Default risk premium + Liquidity premium + maturity risk premium (2) PV of perpetuity

Transcription

1 The Prevew of L Quattatve Method Chapter.The Tme Value of Moey ( Requred rate of retur o a securty Requred rate omal rsk-free rate a + all kds of premums b a omal rsk-free rate real rsk-free rate + expected flato rate b Default rsk premum + Lqudty premum + maturty rsk premum ( PV of perpetuty fxed _ paymet PMT PV perpetury rate I / Y (3 FV ad PV calculatos ~ 考試要點 : a. 注意是 Ordary auty or Auty-due [BG] modes b. 畫圖 tme-le table Example: (Calculato of How may $00 ed-of-year paymet has to be made to accumulate $90 f the dscout rate s 9%? Sol: I/Y9; FV90; PMT-00; CPT years (4Amortzato What s the amortzato schedule of 0000 loa at 0% rate, ed of year paymet? Perod Begg Balace Paymet Iterest Compoet( Prcpal Compoet( Edg Balace( Chapter. Dscouted Cash Flow Applcatos ( PV (et Preset Value PV CF 0 CF + ( + r CF + ( + r CF ( + r CF t t 0 ( + r a. r Dscout rate / Requred rate of retur / Cost of captal b. 固定 r, 看看 PV 的值, 若 PV>0, 值得投資 t

2 ( IRR (Iteral Rate of Retur 0 CF 0 CF + ( + IRR CF + ( + IRR CF ( + IRR t 0 ( + CF t IRR a. 固定等號左方為 0, 看看 IRR 的值, 若 IRR> requred rate of retur, 則值得投資 (3 Coflctg decsos betwee PV ad IRR a. For oe tme, sgle vestmet ~Both PV ad IRR wll lead to exactly the same accept/reject decso. (lke above example 即若 IRR>Req. dscout rate PV > 0 b. For others ~the coflcto s possble. c. Detal explaatos ( PV method assumes the revestmet rate s opportuty cost of captal, whle IRR method assumes the revestmet rate s IRR tself. ( The, why stll IRR? I. Bod s YTM (yeld II. Oly oe tme (3 I the CFA exam, use PV to make the decso. (4 Moey-weghted rate of retur Defto: the moey-weghted rate of retur s defed as the teral rate of retur (IRR o a portfolo, takg to accout all cash flows ad outflows. (5 Tme-weghted rate of retur Defto: Tme-weghted rate of retur measures compoud growth. It s the rate at whch $.00 compouds over a specfed performace horzo. If the total vestmet s greater tha oe year, you must take the geometrc mea of the measuremet perod to fd the aual tme-weght rate of retur. Detal dscusso: the tme-weghted rate of retur s the preferred method of performace measuremet (e: GIPS, because t s ot affected by the tmg of cash flows ad outflows. It s less proe to the possblty of mapulato. 一般判斷標準 : 除非經理人可決定何時加 減碼,Moey-W 才會合適的績效指標 不然 (Tme-W 會是比較好的評量標準 t

3 Example: Assume a vestor buys a share of stock for $00 at t0 ad at the ed of frst year (t, she buys a addtoal share for $0. At the ed of year, the vestor sells both shares for $30 each. At the ed of each year, the stock pad a $ per share dvded. What s ( Moey -Weghted rate of retur, ( Tme -Weghted rate of retur? (Assume all the dvdeds wll ot be revested. T0 purchase share -00 T cash dvded for share + Purchase share -0 T cash dvded for share +4 Sell share +60 ( Moey-weghted rate of retur ( ( IRR + IRR ( 用 IRR CF 0-00, C , C , IRR 3.8 3

4 Holdg perod : Begg prce $00 Dvded Edg prce0 Holdg perod : Begg prce $40 Dvded4 Edg prce60 (Tme-weghted rate of retur HPR (0+/00-% HPR (60+4/40-0% (+T-W rate (.(.0 移項通分 [(.(.0] / -5.84% Chapter.5 Dfferet Rats Covetos ( Bak dscout yeld (r BD D 360 r BD * 單利計算,360 天 bass,d 是 dscout,f 是 face value, 年化值 F t ( Holdg perod yeld (HPY ( P + D P0 HPY HPR 實拿的利率, 沒有 dscout factor,d 是 dvded, 未年化值 P 0 (3 Moey Market yeld (r mm r MM F HPY * rbd *, where P purchase prce (F - D t 360 P ( tdays r BD (Face - Dscout 單利計算, 一統 r BD 和 HPY 了, 把 HPY 年化, 也把 r BD 調整成用 SPOT 而非 Face 來算報酬, 如此才 符合一般定存 (C.D. 算法,360 天 bass (4 Effectve Aual yeld (EAY ( 要熟記 365 t t EAY ( + HPY 複利計算,365 天 bass, 是從 ( + EAY ( + HPY 移項通分來的 365 (5 Bod Equvalet yeld (BEY ( BEY + EAY ( + 任何投資的報酬, 現在可以拿來和一般 Bod 算出的 yeld 做同基礎比較 4

