1. Andrei Simonov (HHS) Finance I – HHS MBA
Risk and Return
Andrei Simonov (HHS)
Finance I – HHS MBA

2. Rates of Return: Single Period- P + D
HPR = 1 1 P
HPR = Holding Period Return
P1 = Ending price
P0 = Beginning price
D1 = Dividend during period one

3. Rates of Return: Single Period Example
Ending Price = 24
Beginning Price = 20
Dividend =
HPR = ( )/ ( 20) = 25%

4. Averaging Multiperiod Returns: Arithmetic vs. Geometric
ra = (r1 + r2 + r rn) / n
ra = ( ) / 4
= .10 or 10%
Geometric
rg = {[(1+r1) (1+r2) .... (1+rn)]} 1/n - 1
rg = {[(1.1) (1.25) (.8) (1.25)]} 1/4 - 1
= (1.5150) 1/4 -1 = = 8.29%

5. Annual Holding Period Returns
Geom. Arith. Stan.
Series Mean% Mean% Dev.%
Lg. Stk
Sm. Stk
LT Gov
T-Bills
Inflation

6. Annual Holding Period Risk Premiums and Real Returns
Excess Real
Series Returns% Returns%
Lg. Stk
Sm. Stk
LT Gov
T-Bills
Inflation

7. Real vs. Nominal Rates Fisher effect: Approximation
nominal rate = real rate + inflation premium
R = r + i or r = R - i
Example: r = 3%, i = 6%
R = 9% = 3% + 6% or r = 9% - 6% = 3%
Fisher effect: Exact
r = (R - i) / (1 + i)
= (9%-6%) / (1.06) = 2.83%

8. Averaging Multiperiod Returns: Dollar Weighted Returns
Internal Rate of Return (IRR)
Discount rate that equates present value of the future cash flows with the current investment amount
Initial Investment is regarded as an cash outflow
Ending value is considered as an inflow
Additional investment is considered as a negative flow
The IRR is an average rate of return over the period, considering changes in investment amount over time
CF1
CF2
CF3
CFT = + +
Inv0
+  +
(1+IRR)
(1+IRR)2
(1+IRR)3
(1+IRR)T

9. Dollar Weighted Average
Net CFs 0 Y1 Y2 Y3 Y4
$ (mil)
Solving for IRR
1.0 = -.1/(1+r) /(1+r)2 + .8/(1+r) /(1+r)4
r = or 4.17%

10. Quoting Conventions APR = annual percentage rate
= (per-period rate) X (periods per year)
Simple interest approach, ignoring compound interest
EAR = effective annual rate
= (1+ per-period rate)Periods per yr – 1
= (1+ (APR/n) )n – 1
Example: Annualize monthly return of 1%
APR = 1% X 12 = 12%
EAR = (1.01) = 12.68%

11. Characteristics of Probability Distributions
Mean
Most likely value (expected value)
Variance or standard deviation (volatility)
Degree of deviation from the mean value
Skewness
Degree of asymmetry in distribution
Kurtosis
Degree of fatness in tail area
...., etc.
If a distribution is approximately normal, the distribution is described simply by characteristics 1) and 2)

12. Normal Distribution r = E(r) s.d. s.d. Symmetric distribution
68.3% of observations within one std. dev.
95.4% of observations within two std. dev., or 95% of observations within 1.65 std.dev.
99% of observations within 2.33 std.dev.
“Fat tails” if more observations are away from the normal distance

13. Annual vs. Daily Volatility
Annual volatility
= daily volatility × (number of trading days per year)1/2
(Ex) S&P 500 index return’s annual std. dev. = 10.88%
Then, what is the daily volatility?
=  × (252)1/2
  = (or 0.68%)
Therefore, we would expect about 11.5 days a year when returns exceeded 1.36% in absolute value.
Or, we would expect that returns would be between
-0.68% and +0.68% in 174 days a year.

