Capital Market Theory: An Overview. Return Measures

Transcription

1 Capital Market Theory: An Overview (Text reference: Chapter 9) Topics return measures measuring index returns (not in text) holding period returns return statistics risk statistics AFM Capital Market Theory: An Overview Slide 1 Return Measures total dollar return: total dollar return = dividend income + capital gain/loss = Div t+1 + P t+1 P t note that this formula assumes a single holding period; otherwise we must consider the time value of money example: suppose you bought a share of Widget Corp. at the beginning of January 2004 when its price was $19.75 per share. By the end of the year, the price of the stock had appreciated to $ In addition, the firm paid a dividend of $1.50 per share. What is your total dollar return? AFM Capital Market Theory: An Overview Slide 2

2 percentage return: total dollar return R t+1 = P t = Div t+1 + (P t+1 P t ) P t this formula also assumes a single holding period, and it gives the return on an investment of $1 example: what was the percentage return for Widget Corp. s stock? percentage return is preferred since it is independent of the amount of money invested AFM Capital Market Theory: An Overview Slide 3 Measuring Index Returns a security index tracks performance of a specific portfolio over time (e.g. S&P/TSX, NASDAQ, Scotia Capital Long Term Bond Index, etc.) An index portfolio is chosen to represent a group of securities (e.g. S&P/TSX represents Canadian publicly traded stocks). There are three different index calculation methods: 1. A price-weighted index assumes you purchase an equal number of shares (one) of each stock in the index: (price-weighted index) t = 1 n n P i,t i=1 where n is the number of stocks in the index and P i,t is the price of stock i at time t like a portfolio having 1 share of each stock in the index continuous adjustments for splits/additions/deletions AFM Capital Market Theory: An Overview Slide 4

3 2. A value-weighted index assumes you make a proportionate market value investment in each company in the index (v-w index) t = [ n i=1 (P i,t N i,t ) n i=1 (P i,0 N i,0 ) ] (v-w index) 0 where N i,t is the number of shares outstanding of stock i at time t this type of index is just the total market capitalization of all stocks in the index at time t divided by the total market capitalization at time 0 (and then multiplied by some base level e.g. the TSE 300 started in 1977 with a base level of 1,000 for the year 1975) a problem: firms with large market capitalizations AFM Capital Market Theory: An Overview Slide 5 Cont d 3. An equal-weighted index assumes you make an equal dollar investment in each stock in the index [ ] Pi,t (equal-weighted index) t = average of P i,0 in effect, working with % price changes can be either arithmetic average ((1/n) n i=1 x i) or geometric average ((Π n i=1 x i) 1/n ), depending on index definition appropriateness of performance comparison to index depends on how your portfolio is structured, e.g. if you hold an equal number of shares of many Canadian companies, it is not really appropriate to compare your performance to the S&P/TSX AFM Capital Market Theory: An Overview Slide 6

4 example: stock initial price final price shares o/s (millions) A $25 $30 20 B $100 $110 1 what is the return on the price-weighted, value-weighted, and arithmetically averaged equal-weighted indexes? AFM Capital Market Theory: An Overview Slide 7 Holding Period Returns T -period percent holding return: (1 + R 1 ) (1 + R 2 ) (1 + R T ) 1 e.g. calculate the holding period return on TSE 300 stocks from Jan. 1, 1997 to Dec. 31, 2000 (see data on p. 249 of text): the geometric average of returns over the holding period is the effective annual compounded rate over the holding period R g = [(1 + R 1 ) (1 + R 2 ) (1 + R T )] 1/T 1 AFM Capital Market Theory: An Overview Slide 8

5 e.g. calculate the effective annual compound rate of return on TSE 300 stocks from Jan. 1, 1997 to Dec. 31, 2000 (p. 249 of text), and the arithmetic average return per year over that period: note that the arithmetic mean R a R g (and R a = R g only if returns are the same in every period) the geometric mean is a good measure of past performance (it represents the constant rate of return per year needed to match actual investment performance over several years), but R g is a downward-biased measure of expected return in any single future year, e.g. suppose that a stock has three equally likely returns, 20%, 10%, and 15%. What is the expected return? AFM Capital Market Theory: An Overview Slide 9 Return Statistics frequency distribution histogram: divides observations into intervals, and displays the number of observations falling in each interval indicates the magnitude and volatility of returns histograms for Table 9.1, p. 249 (see spreadsheet handout) arithmetic mean: the arithmetic mean of n observed returns R i is R = n i=1 R i n arithmetic mean calculations for Table 9.1, p. 249 (see spreadsheet handout) AFM Capital Market Theory: An Overview Slide 10

6 returns, volatility and risk premiums: historical return observations (Table 9.1, p. 249, Figure 9.4, p. 251, and histograms): implications: Figure 9.4 (text), holding period returns: TSE 300 > long term bond > 91 day T-bill spreadsheet handout, average returns: TSE 300 > long term bond > 91 day T-bill histograms, volatility: TSE 300 > long term bond > 91 day T-bill can think of expected return on a risky security as current risk free rate plus a risk premium; more risk higher risk premium higher expected return (compensation for taking on risk) AFM Capital Market Theory: An Overview Slide 11 Risk Statistics want to measure the dispersion of a distribution the variance (denoted by σ 2 ) of a single random variable is σ 2 = n i=1 (R i R) 2 n 1 a related measure is the standard deviation σ (i.e. the square root of variance) individual security variance (or s.d.) can be an appropriate risk measure of a security only if an investor s portfolio consists of just one security AFM Capital Market Theory: An Overview Slide 12

7 intuitively: R i e.g. calculate the variance and standard deviation for TSE 300, long term bonds, and 91-day T-bills from Table 9.1 n AFM Capital Market Theory: An Overview Slide 13 normal distribution and standard deviation 0.4 Plot of Density Function for Normal (0,1) Variable symmetric about its mean, probability of having an observation different from the mean depends only on s.d. after (infinitely) many observations, 68.26% (95.44%; 99.74%) of observations fall within one (two; three) s.d. of the mean equivalently, for any given observation, there is a 68.26% (95.44%; 99.74%) chance that the observation will fall within one (two; three) s.d. of the mean AFM Capital Market Theory: An Overview Slide 14

8 higher risk (i.e. larger s.d.) is typically associated with higher average returns in financial markets of course, this also means more chance of a big loss (or a big gain): 3 Normal(0.05,0.10) Normal(0.15,0.30) AFM Capital Market Theory: An Overview Slide 15