5 (6 Covertg amog all kds of Compouded Rate of Retur( 要熟記 ( r EAY ( m m + + where ( r: 單利年率 : ( m:umber of perods per aual a. Bod Equvalet Yeld (rate b. Mothly Cash Flow Yeld (rate c. Daly Cash Flow Yeld (rate ( + EAY ( + BEY ( + EAY ( + mothly _ rate ( + EAY ( + daly _ rate d. Cotuous Cash Flow Yeld (rate ( + EAY e 365 cotuous compouded rate Example: Gve oe 3 moth vestmet whch ca ear percet, what s ts Bod Equvalet yeld(bey? 3 As: ( + EAY ( /,so EAY ( % ( BEY + EAY ( ( + so, BEY8.08% Example: A stock was purchased for $00 ad sold oe year later for $0. Calculate the vestor s aual rate of retur o a cotuously compouded bass? 0 00 ( + EAY ( + e l(. 8.3% Chapter 3. Statstcal cocepts ad Market returs ( Statstcs type Descrptve statstcs 敘述統計 ( 汲取特質 e: S.D. Iferetal statstcs 推論統計 ( 以機率學用樣本來推估母體的某種特性 e: C.I. ( Populato VS. Sample ~ Populato 母體 ~ All members cluded,e: 人口普查 ~ 做出的 母體統計量 稱 : Parameter 用來描述 Populato 特性,e: ~ Sample 樣本 ~ Subset of populato,e: 產品抽驗 ( X μ ~ 做出的 樣本統計量 稱 : Sample statstc 用來描述 Sample 之特性,e: s ( X μ 5

6 (3 Dfferet measuremet scales ( 少考. omal Scales 名目尺度 ( 隨便分, 無特定 rak 關係, 表 small-cap fud, 表 large-cap fud,3 表 bod fud ( 僅眾數有意義. Ordal Scales 順序尺度 ( 依某特性分順序,e: 夏 商 周 元 明 清 ( 僅眾數 中位數有意義 3. Iterval Scales 區間尺度 ( 有了量化,e: 3 0 C~34 0 C VS 45 0 C~47 0 C, 但 30 0 C 三倍的 0 0 C, 因 0 0 C 不為 oe ( 眾數 中位數 平均數皆有意義 4. Rato Scales 比率尺度 ( 有了量化, 也把 0 當原點,e: 5 元 三倍的 5 元 ( 眾數 中位數 平均數皆有意義 (4 Frequecy dstrbuto Stock Retur table 0.40%.50%.0% -.40% 9.80% 7.00%.80% 8.40% 34.60% -8.60% 0.60% 5.00% -7.60% 5.60% 8.90% 40.40% -.00% -4.0% -5.0%.00% 各種名詞 Sort -8.60% -.00% 8.40% 7.00% -7.60% 0.60% 8.90%.00% -.40%.80% 9.80%.50% -5.0% 5.00% 0.40% 34.60% -4.0% 5.60%.0% 40.40% Absolute Iterval Talles Frequecy Relatve Frequecy Cumulatve Cumulatve Absoulte Relatve Frequecy Frequecy [ -30% ~ -0% 5.00% 5.00% [ -0% ~ -0% 0.00% % [ -0% ~ 0% % % [ 0% ~ 0% % % [ 0% ~ 0% % % [ 0% ~ 30% 0.00% % [ 30% ~ 40% 5.00% % [ 40% ~ 50% 5.00% % Total % 6

7 Hstogram Polygram Absolute Frequecy (Hstogram Absolute Frequecy (Polygo -0% -0% 0% 0% 0% 30% 40% 50% ~ ~ ~ ~ ~ ~ ~ ~ -30% -0% -0% 0% 0% 0% 30% 40% 0 a. Quatle ~ Defto: the geeral term for a value at or below whch a stated proporto of the data a dstrbuto les. ~ Example: ( quartles (dvded to quarters ( qutles (dvded to ffths (3 decles (dvded to teths (4 percetle (dvded to hudredths y L y ( + ~ Formula: 00 Example: What s the thrd quartle for the followg dstrbuto of returs? 8% 0% 0% 0% 5% 6% 7% 7% 9% 3% 4% 6% Soluto: Sce total samples ad thrd quartle 75% 75 So, L y ( 表示 :the 3 rd quartle s the 9 th data pot plus 0.75 * ( dstat 00 betwee 9 th to 0 th. Thus, the 3 rd quartle s *(3-9 (%, 即 % 以下含括 75% 的資料 b. Cetral tedecy ~ Measure of cetral tedecy detfes the ceter, or average, of the data set. Ths s cetral pot ca be used to represet the typcal, or expected, value the data set. ( Populato arthmetc mea: 母體算術平均數 μ ( Sample arthmetc mea: 樣本算術平均數 X ~PS: the Arthmetc mea s the oly measure to make the sum of mea devatos ( X X 0 X X 7

