# Investment Math

To profit from investments, you must know how much you are earning or losing. To know which investments to buy, 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 = 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 = End Value of Investment + Income Received Initial Price
 Principal: \$1,000 Coupon Rate: 6% Initial Price: \$945 Sale Price: \$1,005 Total Return: 12.70% Return Relative: 1.1270
 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 in Domestic Currency = Return Relative × Foreign Exchange Rate at End of TermForeign Exchange Rate at Beginning of Term – 1
 Buy European Stock in Euros 100 Dollar per Euros at Initial Investment 1.25 Investment Value When Sold in Euros 200 Dollar per Euros at End of Investment 1.35 Return Relative for Investment 2 = End Investment Value / Start Investment Value Total Return in USD 1.16 = 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:

 X = ∑Xkn 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

• TR = Total Return

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

 Year Return Return Relative 2000 -10.14% 89.86% 2001 -13.04% 86.96% 2002 -23.37% 76.63% 2003 26.38% 126.38% 2004 8.99% 108.99% 2005 3.00% 103.00% 2006 13.62% 113.62% Total of Returns 5.44% Arithmetic Mean = 0.78% Geometric Mean = -0.50% Index at Start of 2000 1469.25 Index at End of 2006 1418.30 Predicted Ending Index using Arithmetic Mean = 1551.06 = Initial Value * (1 + Average Return Rate)n Predicted Ending Index using Geometric Mean = 1418.20 = Initial Value * (1 + Geometric Mean)n n = 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.

 TRa = 1 + TR1 + IR – 1 TRa = Total Return after adjusting for inflation IR = Inflation Rate
 TRa = 1 + 10%1 + 3% – 1 = 1.11.03 – 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:

 σ2= n ∑ k=1 (Xk– X) n – 1 Xk = Total Return for kth sample asset X = Sample mean n = Sample size σ2= Variance

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).

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