14. Histogram of S&P 500 index returns
Sample period = 1,250 trading days
Number of days with returns outside of 2 std. dev. = 65 days
How many days would you observe returns exceeding 1.36% in absolute value in normal distribution?

15. Pros of Std. Dev. As a Measure of Risk
Easier to calculate and implement.
If return distribution is symmetric, the upside risk is the same as downside risk, and therefore standard deviation is a good measure of downside risk.
if returns are normally distributed, standard deviation would be adequate in characterizing the risk.

16. Cons of Std. Dev. As a Measure of Risk
Investors are concerned about downside risk.
Standard deviation includes both the above-average returns (upside risk) and the below-average returns (downside risk). If returns are skewed, standard deviation is not the only relevant measure of risk.
Holding expected return and standard deviation constant, investor would prefer positive skewed distribution.

17. Skewed Distribution Negative Positive r Left-skewed distribution
Large negative returns possible
Bad surprises are more likely to be extreme
A risk-averse investor would prefer right-skewd distribution to left-skewd distribution
Median
Negative
Positive r

18. Skewed Distribution Negative r Positive Right-skewed distribution
Large positive returns possible
Bad surprises are more likely to be small
Median
Negative r
Positive

19. Measuring Mean: Scenario or Subjective Returns( r ) = p s S
p(s) = probability of a state
r(s) = return if a state occurs
1 to s states

20. State Prob. of State rin State
Numerical Example:
Subjective or Scenario Distributions
State Prob. of State rin State
E(r) = (.1)(-.05) + (.2)(.05)...+ (.1)(.35)
= .15

21. S Measuring Variance Subjective or Scenario Variance = p ( ) [ r - E)] 2
Standard deviation = [variance]1/2
Using Our Example:
Var =[(.1)( )2+(.2)( ) ( )2]
Var=
S.D.= [ ] 1/2 = .1095

22. Sample Statistics 1 Sample mean return
where rt = rate of return on the stock for period t
T = number of periods
Sample variance of returns

23. Useful Formulas for Variance

24. Sample Statistics 2 Sample covariance
where r1,t = rate of return on the stock 1 for period t
r2,t = rate of return on the stock 2 for period t
T = number of periods
Sample correlation

25. Correlation pattern 1
rAB = +1
rAB = -1

26. Correlation pattern 2

27. Asset Allocation Across Risky & Risk-Free Assets
Possible to split investment funds between safe and risky assets
Risk-free asset: T-bills (proxy)
Risky asset: stocks (or a portfolio)
Issues
Examine risk-return tradeoff
Demonstrate how different degrees of risk aversion will affect allocations between risky and risk-free assets

28. Expected Return on the Combined Portfolio
Example:
1. Risk-free asset:
rf = 7%
srf = 0%
E(rp) = 15%
sp = 22%
y = % in p
(1-y) = % in rf +
2. Risky asset portfolio p:

29. Expected Return on the Combined Portfolio
E(rc) = yE(rp) + (1 - y)rf
rc = combined portfolio return
For example, assume y = .75
E(rc) = y(0.15) + (1-y)(0.07)
= y =0.13 (13%)

30. Variance on the Combined Portfolio
= 0, then s p c =
Since rf y s s
This is because
Var(rc) = Var(y rp + (1-y) rf )
= Var(y rp) + Var((1-y) rf)
+ 2Cov(y rp, (1-y) rf)
= y2 sp2

31. Combinations Without Leverage ( y ≤ 1: No borrowing)
= .75(.22) = .165 (16.5%)
If y = .75, then
= 1(.22) = .22 (22%)
If y = 1 (all in risky assets), then
= 0(.22) = .00 (0%)
If y = 0 (all in risk-free asset), then s s s

32. sc E(rc) CAL(Capital allocation line) = Possible Combinations
with no leverage (y ≤ 1)
= Lending portfolio
E(rp) = 15% P
13%
rf = 7% F sc
(y=0)
16.5%
(y=0.75)
sp=22%
(y=1)

