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Financial Risk
Learning Objec-ves
¨ Risk and uncertainty ¨ U-lity and indifference ¨ Probability of return rate
¤ Discrete periods
¨ Intro to por$olio theory
2
Financial Risk -‐ Frank Knight’s Insight
¨ University of Chicago , 1921 ¨ Dis-nguished between risk and uncertainty ¨ Risk – future financial outcomes can be quan-fied and
managed via probabili-es due to sufficient frequency of relevant historical events ¤ Risk is quan-fied and managed via mathema-cal models
¨ Uncertainty – future financial outcomes cannot be quan-fied and managed with probabili-es due to infrequency of relevant historical events ¤ Uncertainty is managed via other means
n managerial judgment n long-‐term or other risk reducing contracts n etc
Return Rate Probability
¨ Compute future return rate probabili-es from natural log rate normal pdf
¨ What is the probability of the return rate next month being less than some cri-cal rate, k, with z variate zk ? ¤ Expected monthly mean natural log rate u and
variance, s2, are known ¤ The area under the standard normal pdf to the leT of zk
4
( ) ( )
sukz
s
uSSln
z
szuSSln
szuSlnSln
k
0
1
0
1
01
−=
−⎟⎟⎠
⎞⎜⎜⎝
⎛
=
⋅+=⎟⎟⎠
⎞⎜⎜⎝
⎛
⋅+=−
normal pdf ~N(u, s2)
zk·∙s
zk·∙s u
standard normal pdf ~N(0,1)
zk zk 0
Return Rate Probability: Example
¨ The monthly natural log return rate es-mate, u, for an asset is 1.00% and the monthly vola-lity, es-mate, s, is 1.25%. What is the probability that next month’s return, , is less than .5% ?
¨ is the cumula-ve standard normal distribu-on, cdf ¤ Normsdist() in Excel
5
34.5% .40000)(N~
.0125.01.005N~
5%].0uPr[)(zN~k]uPr[ k
≈
−=
⎟⎠⎞
⎜⎝⎛ −
=
<
=<
u
N~
h_p://davidmlane.com/hyperstat/z_table.html
Another Example 6
¨ The monthly natural log return rate es-mate, u, for an asset is 1.00% and the monthly vola-lity es-mate, s, is 1.25%. What is the probability that next month’s return, , is actually a loss ?
%2.12 .80000)(N~
.0125.01.00N~
0%].0uPr[)(zN~k]uPr[ k
≈
−=
⎟⎠⎞
⎜⎝⎛ −
=
<
=<
u
And Another Example
¨ The monthly natural log return rate es-mate, u, for an asset is 1.00% and the monthly vola-lity, s, is 1.25%. What is the probability that the total return rate over the next year is greater than 20% ?
7
nsnuk z
ns
unSSln
z
nszunSSln
k
0
n
0
n
⋅
⋅−=
⋅
⋅−⎟⎟⎠
⎞⎜⎜⎝
⎛
=
⋅⋅+⋅=⎟⎟⎠
⎞⎜⎜⎝
⎛
( )( )
( )%2.3
847521.1N~1
12%25.1%121%20N~1
%]20μPr[zN~1]μμPr[ kk
=
−=
⎟⎠
⎞⎜⎝
⎛⋅
⋅−−=
>
−=>
Probability of a Price Decline 8
82193.3 501619.0
500031.44.10375.87ln
nσ
nuSSln
z 0
n
−=⋅
⋅−⎟⎠⎞
⎜⎝⎛
=⋅
⋅−⎟⎟⎠
⎞⎜⎜⎝
⎛
=
What was the probability of the drop in IBM stock price during the week ending October 10, 2008? Prior to Oct 6, IBM’s natural log daily return rate was .031% and standard devia-on was 1.619%. IBM stock closed Friday October 3rd at $103.44 and closed Friday October 10th at $87.75.
That 5 day decline was expected once in 60 years
[ ]%00662.
