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# Financial risk pdf

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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/

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### Financial risk pdf

1. 1. Financial  Risk
2. 2. Learning  Objec-ves       ¨  Risk  and  uncertainty   ¨  U-lity  and  indiﬀerence   ¨  Probability  of  return  rate   ¤  Discrete  periods     ¨  Intro  to  por\$olio  theory   2
3. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 18. Indiﬀerence  Curves   Lost  reference,  but  these  were  not  developed  by   Surprise  Investments
19. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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