# Returns on financial assets - Security Analysis and Investment Management

People want to maximize expected returns subject to their tolerance for risk. Return is the principal reward in the investment process, and it provides the basis to investors in comparing alternative investments. Measuring historical returns allows investors to assess how well they have done, and it plays a part in the estimation of future, unknown returns. We often use two terms regarding return from investments, realized return and expected return. Realized return is after the fact return that was earned. Realized return is history. Expected return is the return from an asset that investors anticipate they will earn over some future period. It is a predicted return, and it may or may not occur.

Components of return

Stock returns consist of both a capital gain and a dividend yield component, and we show that predictability of stock returns by lagged dividend-price ratios mainly reflects predictability of future dividend yields, which make up a significant component of average returns. We propose a novel log linear approximation of stock returns into a capital gain and a dividend yield component and derive testable restrictions of non predictability of capital gains.

We often use the term yield to express return. Yield refers to the income component in relation to some price for a security. For our purposes, the price that is relevant is the purchase price of the security. The yield on a Rs. 1,000 par value, 6 per cent coupon bond purchased for Rs. 950 is 6.31 per cent (Rs. 1,000 par value, 6 percent coupon bond purchased for Rs. 950 is 6.31 percent (Rs. 60/Rs. 950). However, we need to remember that yield is not, for most purposes, The proper measure of return from a security. The capital gain or loss must also be considered. Equation 2.1 is a conceptual statement for total return.

Total return = Income + Price change (+/-) …(2.1)

Note that either component of return can be zero for a given security over any given time period. A bond purchased for Rs. 800 and held to maturity provides both type of income: interest payments and a price change. The purchase of a non-dividend-paying stock that is sold four months later produces either a capital gain or a capital loss, but no income.

Thus, a measure of return must consider both dividend/interest income and price change. Returns over time or from different securities can be measured and compared using the total return concept. The total return for a given holding period relates all the cash flows received by an investor during any designated time period to the amount of money invested in the asset. Total return is defined as

Where, r = total return, P(t) = price of an asset at time (t), P(t–1) = price of an asset at time (t-1), D = dividend or interest income in simple terms.

Example: Jindal Steels share’s price on June 10, 2004 is Rs. 900 (Pt–1) and the price on June 9, 2005 (Pt), is Rs. 950. Dividend received is Rs. 76 (D). Determine the rate of return.

Solution:

Calculation of average returns

The total return is an acceptable measure of return for a specified period of time. But we also need statistics to describe a series of returns. For example, investing in a particular stock for ten years or a different stock in each of ten years could result in 10 total returns, which must be described mathematically. There are two generally used methods of calculating the average return, namely, the arithmetic average and geometric average. The statistics familiar to most people is the arithmetic average. The arithmetic average, customarily designated by the symbol X = ΣX /n, or the sum of each of the values being considered divided by the total number of values.

Example: The return of stock A for four quarters is as follows: Quarter-I = 10%; Quarter-II= 8%; Quarter-III= -4%; and Quarter IV= 20%. The average return is

X = 10 + 8 + (-4) + 20 /4 = 8.5%

The arithmetic average return is appropriate as a measure of the central tendency of a number of returns calculated for a particular time, such as a year. However, when percentage changes in value over time are involved, the arithmetic mean of these changes can be misleading. The geometric average return measures compound, cumulative returns over time. It is used in investments to reflect the realized change in wealth over multiple periods. The geometric average is defined as the nth root of the product resulting from multiplying a series of returns together, as in Equation 2.2.

G = [(1 + r1) (1 + r2) … (1 + rn)]1/n – 1 … (2.2)

Where, r = total return, n = number of periods.

Return relative: On adding 1.0 to each return (r), we shall get a return relative. If the return for a period is 10 percent (.10), then the return relative is 1.10. The investor has received Rs. 1.10 relative to each Rs. 1 invested. If the return for a period is –15 percent (-.15) then the return relative is .85 (1-.15). Return relatives are used in calculating geometric average returns because negative total returns cannot be used in the math. Here, we also need to note that the geometric average rate of return would be lower than the arithmetic average rate of return because it reflects compounding rather than simple averaging.