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