Tài liệu tham khảo về phân tích rủi ro

Real vs. Nominal Rates
Fisher effect: Approximation
R = r + i or r = R - i
Example: r = 3%, i = 6%
R = 9% = 3%+6% or r = 3% = 9%-6%
Fisher effect: Exact

R −i
r =
;
or
1+i
0.09 − 0.06
Numerically: r = 2.83% =
1 + 0.06

1+R
1+r =
1+i


Rates of Return:
Single Period
HPR = P

1

− P0 + D1

P

0

HPR = Holding Period Return
P0 = Beginning price
P1 = Ending price
D1 = Dividend during period one


Rates of Return:
Single Period Example
Ending Price =

48

Beginning Price =

40

Dividend =

2

48 − 40 + 2
HPR =
= 25%
40


Characteristics of
Probability Distributions

1) Mean: most likely value
2) Variance or standard deviation
3) Skewness

* If a distribution is approximately normal, the distribution is described by
characteristics 1 and 2


Normal Distribution

s.d.

s.d.

r
Symmetric distribution


Measuring Mean: Scenario
or Subjective Returns
Subjective returns

E(r) =

s

∑p
i =1

i

⋅ ri

‘s’
= number of scenarios considered
pi = probability that scenario ‘i’ will occur
ri
= return if scenario ‘i’ occurs


Numerical example:
Scenario Distributions
Scenario
1
2
3

Probability
0.1
0.2
0.4

Return
-5%
5%
15%

4
5

0.2
0.1

25%
35%

E(r) = (.1)(-.05)+(.2)(.05)...+(.1)(.35)
E(r) = .15 = 15%


Measuring Variance or
Dispersion of Returns
Subjective or Scenario Distributions

2

Variance = σ =

s

2

∑ p(i) ⋅ [r(i) − E(r)]
i =1

Standard deviation = [variance]1/2 = σ
Using Our Example:
σ2=[(.1)(-.05-.15)2+(.2)(.05- .15)2+…]
=.01199
σ = [ .01199]1/2 = .1095 = 10.95%

Risk - Uncertain
Outcomes
p = .6

W = 100
1-p = .4

W1 = 150; Profit = 50
W2 = 80; Profit = -20

E(W) = pW1 + (1-p)W2 = 122

σ2 = p[W1 - E(W)]2 + (1-p) [W2 - E(W)]2
σ2 = 1,176

and

σ = 34.29%


Risky Investments
with Risk-Free Investment
p = .6

W1 = 150 Profit = 50

1-p = .4

W2 = 80 Profit = -20

Risky
Investment
100

Risk Free T-bills
Risk Premium = 22-5 = 17

Profit = 5


Risk Aversion & Utility


Investor’s view of risk








Risk Averse
Risk Neutral
Risk Seeking

Utility
Utility Function

U = E ( r ) – .005 A σ



A measures the degree of risk aversion

2


Risk Aversion and Value:
The Sample Investment
U = E ( r ) - .005 A σ
=
Risk Aversion
High

22%
A

2
- .005 A (34%)

2

Utility
5

-6.90
3

Low

1

4.66

16.22

T-bill = 5%


Dominance Principle
Expected Return
4
2

3
1

Variance or Standard Deviation

• 2 dominates 1; has a higher return
• 2 dominates 3; has a lower risk
• 4 dominates 3; has a higher return


Utility and Indifference
Curves


Represent an investor’s willingness to trade-off return and risk

Example (for an investor with A=4):

Exp Return
(%)
10
15
20
25

St Deviation U=E(r)-.005Aσ2
(%)
2
20.0
2
25.5
2
30.0
2
33.9


Indifference Curves
Expected Return

Increasing Utility
Standard Deviation

Portfolio Mathematics:
Assets’ Expected Return
Rule 1 : The return for an asset is the probability weighted average return in all scenarios.

E(r) =

s

∑p
i =1

i

⋅ ri


Portfolio Mathematics:
Assets’ Variance of Return
Rule 2: The variance of an asset’s return is the expected value of the squared deviations from the expected return.

2

σ =

s

∑p
i =1

i

2

⋅ [ri − E(r)]


Portfolio Mathematics:
Return on a Portfolio
Rule 3: The rate of return on a portfolio is a weighted average of the rates of return of each asset comprising the portfolio, with the portfolio
proportions as weights.

rp

= w1r1 + w2r2


Portfolio Mathematics:
Risk with Risk-Free Asset
Rule 4: When a risky asset is combined with a risk-free asset, the portfolio standard deviation equals the risky asset’s standard deviation multiplied
by the portfolio proportion invested in the risky asset.

σ

p

= wrisky asset × σrisky asset


Portfolio Mathematics:
Risk with two Risky Assets

Rule 5: When two risky assets with variances

σ12 and σ22 respectively, are combined into a portfolio with portfolio weights w1 and w2,

respectively, the portfolio variance is given by:

σ

p

2

= w12 σ12 + w22 σ22 + 2w1w2Cov(r1, r2)


Allocating Capital Between
Risky & Risk Free Assets





Possible to split investment funds between safe and risky assets
Risk free asset: proxy; T-bills
Risky asset: stock (or a portfolio)


Allocating Capital Between
Risky & Risk Free Assets




Examine risk/return tradeoff
Demonstrate how different degrees of risk aversion will affect allocations between risky and risk free assets


The Risk-Free Asset




Perfectly price-indexed bond – the only risk free asset in real terms;
T-bills are commonly viewed as “the” risk-free asset;
Money market funds - the most accessible risk-free asset for most investors.


Portfolios of One Risky Asset
and One Risk-Free Asset
 Assume a risky portfolio P defined by :
E(rp) = 15% and σp = 22%
 The available risk-free asset has:
rf = 7% and σrf = 0%
 And the proportions invested:
y% in P and (1-y)% in rf


Expected Returns for
Combinations
E(rc) = yE(rp) + (1 - y)rf
rc = complete or combined portfolio

If, for example, y = .75
E(rc) = .75(.15) + .25(.07)
= .13 or 13%