Introduction to Risk, Return and the Historical Record
Rates of return Investors pay attention to the rate at which their fund have grown during the period The holding period returns (HDR) measure the percentage change in the price of the share over the investment period HPR assumes dividend paid at the end of holding period
Example : HPR single period Ex. You invested in a FTSE500 index fund, the price of a share in the fund is currently $100, and your time horizon is a year. You expect cash dividend during the year to be $4 (so dividend yield = 4%), what is your holding-period return? Sol. Ending Price = 110 Beginning Price = 100 Dividend = 4 Calculate Holding Period Return (HPR) HPR = (110-100+4) / 100 = 14%
Rates of return over multiple periods HPR measure return over a single period But often you will be interested in average returns over a longer periods. Hence, one can calculate arithmetic average and geometric average over the investment period.
Geometric Average Return By solving eq. (1+g) n = TV n, we will get ;
Example : Geometric Average Return Ex. You invested in a FTSE500 index fund, the holding period return of the fund was 10%, 25%, -20% and 20% over the past 4 quarters. What is the average geometric return over the year? Sol. (1+0.10)*(1+0.25)*(1-0.20)*(1+0.20) = (1+g) 4 Calculate average geometric return g = [(1+0.10)*(1+0.25)*(1-0.20)*(1+0.20)] 1/4-1 = 7.19%
Annual Percentage Rates : APR In the same example, fund may choose to annualize its return using annual percentage rate (APR) Returns on assets with regular cash flows (ex. mortgages (monthly payments) & bonds (semi annual coupons)), are usually quoted as annual percentage rate (APR) APR : annualizing per-period rate using simple interest, ignoring the compound interest during the period. Hence, APR does not equal the rate at which your investment actually grow. This is called the effective annual rate (EAR) EAR : actual percentage increase in funds invested over a year horizon
Example : APR and EAR Ex. You buy UK treasury certificates with 100,000 face value maturing in 1 month for 99,000. On the maturity date, you reinvest the amount in the treasury certificates. What is your EAR and APR? Sol. HPR = (100,000-99,000) / 99,000 = 1000 / 99,000 = 1.0101% per month The APR is 1.0101*12 = 12.12% per year But the effective annual rate is higher due to compounding rate (1+EAR) = 1.0101 12 = 1.1282 ; hence EAR = 12.82% per year
Scenario Analysis : Expected Return As we attempt to quantify risk, we could begin with a question : What HPRs are possible and how likely are they? The answer to the question require scenario analysis : expected returns? And likely hood? In a scenario, we can compute the expected returns as
Example : Expected Return
Scenario analysis : Variance and Standard Deviation So we can calculate the expected returns, but how about risks? Risk (uncertainties) can be quantified as Variance or SD -> Weight VAR with prob. p(s)
Example : Scenario VAR & SD
Example2 : Scenario VAR & SD Bear Market Normal Market Bull Market Prob. 0.2 0.5 0.3 Stock X -20% 18% 50% Stock Y -15% 20% 40% 1) What are the expected returns for stock X and Y? 2) What are the standard deviation of returns on stock X and Y? 3) Assume that of your $10,000 portfolio, you invest $8000 in stock X and $2000 in stock Y. What is the expected return of your portfolio? Sol. 1) E(R x ) = 0.2*-0.2 + 0.5*0.18 + 0.3*0.50 = 20% E(R y ) = 0.2*-0.15 + 0.5*0.20 + 0.3*0.40 = 19% 2) Var x = 0.2*(-.2-.2) 2 + 0.5(.18-.20) 2 + 0.3 (.5-.2) 2 = 0.0592 Var y = 0.2*(-.15-.19) 2 + 0.5(.20-.19) 2 + 0.3 (.4-.19) 2 = 0.037 σ x = 0.0592 = 24.33%, σ y = 0.037 = 19.25% 3) E(r p ) = 0.8*20% + 0.2*19% = 19.8%
The normal distribution So far, we talked about quantifying portfolio risk by Var and SD of returns. But why Var and SD as measurement for risks? The distribution of returns over a short period are approximately normal. The distribution of returns over longer period (ex. a year), if expressed as compounded return, are also close to normal. Two important properties of normal distribution 1. The returns on portfolio comprising two or more assets whose returns are normally distributed also will be normally distributed 2. The normal distribution can be entirely described by mean and variance. No other statistic needed to learn about the distribution Will lead to an important conclusion The SD is an appropriate measure of risk for a portfolio of assets with normally distributed returns. No other statistic can improve the risk assessment conveyed by the SD of a portfolio.
The normal distribution Investment management is easier when returns are normally distributed. Standard deviation is a good measure of risk when returns are normally distributed If security s returns are normally distributed, portfolio returns will be, too. Future scenarios can be estimated using only the mean and standard deviation
Figure : The normal distribution
Normality and Risk Measures What if excess returns are not normally distributed? Standard deviation is no longer a complete measure of risk Sharpe ratio is not a complete measure of portfolio performance Need to consider skewness and kurtosis
Skewness and Kurtosis Skewness Kurtosis The "minus 3" at the end of this formula is often explained as a correction to make the kurtosis of the normal distribution equal to zero
Skewness skewness is a measure of the asymmetry of the probability distribution of a random variable about its mean. skewness does not determine the relationship of mean and median. Negative skew : The left tail is longer. The distribution is said to be left-skewed, left-tailed or skew to the left. Positive skew : The right tail is longer. The distribution is said to be right-skewed, right-tailed or skew to the right.
Normal and Skewed Distributions
Kurtosis kurtosis is any measure of the "peakedness" of the probability distribution of a random variable. Negative excess kurtosis distribution are called platykurtic distributions. Positive excess kurtosis distribution are called leptokurtic distributions.
Normal and fat-tailed Distributions Kurt = 0.35
Inflation and real rate of returns A 10% annual rate of returns mean that your investment worth 10% at the end of the year. However, this does not necessary mean that you could have bought 10% more goods with that. The reason is the effect of Inflation rate. Inflation rate measured by consumer price index (CPI) -typical consumption basket of an urban family of four
Real and Nominal Rates of Interest Nominal interest rate: Growth rate of your money Real interest rate : Let R = nominal rate r = real rate I = inflation rate Then : Growth rate of your purchasing power (PP)
Example : Real interest rate A 10% annual rate of returns mean that your investment worth 10% at the end of the year. However, your research reveal that, over the same period, CPI increased by 6% Sol. Nominal rate = 10%, Inflation = 6%, r =? the real interest rate (r) = (0.10-0.06) / (1+0.06) = 0.04 / 1.06 = 3.77% In this case, you purchasing power increased by 3.77% while nominal rate was 10%
Equilibrium Nominal Rate of interest As the inflation rate increases, investors will demand higher nominal rates of returns If E(i) denotes current expectation of of inflation, then we get the Fisher Equation : Fisher equation stated a one-on-one relationship between nominal rate and inflation rate Nominal rate = real rate + inflation forecast Ideally, the correlation between nominal rate and expected inflation should be 1. In contrast, correlations between real rate and inflation should be zero.
Example2 : Arithmetic and Geometric Average Return Ex. A portfolio of nondividend-paying stocks earned a geometric mean return of 5% between 1 Jan 2005 and 31 Dec 2011. The arithmetic mean return for the same period was 6%. If the market value of the portfolio on 1 Jan 2005 was $100,000, what was the market value of the portfolio at 31 Dec 2011? Sol. 2005 -> 2011 = 6 years 100,000*(1+0.05) 6 = $134,009.5 Why don t you care about the other estimate?