1 What Practitioners Need to know... By Mark Kritzman Holding Period Return Holding Period Return  HPR = (Ending Price – Beginning Price + Income) / Beginning Price  Purchase a stock for $50  Dividend Income - $2  Selling Price - 55  HPR = (55 – ) / 50 = 14%

2 Dollar Weighted vs. Time Weighted Rate of Return over multiple holding periods. Assume the annual holding period returns for mutual funds. Rate of Return over multiple holding periods. Assume the annual holding period returns for mutual funds. Total Investment = $75,000 Total Investment = $75,000 Ending Value = $103, Ending Value = $103,804.56

3 Dollar Weighted Return Dollar weighted return is the same as IRR. Dollar weighted return is the same as IRR = -10,000/(1+r) – 15,000/(1+r)^2 - …. - 25,000/(1+r)^ ,805 / (1+r)^ = -10,000/(1+r) – 15,000/(1+r)^2 - …. - 25,000/(1+r)^ ,805 / (1+r)^5 r = 14.25% r = 14.25% If the contribution is reversed: If the contribution is reversed: 25,000 = -20,000/ (1+r) – 15,000 / (1+r)^2 - ….+ 103,8905/(1+r)^5 25,000 = -20,000/ (1+r) – 15,000 / (1+r)^2 - ….+ 103,8905/(1+r)^5 r = 9.12% r = 9.12%

4 Time Weighted Rate of Return Same as Geometric Return Same as Geometric Return Return is wealth relative Return is wealth relative TWR = [ n  i=1 (1 + HPR i ) ] 1/n – 1 TWR = [ n  i=1 (1 + HPR i ) ] 1/n – 1 Example: Example:  Invest $10,000; Year 1 HPR = 50%;  Year 2 HPR = -50% ; Ending wealth = $7,500 TWR = [(1+.5)(1-.5)] 1/2 – 1 = -13.4% TWR = [(1+.5)(1-.5)] 1/2 – 1 = -13.4%

5 Time Weighted Rate of Return Arithmetic Return for the same example is: Arithmetic Return for the same example is: [(50%) + (-50%)] / 2 = 0% Arithmetic Return always exceeds Geometric Return. Overestimates performance. Most studies dealing with long-term historical performance include both arithmetic and geometric rates of return. Arithmetic Return always exceeds Geometric Return. Overestimates performance. Most studies dealing with long-term historical performance include both arithmetic and geometric rates of return.

6 Application of Geometric Return Example: You expect to receive a geometric return of 8% over a 20 year horizon. During the past 5 years, the fund’s geometric return had been 6.5%. What must its geometric return be for the remaining 15 year if you are to meet your original goal? 8% 5 Yr FV % r = ? 8% PV

7 Application of Geometric Return At 8% FV 20 = $1,000(1.08) 20 = $466, year 8% intent factor = year 6.5% intent factor = year return = ( / ) 1/15 – 1 = 8.5%

8 What Practitioners Need to know about Uncertainty? Define Risk vs. Uncertainty Define Risk vs. Uncertainty Random variable (Stock Price): an event whose outcome in a given situation depends on chance factors. Because an outcome is influenced by chance does not mean that we are completely ignorant about its possible values. Random variable (Stock Price): an event whose outcome in a given situation depends on chance factors. Because an outcome is influenced by chance does not mean that we are completely ignorant about its possible values.

9 RISK (Contd.) We are interested in predicting the return of S&P 500 over the next 12 months. We are interested in predicting the return of S&P 500 over the next 12 months.  Should it be between 0 and 10% than 10 to 20%? Consider Table 1: Raw data of S&P500 Return Consider Table 1: Raw data of S&P500 Return

10 RISK (Contd.) Table 2: Relative frequency = Frequency /  Frequency Frequency /  Frequency a) 24/40 (~ 2/3) more likely to observe a return within the range of 10 to 20% than a return within the range of 0 to 10% b) 25% chance of experiencing a negative return

11 Normal Distribution Continuous probability distribution – infinite number of observations covering all possible values along a continuous scale. Continuous probability distribution – infinite number of observations covering all possible values along a continuous scale.  [Stock Price is quoted as 1/8 th, therefore it cannot have a continuous scale] Normal Distribution is a normal approximation for stock price movement. Normal Distribution is a normal approximation for stock price movement.

12 Normal Distribution (Contd.) Normal Distribution can be described fully by two values: Normal Distribution can be described fully by two values: 1. Mean of the observations 2. Variance of observations 1. Mean = or 12.9% from Table 1 2.  2 = = 2.9% ; S.D = 16.9% 2.  2 = = 2.9% ; S.D = 16.9% Explain Figure B: S.D. = 12.9  (1) (16.9) S.D. = 12.9  (1) (16.9)

13 Standardized Variable A) Likelihood of experiencing a return of less than 0% or greater than 15%. In order to determine the probabilities of these returns, we can standardize the target return. In order to determine the probabilities of these returns, we can standardize the target return. Standardized returns have zero mean and a SD of 1 Standardized returns have zero mean and a SD of 1 Standardized Value =(0%-12.9%)/16.9% Standardized Value =(0%-12.9%)/16.9% =

14 Standardized Variable 0% is SD below mean 0% is SD below mean How to read Standard Normal Table How to read Standard Normal Table  To find the area under the curve to the left of standardize variable. The probability of experiencing a return less than 0% is Therefore the chance of experiencing return greater than 0% is ( ) =.7764% Therefore the chance of experiencing return greater than 0% is ( ) =.7764%

15 B) To find the likelihood of experiencing an annualized return of less than 0% on average over a five year horizon. (Assume year- by- year return is independent)  = =  Using the Standardized Normal Table  =.0436 = 4.36% Standardized Variable (Contd.)

16 Standardized Variable (Contd.) C) Likelihood that we might lose money in one or more of the five years: This probability is equivalent to = This probability is equivalent to =  (1 – Probability of experiencing a positive return in every one of the five years) Previously obtained: Probability of experiencing a return greater than 0% =.7764 Previously obtained: Probability of experiencing a return greater than 0% =.7764

17 Standardized Variable (Contd.) Likelihood of experiencing five consecutive yearly return each greater than 0% = (.7764) 5 =.2821 Likelihood of experiencing five consecutive yearly return each greater than 0% = (.7764) 5 =.2821 Probability of experiencing a negative return in at least one of the five years = Probability of experiencing a negative return in at least one of the five years = ( ) =.718