8 50 (3 meda: 中位數 ( 依上例, 即 L y ( ,so, meda6+0.5( (4 mode: 眾數 ( 依上例, 因 0% 出現三次, 為最多,so, mode0% (5 Geometrc mea: 幾何平均數 G. M X * X *...* X ( X * X *...* X ( 依上例, G. M 8% *0% *...* 6% 5.% ; Arth. Mea μ 6.5% ~ 特性 : Arthmetc Mea > Geometrc Mea (6 Weghted Mea: 加權平均數 X w w X (7 Harmoc Mea: 調和平均數 X Arthmetc Mea > Geometrc Mea > Harmoc Mea 各樣本皆相等時等號成立 H X c. Dsperso ( Rage: ragemax Value-M Value ( 依上例 : rage6-88(% ( Mea Absolute Devato (MAD ~wthout the ABS calculato, the sum of devatos s 0 for Arth. Mea X X X 6.5% MAD ( 依上例 : MAD % (3 Populato Varace: ( X μ (4 Sample Varace: (5 Sem-varace: s ( X X All _ X< X ( X X sem Var (# _ of _ Xs _ less _ tha _ X ~ e: usually, we care oly the uderperformg stuatos (6 Populato Stadard Devato (S.D: ( X μ (7 Sample Stadard Devato (S.D: s ( X X 8

9 (8 Chebyshev s equalty ~ 一種無母數統計值 ~ Defto: o matter what kd of dstrbuto the data set s, the percetage of the observatos that le wth k stadard devatos of the mea s AT LEAST for k all k>. ( 因為不論為任何分佈情況的資料群皆會合乎此一公式, 故 Chebyshev s equalty s used to measure the MAX amout of dsperso. 以 mea 為 中心, 前後 k 個 S.D., 總計 包含多少個資 料 K Chebyshev s k ormal dstrbuto + 0% 約 68% % % + 75% 約 95% % 約 99% % d. the relato betwee S.D ad mea ( Coeffcet of varato (C.V 變異係數 ~ C.V. s varato per ut of retur s X S. D _ of _ x C. V X mea _ of _ x ( Sharpe Rato of retur ~ Sharpe rato s the excess retur per ut of rsk, where rsk s measured by S.D r portfolo r rsk Sharpe _ rato portfolo free (5 The measuremet to the shape of dstrbutos a. Kurtoss ( 峰態 ~ the Kurtoss s a measure of the degree to whch a gve dstrbuto s more or less peaked tha the ormal dstrbuto. ~ the Kurtoss of the ormal Dstrbuto s equal to 3 ~ excess kurtoss sample kurtoss 3 where the sample kurtoss * ( X X s 4 4 9

10 Sample termology Characterstcs kurtoss(k K>3 Leptokurtc (fat-tal*** 中心多, 次中心少, 旁邊多 K3 K<3 Platykurtc 中心少, 次中心多, 旁邊少 示意圖 : K>3 K<3 b. Skewess ( 偏態 ~ the Skewess of the ormal Dstrbuto s equal to 0 ( X X where the sample Skewess (S k * 3 s Sample Skewess Termology Characterstcs 小 中 大 (S k S k >0 Postve Skew Mode Meda Mea S k 0 Symmetrcal dstrbuto MeaMedaMode S k <0 egatve Skew Mea Meda Mode 3 示意圖 : 記憶法 :meda 放中間, 配合劃圖看 mode Postve Symmetrc egatve Mode Meda Mea MeaMedaMode Mea Meda Mode 0

