1114

    1.1114

    1927

    37.13

    1.3713

    1.5241

    1928

    43.31

    1.4331

    2.1841

    1929

    -8.91

    0.9109

    1.9895

    1930

    -25.26

    0.7474

    1.4870

    • (1.4870)15=1.0826(1.4870)^{\frac{1}{5}} = 1.0826

    • Geometric Average Return: 8.26%

    • Using a financial calculator: N = 5, PV = −1-1, PMT = 0, FV = 1.48701.4870, CPT I/Y = 8.26%

Problems

  • The next two questions pertain to the historical investment returns below:

    Year

    Return

    2001

    8%

    2002

    -10%

    2003

    2%

    1. What is the simple or arithmetic mean of these returns?

      • (8−10+2)/3=0%(8 - 10 + 2) / 3 = 0\%%

    2. What is the geometric mean of these returns?

      • (1.08∗0.9∗1.02)13−1=−0.2861%(1.08 * 0.9 * 1.02)^{\frac{1}{3}} - 1 = -0.2861\%%

Problems

  • 3. An 80% loss on an investment requires what percentage gain in order to be offset (breakeven)?

    • Assume starting with $100. An 80% decline reduces the investment to $20. To get back to $100, a net gain of $80 is needed on an investment of $20. Thus, a gain of 8020=4\frac{80}{20} = 4 or 400% is required.

    • Financial literacy rule: DO NOT TAKE BIG LOSSES

    • Corollary: ALMOST ALL BIG LOSSES ARE VOLUNTARY

Problems

  • 5. You have the following rates of return for a risky portfolio for several recent years. Assume that the stock pays no dividends.
    a) What is the geometric average return for the period?
    * [(1.10)(1+−0.0727)(1.0588)]13−1=2.60%[(1.10)(1 + -0.0727)(1.0588)]^{\frac{1}{3}} - 1 = 2.60\%%

Problems

  • b) What is the dollar weighted return over the entire time period?

    • Solve the equation below:

    • 0=−50(100)+−55(50)(1+r)+51(75)(1+r)2+54(75)(1+r)30 = -50(100) + \frac{-55(50)}{(1 + r)} + \frac{51(75)}{(1 + r)^2} + \frac{54(75)}{(1 + r)^3}

    • Financial calculator: irr(−50∗100-50 * 100, {−55∗50-55 * 50, 51∗7551 * 75, 54∗7554 * 75}) = 0.744%

Averages

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