Investment Math

If you want to succeed at investments, you must know how much you are earning or losing. If you want to know which investments to buy, then you must be able to compare their returns. Some investments earn most of their return through the payment of dividends, distributions or interest, which is often expressed as a yield, a percentage of the value of the asset or the purchase price; other investments, such as growth stocks, yield a return only through changes in their price — the capital gain or loss. The sum of the yield plus capital gain or loss over a given period equals the total return (TR) for that same period, usually 1 year:

Total Return = Income Received ± Price Change

Total Return Formula
Total Return=End Price – Initial Price + Income Received
Initial Price

Obviously, if an investment does not pay income or if the price does not change, then the return will be determined by the other component of the total return. The return relative (RR) is also often used to calculate investment returns, which is simply the total end value of the investment plus income divided by the initial value of the investment. Hence, it is the return relative to the initial price:

Return Relative Formula
Return Relative=End Value of Investment + Income Received
Initial Price
Example: Return Relative for a Bond
Principal:$1,000
Coupon Rate:6%
Initial Price:$945
Sale Price:$1,005
Total Return:12.70%
Return Relative:1.1270
Example: Return Relative for a Stock
Initial Price:$100
Sale Price:$94
Dividend:$4
Total Return:-2.00%
Return Relative:0.9800= (Income + End Price) ÷ Initial Price

The main advantage of calculating the return relative is that it avoids negative values, which cannot be used in some calculations, such as the arithmetic or geometric mean or the cumulative wealth index.

Calculating Investment Returns Involving Foreign Currency

The return relative an also be used to convert a return paid in a foreign currency to the domestic currency. With a foreign investment, changes in the foreign exchange rate will either increase or decrease the total return of an investment in terms of the domestic currency. Thus, to calculate the total return in the domestic currency, multiply the return relative by the end value of the foreign currency divided by the beginning value, then subtracting 1:

Total Return Formula in Domestic Currency in Terms of Foreign Currency
Total Return in Domestic Currency=Return Relative×Foreign Exchange Rate at End of Term
Foreign Exchange Rate at Beginning of Term
– 1
Example: Calculating the Total Return of a Foreign Investment in Domestic Currency
Buy European Stock in Euros100
Dollar per Euros at Initial Investment1.25
Investment Value When Sold in Euros200
Dollar per Euros at End of Investment1.35 
Return Relative for Investment2= End Investment Value / Start Investment Value
Total Return in USD1.16= Return Relative × End Foreign Currency Value/Start Foreign Currency Value - 1

Calculating the Average of Investment Returns

There are 2 primary methods of calculating the average of investment returns: arithmetic mean and geometric mean. The arithmetic mean is simply the sum of the returns for each year, divided by the number of years:

Arithmetic Mean Formula
X=∑ Xk
n
  • Xk = Total Return for kth year
  • n = Number of Years

However, the arithmetic mean is usually not accurate over successive holding periods, because the arithmetic mean does not account for compounding. In some cases, where the returns are negative in some years, the deviation from the actual average investment return can be large. For instance, consider a $10 stock that increases by 100% to $20 after the 1st year, then declines by 50% in the 2nd year. The average return for the 2 years would be (100% – 50%)/2 = 25%, but the actual return is 0%, because the stock is at the same price at the end of the holding period as it was at the beginning.

The geometric mean is more accurate than the arithmetic mean because it accounts for compounding:

Geometric Mean = [(1+ TR1) (1+ TR2) … (1+ TRn)] 1/n – 1

So the geometric mean for the above $10 stock would be √(1 + 1) × (1 - .5) – 1 = √2 × .5 – 1 = 1 – 1 = 0%.

Example: Calculating the Arithmetic and Geometric Mean for the S&P 500 Index from 2000 to 2006
YearReturnReturn Relative
2000-10.14%89.86%
2001-13.04%86.96%
2002-23.37%76.63%
200326.38%126.38%
20048.99%108.99%
20053.00%103.00%
200613.62%113.62%
Total of Returns5.44%
Arithmetic Mean =0.78%
Geometric Mean =-0.50%
Index at Start of 20001469.25
Index at End of 20061418.30
Predicted Ending Index using Arithmetic Mean =1551.06= Initial Value * (1 + Average Return Rate)(Number of Compounding Periods)
Predicted Ending Index using Geometric Mean =1418.20= Initial Value * (1 + Geometric Mean)(Number of Compounding Periods)

Because the S&P example includes some years with negative returns, the discrepancy between the arithmetic mean and the geometric mean is large. The geometric mean will usually yield the correct, accurate result, but the example was off a little because of rounding errors in the data.