)82193.3(N~)z(N~SSPr 0T
=
−==<
[ ]( )zN~
SSPr 0T =≤
Confidence Intervals 9
$81.86 e$87.75
eSS
$94.35 e$87.75
eSS
5s1.959965u
ns1.95996nun
5s1.959965u
ns1.95996nun
0
0
=
=
=
=
=
=
⋅⋅−⋅⋅
⋅⋅−⋅⋅
−
⋅⋅+⋅⋅
⋅⋅+⋅⋅
+
Confidence Level (1-‐α)
α α/2 -‐Z +Z
90% 10% 5.00% -‐1.64485 1.6448595% 5% 2.50% -‐1.95996 1.9599699% 1% 0.50% -‐2.57583 2.57583
What are the upper and lower bounds on 5 day IBM stock price for which one is 95% (=1-‐α) confident? (using pre Oct 2008 data, with price at the Oct 10 Close )
( )95996.1N~ −( )95996.1N~1−
Value at Risk (VaR) 10
What is the maximum loss that an investor would expect over n periods ? What is the maximum loss expected with 95% confidence from holding an equity over a 10 day period? Use the historical (expected) mean rate and standard devia-on. Unlike the confidence interval, which uses a two tailed confidence , VaR is a one-‐tail interval.
Confidence Level (1-‐α)
α -‐Z
90% 10% -‐1.2815595% 5% -‐1.6448599% 1% -‐2.32635
%619.1s %031.u
91.80$ e7.858$
eSS10s1.6448510u
ns1.64485nu-‐0n
==
=
=
=
⋅⋅−⋅⋅
⋅⋅−⋅⋅
( )64485.1N~ −
Value at Risk (VaR) 11
The minimum 95% confident price is $37.67, thus the 95% maximum expected loss is $3.63 or value at risk, VaR
And commonly approximated for short -me periods as follows
$6.8491.80$85.87$VaR =−=
( )( )
$6.84 e17.858$
e1SVaR10s1.6448510u.
nsznu0
=
−⋅=
−⋅=⋅⋅−⋅
⋅⋅−⋅
( )( )
$7.09 e17.858$
e1SVaR10s1.64485
0
nsznu
=
−⋅=
−⋅=⋅⋅−
⋅⋅−⋅
VaR is computed directly as follows
U-lity
An economic term referring to the total sa-sfac-on received from consuming a good or service. A consumer's u-lity is hard to measure. However, we can determine it indirectly with consumer behavior theories, which assume that consumers will strive to maximize their u-lity. U-lity is a concept that was introduced by Daniel Bernoulli. He believed that for the usual person, u-lity increased with wealth but at a decreasing rate. Investopedia
12
Exposi-on of a New Theory on the
Measurement of Risk -‐ 1738
U-lity and Risk Aversion
¨ An individual may value expected outcome differently based on their risk aversion which may be based on wealth or preferences
¨ The u-lity of a financial gain or loss to an individual is likely dependent on current wealth
0
1
2
3
4
5
6
7
8
$0 $250 $500 $750 $1,000 $1,250 $1,500
U(w)
w
U(w)=ln(1+w)
U-lity and Risk Aversion
¨ An individual has wealth of 1000 and has the opportunity to par-cipate in a fair ‘financial game.’ 50% chance to gain 100 or lose 100. Assume her u-lity func-on is the natural log of her wealth
904.6)11100ln(5.)1900ln(5.)w(U =+⋅++⋅=
00.1005$1100525.900475.w$1005.00 is game after wealth expected Her
909.6)11100ln(525.)1900ln(475.)w(Uwinningof yprobabilit 52.5% a needs She
909.6)11100ln(p)1900ln()p1()w(U
=⋅+⋅=
=+⋅++⋅=
=+⋅++⋅−=
909.6)11000ln()w(U =+=
What probability of winning 100, p, would mo-vate her to play the financial game?