33. Combinations With Leverage ( y > 1: With borrowing)
Borrow at the risk-free rate and invest in stocks
(Ex) With 50% Leverage,
E(rc) = (-.5) (.07) + (1.5) (.15) = .19 (19%)
sc = (1.5) (.22) = .33 (33%)

34. sc E(rc) CAL with leverage (y>1) = Borrowing portfolio 19% P
E(rp) = 15%
E(rp) - rf = 8%
) Slope = 8/22
rf = 7% F
(=Sharpe ratio) sc
sp=22%
(y=1)
33%
(y=1.5)

35. Two-Risky-Asset Case Two risky-asset portfolio expected return, rp
r1 = Expected return on Security 1
r2 = Expected return on Security 2
w1 = Proportion of funds in Security 1
w2 = Proportion of funds in Security 2

36. Two-Risky-Asset Portfolio Variance
Var(rp) = Var(w1 r1 + w2 r2 )
= Var(w1r1) + Var(w2r2) + 2Cov(w1r1, w2r2)
which can be simply rewritten as,
sp2 = w12s12 + w22s22 + 2w1w2 Cov(r1,r2)
s12 = Variance of Security 1
s22 = Variance of Security 2
Cov(r1,r2) = Covariance of returns for
Security 1 and Security 2

37. Three-Security Portfolio
rp = w1r1 + w2r2 + w3r3
s2p = w12s12 + w22s22 + w32s32
+ 2w1w2 Cov(r1,r2)
+ 2w1w3Cov(r1,r3)
+ 2w2w3Cov(r2,r3)

38. In General, n-Security Portfolio
rp = Weighted average of the
n-securities’ returns
sp2 = Own variance terms + all pair-wise covariances
: N variances, N(N-1) covariances
if wi = 1/N (i.e., equal-weighted)
Converge to average covariance
(Diversification effect)

39. Diversification Effect
Diversifiable risk
= Unsystematic risk
= Idiosyncratic risk
= Firm-specific risk
Average covariance
Nondiversifiable risk
= Systematic risk
= Market risk
The risk of a random portfolio of stocks as a percentage of the risk of an individual security.
That is, about 73% of the risk of an individual security can be eliminated by holding a random portfolio of stocks in U.S.

40. Diversification Benefits
The combined portfolio of domestic and foreign stocks reduces risk substantially.

41. Risk Aversion and Asset Allocation
Greater levels of risk aversion lead to larger proportions in the risk-free assets
Lower levels of risk aversion lead to larger proportions in the risky asset portfolio
Willingness to accept high levels of risk for high levels of returns would result in leveraged combinations and larger investment in risky assets
How to measure the risk aversion?
Many kinds of models

42. One Way of Quantifying Risk Aversion( ) rp - =
s 2 E rf
0.5 A
Assume p
E(rp) = Expected return on portfolio p
rf = the risk free rate
0.5 = Scale factor
A x s2 = Proportional risk premium
The larger A is, the larger return will be required for the risk

43. One Way of Quantifying Risk Aversion
Rearranging the equation and solving for A, r - r E ( ) p f A = 2
0.5 σ p
Many studies have concluded that investors’ average risk aversion is between 2 and 4

44. Finding an optimal weight, y*
Assume a utility function: U(r) = E(r) – 0.5 A s2
Note that it has a trade-off between E(r) and s2
Using the previous example:
U(ry) = ( y) – 0.5 A (0.222 y2) = f(y)
Find y* that maximizes the utility function
f´(y) = 0.08 – A (0.222) y = 0
f´´(y) = – A (0.222) < 0 (Optimality condition satisfies)
 y* = 0.08 / [A (0.222)]
= [ E(rp) – rf ] / [A s2 ]
A more risk-averse investor will invest less in risk asset
If risk premium is negative, a risk-averse investor will sell the asset short

45. Finding an optimal portfolio for y*
E(rc)
U(r) with A=4
CAL
E(rp) = 15% P
E(rc) = 10.28%
rf = 7% F sc
sp=22%
(y=1)
sp= 9.02%