This slide set is a work in progress and is embedded in my Principles of Finance course site (under construction) that I teach to computer scientists and engineers http://awesomefinance.weebly.com/

1. Financial
Risk

2. Learning
Objec-ves
¨ Risk
and
uncertainty
¨ U-lity
and
indifference
¨ Probability
of
return
rate
¤ Discrete
periods
¨ Intro
to
por$olio
theory
2

3. 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

4. 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
( ) ( )
s
uk
z
s
u
S
S
ln
z
szu
S
S
ln
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

5. 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
.01.005
N~
5%].0uˆPr[
)(zN~k]uˆPr[ k
≈
−=
⎟
⎠
⎞
⎜
⎝
⎛ −
=
<
=<
uˆ
N~
h_p://davidmlane.com/hyperstat/z_table.html

6. 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
.01.00
N~
0%].0uˆPr[
)(zN~k]uˆPr[ k
≈
−=
⎟
⎠
⎞
⎜
⎝
⎛ −
=
<
=<
uˆ

7. 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
ns
nuk
z
ns
un
S
S
ln
z
nszun
S
S
ln
k
0
n
0
n
⋅
⋅−
=
⋅
⋅−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
⋅⋅+⋅=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
( )( )
( )
%2.3
847521.1N~1
12%25.1
%121%20
N~1
%]20μˆPr[
zN~1]μμˆPr[ kk
=
−=
⎟
⎠
⎞
⎜
⎝
⎛
⋅
⋅−
−=
>
−=>

8. Probability
of
a
Price
Decline
8
82193.3
501619.0
500031.
44.103
75.87
ln
nσ
nu
S
S
ln
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 =≤

9. Confidence
Intervals
9
$81.86
e$87.75
eSS
$94.35
e$87.75
eSS
5s1.959965u
ns1.95996nu
n
5s1.959965u
ns1.95996nu
n
0
0
=
=
=
=
=
=
⋅⋅−⋅
⋅
⋅⋅−⋅
⋅
−
⋅⋅+⋅
⋅
⋅⋅+⋅
⋅
+
Confidence
Level
(1-­‐α)
α α/2 -­‐Z +Z
90% 10% 5.00% -­‐1.64485 1.64485
95% 5% 2.50% -­‐1.95996 1.95996
99% 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−

10. 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% -­‐1.28155
95% 5% -­‐1.64485
99% 1% -­‐2.32635
%619.1s
%031.u
91.80$
e7.858$
eSS
10s1.6448510u
ns1.64485nu-­‐
0n
==
=
=
=
⋅⋅−⋅
⋅
⋅⋅−⋅
⋅
( )64485.1N~ −

11. 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 =−=
( )
( )
$6.84
e17.858$
e1SVaR
10s1.6448510u.
nsznu
0
=
−⋅=
−⋅=
⋅⋅−⋅
⋅⋅−⋅
( )
( )
$7.09
e17.858$
e1SVaR
10s1.64485
0
nsznu
=
−⋅=
−⋅=
⋅⋅−
⋅⋅−⋅
VaR
is
computed
directly
as
follows

12. 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

13. 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)

14. 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(U
winningof
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?

15. 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%
( )
2
sA
uuU
2
⋅
−=

16. Risk
–
Return
U-lity
Curve
( )
2
s3
uuU
2
⋅
−=
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
Return
&
Utility
of
Expected
Return
[%]
A=3

17. 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

18. Indifference
Curves
Lost
reference,
but
these
were
not
developed
by
Surprise
Investments

19. 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
( )
2
sA
uu
2
)E(sA
uuE
2
CE
2
CE
⋅
+=
⋅
+=

20. 0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
0 2 4 6 8 10 12 14 16 18 20
Expected
Risk
[Std
Dev
%]
Expected
Return
%
Risk
–
Return
Indifference
Curve
2
s3
uu
2
CE
⋅
+=
Note
the
investor’s
indifference
between
these
assets
uCE
=
11%
s
=
0%
u
=
12%
s
=
8%
u
=
14%
s
=
14%

21. 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?”

22. 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

23. 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
s
uu
λ
−
=
2
sA
uu
2
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

24. 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%
A
λ
sP =
PFP sλuu ⋅+=
2
sA
uu
2
P
PCE
⋅
−=
A
P
A
s
s
w =

25. Probability
of
a
Loss
Over
1
Quarter
%0u
sZuu
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.
s
u
Z
P
P
P
=<
−=
−=−=

26. 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

27. 0.0%
0.2%
0.4%
0.6%
0.8%
1.0%
1.2%
1.4%
0% 5% 10% 15% 20%
Expected
Return
Rate
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
Rate
Expected
Std
Dev
A
B
F
A
B
F

28. 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
ABBABA
2
B
2
B
2
A
2
A
2
p
ABBA
2
B
2
B
2
A
2
A
2
p
ABBABB
2
BAA
2
A
2
p
ρssww2swsws
sww2swsws
sww2swsws
⋅⋅⋅⋅⋅+⋅+⋅=
⋅⋅⋅+⋅+⋅=
⋅⋅⋅+⋅+⋅=
AAAA
2
A ssss ≡⋅≡

29. 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
AB
2
B
2
A
AB
2
B
V
w-­‐1w
2sss
ss
w
=
−+
−
=
ABBABA
2
B
2
B
2
A
2
A
2
p 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%
Expected
Return
Rate
Expected
Std
Dev

30. 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%
Expected
Return
Rate
Expected
Std
Dev
30
Poruolios
With
Two
Risky
Assets
ρAB=1
ρAB=0
ρAB=-­‐.5
ρAB=-­‐1
A
B
ABBABA
2
B
2
B
2
A
2
A
2
p ρssww2swsws ⋅⋅⋅⋅⋅+⋅+⋅=

31. 31
Two
Risky
and
One
Risk
Free
Asset
( ) ( )
( ) ( ) ( ) ( )[ ] ABA TT
ABFBFA
2
AFA
2
BFA
ABFB
2
BFA
T w-­‐1w
σuuuusuusuu
suusuu
w =
⋅−+−−⋅−+⋅−
⋅−−⋅−
=
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%
Expected
Return
Rate
Expected
Std
Dev

32. 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%
Expected
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. 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

34. 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

35. 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