Introduce u-lity and risk aversion to expected rate of return and expected risk, which is represented by standard devia-on (vola-lity.) Vola-lity detracts from the u-lity of the expected return. We use expected quarterly natural log return rate and standard devia-on in all illustra-ons. Avoid mul--‐period considera-ons for now For single period analyses ok to use r & d considera-on, any IID/FV expected value and expected standard devia-on could be used A is the Pra_-‐Arrow measure of risk aversion Based on an individual’s aversion to risk The parameter, A, captures the slope and curvature of a u-lity curve
U-lity of Expected Return and Risk 15
-‐75% -‐50% -‐25% 0% 25% 50% 75% 100% 125% 150% 175% 200% 225% 250% 275% 300%
( )2sAuuU2⋅
−=
Risk – Return U-lity Curve
( )2s3uuU2⋅
−=
Note the same u-lity for these assets
u = 10% s = 20%
u = 7% s = 14%
u = 4% s = 0%
0%
1%
2%
3%
4%
5%
6%
7%
8%
9%
10%
11%
0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20%
Expected Risk [Std Dev %]
Expected Re
turn &
Utility of E
xpected Re
turn [%
] A=3
Aqtude Towards Risk
¨ A>0 ¤ Risk decreases u-lity of return ¤ Individual is risk averse and is thus an ‘investor’ ¤ Investor will not par-cipate in a ‘fair financial game’
¨ A=0 ¤ Risk does not effect the u-lity of return ¤ Individual is risk neutral and will par-cipate in a ‘fair financial game’
¨ A<0 ¤ Risk increases u-lity of return ¤ Individual will par-cipate in an “unfair financial game”
n Las Vegas
Indifference Curves
Lost reference, but these were not developed by Surprise Investments
Risk – Return Indifference Curve
¨ Combine indifference curve with risk – return expecta-on
¨ Where uCE is the (certain) return in the case of no expected vola-lity ¤ E(s) = 0 ¤ uCE the ‘certainty equivalent’ rate of return
( )
2sAuu
2)E(sAuuE
2
CE
2
CE
⋅+=
⋅+=
0123456789101112131415161718
0 2 4 6 8 10 12 14 16 18 20
Expected Risk [Std Dev %]
Expected Re
turn %
Risk – Return Indifference Curve
2s3uu2
CE⋅
+=
Note the investor’s indifference between these assets
uCE = 11% s = 0%
u = 12% s = 8%
u = 14% s = 14%
Capital Alloca-on Line
¨ A “line” of poruolios of consis-ng of two assets – a risk free asset, F, and a risky asset, A ¤ wA + wB = 1 ¤ Example: total stock market index fund and a money market fund (or a fund of treasury bills)
¨ So if all possible investments are on one straight line, how does an investor chose the op-mal alloca-on to each asset? ¤ “How much in stocks and how much in cash?”
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
0% 5% 10% 15% 20% 25% 30% 35% 40%
Expected Std Dev
Expected Return
Op-mal Poruolio
¨ CAL line contains all possible poruolios
¨ What’s your alloca-on of funds between assets
¨ Depends on your “A” and say its 5 ¤ Set the shape and
orienta-on of indifference curve
¨ Your op-mal poruolio is at the tangent point ¤ Equal slopes
Asset A
Asset P
Asset F
CAL
Indifference curve with A=5 tangent
to the CAL
uCE
λ
Op-mal Poruolio
¨ Sta-s-cs for two assets ¤ Asset A: uA , sA ¤ Asset F: uF with no risk, sF=0
¨ Equa-on for CAL: u = uF + λ·∙s
¤ Slope of CAL:
¨ Equa-on for indifference curves
¨ Slope of indifference curves: A·∙s ¨ Set slopes of CAL and indifference
curve equal ¤ λ = A·∙sP
A
FA
suuλ −
=
2sAuu2
CE⋅
+=
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
0% 5% 10% 15% 20% 25% 30% 35% 40%
Expected Std Dev
Expected Return
Op-mal Poruolio
¨ Op-mal poruolio has sta-s-cs uP and sP
¨ Frac-on of poruolio in risky asset A
Input Computed
uA 25% λ .6333
sA 30% sP 12.67%
uF 6% uP 14.02%
A 5.0 uCE 10.01%
wA 42.2%
wF 57.8%
AλsP =
PFP sλuu ⋅+=
2sAuu2P
PCE⋅
−=
A
PA s
sw =
Probability of a Loss Over 1 Quarter
%0usZuu
T
PPT
=
⋅+=
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20%
Percen
t Risky Asset
Prob of Negative Return
[ ] %4.13%0uPr
1070.1 1402.1267.