Total returns can also be adjusted for inflation by dividing the total return over a given time period by the inflation rate over that same period, usually 1 year.

Inflation-Adjusted Total Return Formula
TRa=1 + TR
1 + IR
– 1
  • TRa = Total Return after adjusting for inflation
  • IR = Inflation Rate
Example: Inflation-Adjusted Total Return Formula
TRa =1 + 10%
1 + 3%
– 1=1.1
1.03
– 1≈ 6.8%
  • Total Return before adjusting for inflation = 10%
  • Inflation Rate = 3%

Cumulative Wealth Index

The cumulative wealth index (CWI) is simply the return, expressed as a decimal multiple of the initial amount, earned by a certain initial amount of money over a period of years. The calculation usually uses $1 as the initial investment and the returns are compounded annually:

CWIn = WI0 × (1 + TR1) × (1 + TR2) × … × (1 + TRn)

When the initial wealth (WI0) is set to $1, the cumulative wealth index reduces to:

CWIn = (1 + TR1) × (1 + TR2) × … × (1 + TRn)

For example, if your initial investment is $100 and you earned 25% in the 1st year, -10% in the 2nd year, and 12% in the 3rd year, then the cumulative wealth index would be equal to: $100 × 1.25 × .9 × 1.12 = $126. Note that if you set the initial value to $1, then the cumulative wealth index would have been equal to 1.26, so to find the cumulative wealth index for any amount, simply multiply the initial investment by the cumulative wealth index. So if the initial investment was $1 million, then the ending amount would be $1 million × 1.26 = $1.26 million.

Measuring Investment Risk

Investment risk is the probability that investment returns will be less than what was desired or that losses will be incurred. The greater the probability, the greater the risk. If an investment has greater risk, then it should potentially reward the investor with a greater return; otherwise, the investor would never assume the risk if the probability of a greater return was nil. The dispersions of investment returns can usually be represented by the normal distribution curve, so another way to look at this is that the probability of any losses is equal to the probability of any gains, since the area under the normal distribution curve before the mean is equal to the area of the distribution curve after the mean.

Statistical methods are used to measure risk. Because the investment returns of riskier assets has a greater dispersion — meaning that investment returns vary more widely than for less risky assets — variance and standard deviation are used to measure this dispersion, and, therefore, risk. By measuring the actual historical returns of various assets, such as stocks or bonds, the degree by which they fluctuate in value can easily be measured. These values are used to measure the variance of the investment returns. Variance is measured by the specific investment returns that deviate from the mean, which is the average return of a sample selection of assets. If an asset always returned the mean, then there would be no dispersion, and therefore, no risk in undertaking the investment, since it would always pay the expected rate of return. Treasury bonds, for instance, have the lowest dispersions of any other type of investment because they pay a fixed rate of interest and, backed by full faith and credit of the United States government, there is no credit default risk. Consequently, US treasury bonds pay the lowest rate of interest. Variance is measured by the following equation:

Variance Formula
σ2=n

k=1
(XkX)
  • Xk = Total Return for kth sample asset.
  • X = Sample mean
  • n = Sample size
  • σ2= Variance

n – 1

Deviations from the mean are squared to eliminate negative values, so the standard deviation, denoted by σ, which is simply the square root of the variance (σ2), is used to measure dispersion, and therefore risk:

σ = Standard Deviation = σ2

The riskiness of any asset class is commensurate with the standard deviation of the historical returns of that type of investment. Additionally, the return of an investment can be divided into a risk-free return plus a risk premium, which is the excess of return required for assuming a greater risk. The risk-free rate is generally measured by the return on Treasuries, since they are considered free of credit default risk. More about the measurement of risk can be found in Beta, Capital Asset Pricing Model (CAPM), and the Security Market Line (SML).