1. Expected Return and Variance
Risk and Return
Expected Return and Variance

2. Probability of State of Economy Stock Returns if State Occurs
Expected Return
 State of Economy
 Probability of State of Economy
Stock Returns if State Occurs
Stock A
Stock B
Boom
50%
0.70
0.10
Recession
-0.20
0.30
Which one would you like?

3. Expected Return 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑅𝑒𝑡𝑢𝑟𝑛
= 𝑃 𝑏𝑜𝑜𝑚 × 𝑅 𝑏𝑜𝑜𝑚 + 𝑃 𝑅𝑒𝑐𝑒𝑠𝑠𝑖𝑜𝑛 × 𝑅 𝑟𝑒𝑐𝑒𝑠𝑠𝑖𝑜𝑛
𝑃 𝑏𝑜𝑜𝑚 = Probability of boom
𝑅 𝑏𝑜𝑜𝑚 = Return if boom occurs
𝑃 𝑟𝑒𝑐𝑒𝑠𝑠𝑖𝑜𝑛 = Probability of recession
𝑅 𝑟𝑒𝑐𝑒𝑠𝑠𝑖𝑜𝑛 = Return if recession occurs

4. Stock A and B Investors expect to earn from A
50%× %×(−0.20)=0.25
How much do investors expect to earn from B?
0.20

5. Problem 11-6 Calculating Expected Return
Consider the following information: Calculate the expected return.
 State of Economy
 Probability of State of Economy
Stock Returns if State Occurs
Recession
0.22
-0.12
Normal
0.48
0.14
Boom
0.30
0.33
E(R) = .22(–.12) + .48(.14) + .30(.33) = .1398, or 13.98%

6. How Volatile Stock Return Is?
Variance measures the average squared difference between the actual returns and the expected return
Standard deviation is the square root of variance
The larger the variance or standard deviation is, the more spread out the returns will be

7. Variance and Standard Deviation
𝑉𝑎𝑟= 𝑝 𝑏𝑜𝑜𝑚 𝑅 𝑏𝑜𝑜𝑚 −𝑅 𝑝 𝑟𝑒𝑐𝑒𝑠𝑠𝑖𝑜𝑛 𝑅 𝑟𝑒𝑐𝑒𝑠𝑠𝑖𝑜𝑛 −𝑅 2
𝑃 𝑏𝑜𝑜𝑚 = Probability of boom
𝑅 𝑏𝑜𝑜𝑚 = Return if boom occurs
𝑃 𝑟𝑒𝑐𝑒𝑠𝑠𝑖𝑜𝑛 = Probability of recession
𝑅 𝑟𝑒𝑐𝑒𝑠𝑠𝑖𝑜𝑛 = Return if recession occurs
𝑅= Expected return
Standard Deviation (𝑆𝐷)
𝑆𝐷= 𝑉𝑎𝑟

8. Variance
Stock A
0.10 stock B

9. Problem 11-7 Calculating Standard Deviation
Consider the following information: Calculate the expected return and standard deviation for two stocks
 State of Economy
 Probability of State of Economy
Stock Returns if State Occurs
Stock A
Stock B
Boom
0.24
0.055
-0.34
Normal
0.64
0.135
Recession
0.12
0.230
0.47
E(RA) = .24(.055) + .64(.135) + .12(.23) = .1272, or 12.72%
E(RB) = .24(–.34) + .64(.24) + .12(.47) = .1284, or 12.84%
σA^2 = .24(.055 – .1272)^ (.135 – .1272)^ (.23 – .1272)^2 =
σA = (.00256)^1/2 = .0506, or 5.06%
σB^2 = .24(–.34 – .1284)^ (.24 – .1284)^ (.47 – .1284)^2 =
σB = (.07463)^1/2 = .2732, or 27.32%