suZ
P
P
P
=<
−=
−=−=
Risk Aversion Equivalents
For poruolios of assets A & F The op-mal poruolio corresponds to A = 5
0
5
10
15
20
0% 20% 40% 60% 80% 100%
A
Percent Risky Asset
0
5
10
15
20
0% 5% 10% 15% 20% 25% 30% 35% 40% 45%
A
Expected Std Dev For Portfolio
0.0%
0.2%
0.4%
0.6%
0.8%
1.0%
1.2%
1.4%
0% 5% 10% 15% 20%
Expected Return Ra
te
Expected Std Dev
27
A Poruolio With Two Risky Assets
0.0%
0.2%
0.4%
0.6%
0.8%
1.0%
1.2%
1.4%
0% 5% 10% 15% 20%
Expected Return Ra
te
Expected Std Dev
A
B
F
A
B
F
28
A Poruolio With Two Risky Assets
¨ uP = wA·∙uA + wB·∙uB ¤ wA + wB =1
n requires that the poruolio is fully invested in the 2 assets A and B
¤ wA ≥ 0, wB ≥ 0 n prohibits short selling or borrowing an asset
¤ 1 ≥ wA, 1 ≥ wB n Restricts buying an asset on margin
ABBABA2B
2B
2A
2A
2p
ABBA2B
2B
2A
2A
2p
ABBABB2BAA
2A
2p
ρssww2swsws
sww2swsws
sww2swsws
⋅⋅⋅⋅⋅+⋅+⋅=
⋅⋅⋅+⋅+⋅=
⋅⋅⋅+⋅+⋅=
AAAA2A ssss ≡⋅≡
29
Poruolios With Two Risky Assets
¨ sA= 8.3% ¨ sB= 16.3% ¨ sAB = .004 ¨ uA =0.9% ¨ uB = 2.3%
¨ ρAB = .28
A
( )
AB
A
VV
AB2B
2A
AB2B
V
w-‐1w
2sssssw
=
−+
−=
ABBABA2B
2B
2A
2A
2p ρssww2swsws ⋅⋅⋅⋅⋅+⋅+⋅=
0.50%
0.75%
1.00%
1.25%
1.50%
1.75%
2.00%
2.25%
2.50%
0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20%
Expe
cted
Return Rate
Expected Std Dev
0.0%
0.2%
0.4%
0.6%
0.8%
1.0%
1.2%
1.4%
1.6%
1.8%
2.0%
2.2%
2.4%
0% 2% 4% 6% 8% 10% 12% 14% 16% 18%
Expe
cted
Return Rate
Expected Std Dev
30
Poruolios With Two Risky Assets
ρAB=1 ρAB=0 ρAB=-‐.5
ρAB=-‐1
A
B
ABBABA2B
2B
2A
2A
2p ρssww2swsws ⋅⋅⋅⋅⋅+⋅+⋅=
31
Two Risky and One Risk Free Asset
( ) ( )( ) ( ) ( ) ( )[ ] ABA TT
ABFBFA2AFA
2BFA
ABFB2BFA
T w-‐1w σuuuusuusuu
suusuuw =⋅−+−−⋅−+⋅−
⋅−−⋅−=
0.0%
0.2%
0.4%
0.6%
0.8%
1.0%
1.2%
1.4%
1.6%
1.8%
2.0%
0% 2% 4% 6% 8% 10% 12% 14% 16% 18%
Expe
cted
Return Rate
Expected Std Dev
0.0%
0.5%
1.0%
1.5%
2.0%
2.5%
3.0%
0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20%
Expe
cted
Return Rate
Expected Std Dev
32
Now Determine Your Op-mal Poruolio
Indifference curves
A=2 , 4, 7
T: Op-mal Risky Poruolio
F P: Your op-mal poruolio
A
B
V
33
Poruolio with 2 Risky Assets
0.0%
0.5%
1.0%
1.5%
2.0%
2.5%
3.0%
0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20%
Std Dev
Return
Indifference curves A=4
T: Op-mal Risky Poruolio
F P: Your op-mal poruolio
A
B
V
Essen-al Points
¨ Dis-nc-on between the ‘uncertainty’ and ‘risk’ ¤ One can be modeled and managed with ‘probabili-es’
¨ When probabili-es are computed the natural log rate of return measure must be used – not the simple rate of return
¨ U8lity includes subjec-vity – value and risk aversion ¨ The probability distribu-ons in the chapter must only be quadra-c
which are two parameter distribu-ons including the normal distribu-on ¨ Specula-on means taking risk
¤ It is not necessarily equivalent to gambling, which is taking risk with insufficient considera-on of the expected return
¨ One risk free asset and one risky asset is the simplest investment poruolio ¤ σA = 0 and ρAF = 0
Essen-al Points
¨ There is an op-mal poruolio -‐ comprised of the risk free and the op-mal risky asset -‐ given the available investments and the investor’s
¨ The tangent poruolio is the op-mal risky poruolio ¨ The slope of the CAL line is the called the “Sharpe ra-o” and has the
steepest slope of any line connec-ng the risk free asset and a tangency poruolio on the efficient fron-er
¨ Extension of the CAL beyond the op-mal risky asset requires the investor to borrow the risk free asset and invest in the risky asset ¤ In this case the risk free asset weight will be nega-ve and the weight for the
op-mal risky asset will be greater than 1. ¤ For the CAL to be straight beyond the op-mal risky asset, the borrowing rate
must equal the risk free rate.
35