The rate of return on a security is the natural extension of the notion of an interest rate to securities with randomness. The rate of return over some time period is the change in value divided by the value at the start of the period. For example, for a stock paying a dividend, the rate of return is
P + D - P / P \ D t t t-1 | t | t R = -------------- = | ---- - 1 | + ---- . t P | P | P t-1 \ t-1 / t-1
In the last expression, the first term is the capital gain (loss), and the second term is the income. The rate of return is often stated in percent (also called points), or in basis points (100 basis points = 1 percent). The obvious adjustment is made for stock splits or other distributions.
The term total return is sometimes used to refer to one plus the rate of return. The term return usually refers to the rate of return but can refer to the total return instead. Another useful term: the ex-dividend day is the first day the stock trades without a claim to the dividend.
notation: R-rate of return P-price D-Dividend t-time
Ursidae Corp's common stock was worth $60/share at the end of December, $54/share at the end of January, $50/share at the end of February, and $27/share at the end of March. There was a $4 dividend paid in February, and a 2-for-1 stock split in March. Compute the three monthly rates of return for Ursidae Corp.
Having your portfolio value in dollars increase by 10% does not leave you any better if there is inflation and the spending power of a dollar falls by 10% at the same time. If we have an inflation rate of 5% in a year, then it takes 5% more dollars to buy the same goods. In terms of spending power, we have that the value of your holding in an asset increases by a factor
real 1+Rt 1+R = ---- ~ 1+Rt-It t 1+It
where It is the inflation rate over the same period as the return. The final approximation is due to the mathematical fact that
1+x --- ~ 1+x-y ~ (1+x)(1-y) 1+y
when x and y are close to zero. This tells us that ignoring compounding (interest on interest) is not so important over short periods.
If inflation is running at 1%/month, what are the real rates of return to Ursidae Corp for the three months examined in the previous in-class exercise?
Computed real rates of return do have some drawbacks. One is that the measures of inflation we normally use (for example the CPI or CPI-U in the US) try to give a single summary of much different inflation rates for different goods. Such a summary may be a good measure or a bad measure of the cost-of-living for an individual. For example, buying a house may hedge fairly precisely the housing costs of an individual, and therefore it is essentially riskless for the individual, but looks risky when viewed under the lens of the CPI.
Another difficulty with using real returns is that the CPI is not measured very frequently. We have daily and even intraday observations of stock prices, but the price level and inflation are only measured monthly. Even monthly inflation numbers are imprecise and are based not on prices at a point of time at the end of the month and rather on surveys of prices over time within a month.
The arithmetic mean (or just the mean) is what is relevant for thinking about the trade-off between risk and return:
1 T mean = - Sum R T t=1 t
The geometric mean gives the equivalent constant return to a buy-and-hold strategy with reinvestment of dividends:
T 1/T geometric mean = (Prod 1 + R ) - 1 . t=1 t
Over many short periods, the geometric mean is approximately the arithmetic mean minus half the variance.
The variance is the average of the squared deviation from the mean, and the standard deviation is the square root of the variance. In a sample of returns, we measure the variance and standard deviation of returns as
1 T 2 variance = --- Sum (R - mean) T-1 t=1 t
standard deviation = sqrt(variance)
If T is large, it obviously doesn't make much difference whether we divide by T or T-1. For daily returns, the mean is much smaller than a typical Rt, and it doesn't make much difference if we don't subtract the mean.
Practitioners commonly refer to volatility as vol for short.
In four years, a portfolio of common stocks realized returns of 40%, 0%, -30%, and 10%. Compute the (arithmetic) mean, variance, and standard deviation of returns. For this exercise, divide by T (not T-1) to compute the variance.