10. 𝑊 𝑖 = $ 𝑖𝑛𝑣𝑒𝑠𝑡𝑒𝑑 𝑜𝑛 𝑎𝑠𝑠𝑒𝑡 𝑖 𝑇𝑜𝑡𝑎𝑙 $ 𝑖𝑛𝑣𝑒𝑠𝑡𝑒𝑑 𝑜𝑛 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜
Portfolio
Portfolio = Collection of individual assets
Weight of individual assets 𝑖 in the portfolio
𝑊 𝑖 = $ 𝑖𝑛𝑣𝑒𝑠𝑡𝑒𝑑 𝑜𝑛 𝑎𝑠𝑠𝑒𝑡 𝑖 𝑇𝑜𝑡𝑎𝑙 $ 𝑖𝑛𝑣𝑒𝑠𝑡𝑒𝑑 𝑜𝑛 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜
Total $ invested on portfolio = Sum of $ invested on all individual assets in the portfolio
𝑊 𝑖 = The fraction of portfolio invested on each individual asset

11. If We Invest on A and B…
If we invest $200 on stock A and invest $800 on B, what is the portfolio weight of on two stocks?
𝑊 𝐴 = $200 $200+$800 = =0.2
𝑊 𝐵 = $800 $200+$800 = =0.8
Expected return of stocks A is 0.25 and that of B is 0.20, what is the portfolio’s expected return?
0.2× ×0.20=0.21

12. Problem 11-1 Determining Portfolio Weights
What are the portfolio weights for a portfolio that has 150 shares of Stock A that sell for $87 per share and 125 shares of Stock B that sell for $94 per share?
Total value = 150($87) + 125($94) = $24,800
xA = 150($87) / $24,800 = .5262
xB = 125($94) / $24,800 = .4738

13. Portfolio Expected Return
𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑅𝑒𝑡𝑢𝑟𝑛= 𝑊 1 𝑅 1 + 𝑊 2 𝑅 2 +…+ 𝑊 𝑛 𝑅 𝑛
𝑊 𝑛 = Weight of stock 𝑛
𝑅 𝑛 = Expected return of stock 𝑛

14. Portfolio Expected Return
Suppose we had following investments:
What is the expected return of this portfolio?
Security
Amount Invested
Expected Return
Stock A
1,000 8%
Stock B
2,000
12%
Stock C
3,000
15%
Stock D
4,000
18%

15. Portfolio Expected Return
14.6
1.16

16. Problem 11-4 Portfolio Expected Return
You have $13,000 to invest in a stock portfolio. Your choices are Stock X with an expected return of 15% and Stock Y with an expected return of 9%. Assume your goal is to create a portfolio with an expected return of 12.35%. How much money will you invest in Stock X and Stock Y?
E(Rp) = = .15wX + .09(1 – wX)
.1235 = .15wX – .09wX
.0335 = .06wX
wX = .5583
Investment in X = .5583($13,000) = $7,258.33
Investment in Y = (1 – .5583)($13,000) = $5,741.67

17. Portfolio Variance and Standard Deviation
If we invest $200 on stock A and invest $800 on B, what is the portfolio variance?
 State of Economy
 Probability of State of Economy
Stock Returns if State Occurs
Stock A
Stock B
Boom
50%
0.70
0.10
Recession
-0.20
0.30

18. Portfolio Variance and Standard Deviation

19. Problem 11-9 Portfolio Variance
Consider the following information: What is the variance of a portfolio invested 22 percent each in A and B and 56 percent in C?
 State of Economy
 Probability of State of Economy
Stock Returns if State Occurs
Stock A
Stock B
Stock C
Boom
0.67
0.10
0.04
0.35
Bust
0.33
0.24
0.30
-0.15
Boom: Rp = .22(.10) +.22(.04) + .56(.35) = .2268, or 22.68%
Bust: Rp = .22(.24) +.22(.30) + .56(–.15) = .0348, or 3.48%
E(Rp) = .67(.2268) + .33(.0348) = .1634, or 16.34%
σp^2 = .67(.2268 – .1634)^ (.0348 – .1634)^2 =