1. © Paul Koch 1-1 Background Information I. Measures of Return and Risk. A. Operational defn of Ex Post, Nominal Return. r = (ending value – beginning value + cash flows) / (beginning value) = (ending – beginning)/(beginning) + (cash flows) / (beginning) = (% Capital Appreciation) + (% Dividend Income). 1. This is a measure of ex post, actual return earned in the past. a. Accounting measure of past performance. 2. For investment decisions, consider expected future performance! 3. To get expected future return, consider possible future outcomes. 4. This requires pdf that reflects expectations of possible outcomes. 5. Consideration of pdf introduces uncertainty – risk.

2. © Paul Koch 1-2 Background Information B. Ex Ante, Expected Return and Uncertainty. 1. Reflected in investor’s probability distribution function (pdf). 2. Consider the pdf’s of two possible investments, #1 & #2 : Investment: | #1 | #2. Possible return (r i ) |6%| 2% 6% 10% Probability (p i ) |1.0| 0.2 0.6 0.2 p i p i 1.0 | Investment #1 * 1.0 | | * | Investment #2 0.8 | * 0.8 | | * | 0.6 | * | * | * 0.4 | * | * | * 0.2 | * 0.2 | * * * | *. r i | * * *. r i 0 2% 6% 10% 0 2% 6% 10% 3.Investment #1 is a T.Bill; r = 6% with certainty (no risk). Investment #2 depends on future states of the world; uncertainty!

3. © Paul Koch 1-3 Background Information p i p i 1.0 | Investment #1 * 1.0 | | * | Investment #2 0.8 | * 0.8 | | * | 0.6 | * | * | * 0.4 | * | * | * 0.2 | * 0.2 | * * * | *. r i | * * *. r i 0 2% 6% 10% 0 2% 6% 10% 4.Expected Return = E( r i ) =  p i r i ; i indexes states of world. a. For investment A, E( r i ) = (1.0)(6%) = 6% b. For investment B, E( r i ) = (.2)(2%) + (.6)(6%) + (.2)(10%) = 6% 5. Observe, expected return is the same for 1 & 2, but 2 has more risk. 6. In reality, pdf is continuous over (- ,+  ); smooth bell-shaped curve.  7. For continuous pdf’s, E(r i ) = ∫ f( r i ) r i dr i ; analogous to  p i r i. - 

4. © Paul Koch 1-4 Background Information C. How do we measure Risk? Variance =  2 = E[ r i - E(r i ) ] 2 =  p i [r i - E(r i )] 2. 1.For #1,  2 = (1.0)(.06 -.06) 2 = 0. No uncertainty, no risk. 2.For #2,  2 = (.2)(.02-.06) 2 + (.6)(.06-.06) 2 + (.2)(.10-.06) 2 =.00032 + 0 +.00032 =.00064 3. Standard Deviation, . For #1,  = 0; For #2,  .0253  4. For continuous case,  2 = ∫ f( r i ) [ r i - E(r i ) ] 2 dr i.  5. If r i deviates further from mean, distribution more spread out: [ r i - E(r i ) ] is larger; [ r i - E(r i ) ] 2 is larger;  2 is larger.

5. © Paul Koch 1-5 Background Information D. Covariance. 1. Defn:  12 = Cov(r 1, r 2 ) = E[r 1 - E(r 1 )][r 2 - E(r 2 )] n 2. Operational Defn:  12 = (1/n)  [r 1i - E(r 1 )][r 2i - E(r 2 )] i=1 3. Case i : Suppose most points, (r 1i, r 2i ), are in 1 st & 3 rd quadrants. r 2 -E(r 2 ) First quad: [r 1 - E(r 1 )] > 0|... [r 2 - E(r 2 )] > 0|...  12 > 0.|.... __________ | _________ r 1 -E(r 1 ) Third quad: [r 1 - E(r 1 )] < 0..|.. [r 2 - E(r 2 )] < 0.....|  12 > 0...| |

6. © Paul Koch 1-6 Background Information 4. Case ii : Suppose most points, (r 1i, r 2i ), are in 2 nd & 4 th quadrants. Second quad: [r 1 - E(r 1 )] < 0. r 2 -E(r 2 ) [r 2 - E(r 2 )] > 0....|  12 < 0.....|. __________ | _________ r 1 -E(r 1 ) Fourth quad: [r 1 - E(r 1 )] > 0..|.... [r 2 - E(r 2 )] < 0 |....  12 < 0 |.. 5.Case iii : Suppose points, (r 1i, r 2i ), are scattered in all 4 quadrants.. r 2 -E(r 2 ).. Terms in 1 st & 3 rd quad →  12 > 0....|.... Terms in 2 nd & 4 th quad →  12 < 0.....|.... __________ | _________ r 1 -E(r 1 ) Altogether, terms cancel out,.....|.... so that  12 ≈ 0.....|....... |.. Point: Sign of  12 shows the nature of the relation between r 1 & r 2, but not the strength of the relation.

7. © Paul Koch 1-7 Background Information E. Correlation =  12 =  12 /  1  2. 1. Note:  12 may vary between -  & + . a. If r 1 and/or r 2 vary more widely (if  1 and/or  2 larger), then [r 1 - E(r 1 )] and/or [r 2 - E(r 2 )] are larger in magnitude, and  12 will be larger in mag. (depending on case i, ii, or iii). 2. Thus, magnitude of  12 does not tell us about strength of relation. 3. Correlation fixes this problem; adjusts  12 for size of  1 and  2. a. If  12 is larger (because  1 and/or  2 larger),  12 corrects for this by dividing  12 by (  1  2 ). 4. Result:  12 varies between -1 and +1. a. If  12 = +1, r 1 & r 2 are perfectly positively related. b. If  12 = -1, r 1 & r 2 are perfectly negatively related. c. If  12 = 0, r 1 & r 2 are unrelated.