11 Chapter 4. Probablty Cocepts ( Basc deftos: a. radom varable (R.V. 隨機變數 a ucerta umber b. outcome the realzato of a R.V. c. evet a specfed set of sgle/multple outcome(s d. mutually exclusve evets oly oe evet ca occur at a tme caot both happe at the same tme e. exhaustve evets the evets that cover all possble outcomes ( Probablty Theory a. Emprcal probablty by aalyzg hstorcal data b. Pror probablty by reasog / logcal process c. subjectve probablty (3 Ivestmet cosequece whe cosstet probabltes exst. Ex: evet: Fed wll decrease the terest rate Prce of stock A mply 90% possblty wll decrease. Prce of stock B mply 30% possblty wll decrease. It meas that f the terest rate decrease, the prce of stock A wll oly go up by lttle compare to stock B as t s kd of the surprsg good ews for stock B. ( 真實世界 mply0% 利多出盡 (4 Probablty has two propertes( 要熟記 a. 0<P[Ay Evet]< b. f E,E E are mutually exclusve & exhaustve evets, the P[ ] (5 Probablty theory( 看例子先 a. Ucodtoal probablty P[A] 示意圖 : 三者間關系 : b. Codtoal probablty P[A B] c. Jot probablty P[AB] A B P [ A B] p[ AB] p[ B] E d. Addto rule P[A or B]P[A]+P[B] - P[AB] e. Total probablty P[ R] P[ R S ]* P[ S where S s are mutually exclusve ad exhaustve. ]

12 f. Bayes formula P[ A B] P[ A B] P[ B] P[ A ]* P[ B A ] P[ A ]* P[ B A ] ~~ We ca fd P[A B] by gve all P[A] s ad P[B A] s ~~ 題目給你所有 P[A] s ad P[B A] s, 求 P[A B] g. Idepedet evets for evets: defto: P [ AB] P[ A]* P[ B] for 3 or more evets: property: P [ ABC] P[ A]* P[ B]* P[ C] Example: (Bayes formula 相關題型 You have developed a set of crtera for evaluatg dstressed credts. Frms that do ot receve a passg score are classed as lkely to go bakrupt wth moths. You gather the followg formato whe valdatg the crtera:. Forty percet of the compaes to whch the test s admstered wll go bakrupt wth moths. Ffty-fve percet of the compaes to whch the test s admstered pass t 3. The probablty that a frm wll pass the test (ad be classed as a -moth survvor, gve that t wll subsequetly survve moths, s 0.85 Usg Bayes theory, calculate the probablty that a frm s a survvor, gve that t passes the test? A B.0.80 C.0.0 D.0.85 Soluto: 一樣先用機率式表之, 免得反應不過來 (I P[Bakrupt] P[B]0.40 (II P[PassTest]0.55 (III P[PassTest ot B]0.85 求 P[ot B PassTest]? 貝式定理解題中, 最重要的是, 把問你的問題寫成此機率式 P[ ot _ B & PassTest] P[ ot _ B PassTest] P[ PassTest] P[ ot _ B]* P[ PassTest ot _ B] 0.6 * P[ Bakrupt ]* P[ PassTest Bakrupt ] 0.55 (6 Probablty coutg rules a. Factoral ( 階乘 ~ Used to arrage all tems of oe group.!((-(- ((, ad by defto 0! b. permutato formula ( 排列 ~ the umber of ways that we ca choose r objects from a total objects, where the order whch the r objects s lsted does matter. P r! * ( * ( *...( ( r ( r! c. combato formula ( 組合 ~ the umber of ways that we ca choose r objects from a total objects, where the

13 order whch the r objects s lsted does OT matter.! *( *( *...( ( r ( r!* r! **... r C r * ~ 一副牌有多少種 FULL HOUSE? C d. Multple sub-group combato formula ( 多重組合 ~more tha groups! C * (- C * (-- C 3!*!*...*! k 3 4 C3 * C * * C where 4 k ~ e: you are gog to rak total 8 fuds to low rsk(6, md rsk(8, hgh rsk (4 6 8! 8 (7 Expected value, Varace, ad Covarace!! a. Expected value: E[X] x * P[ ] ( 要熟記 x b. Varace: V[X] X E[(X- μ x (X- μ x ] rage[0~+ ] 4 μ ( x ( x μ * P[ x ] E(X - COV[X,X] c. Stadard Devato: S.D.[X] X V [X ] μ x c. Covarace: COV[X,Y] E[(X- μ x (Y- μ ] rage[- ~+ ] μ ( y j y! ( x μ * P[ x, y ] E(XY- μ x μ y ( 要熟記 COV[ X, Y ] d. Correlato: Corr(X,Y ρ X, Y rage from [-,+] ~ 比較與說明 ( Varace measures how a R.V. moves wth tself ( Covarace measures how oe RV moves wth aother RV. (3 Correlato measures the lear relatoshp betwee RVs. ρ X,Y + Perfect postve correlato ~ 完全正相關 X Y j j j ρ X,Y 0 o lear relatoshp betwee RVs ~ 無關 ( 給你 y 的資訊, 並無助 於你去預測 x 值 3

14 ρ X,Y - Perfect egatve correlato ~ 完全負相關 (4 Covarace Matrx: 因任兩個 asset 間都可算出 Covarace, thus, for example, f you have 4 assets your portfolo, you ca costruct 4*4 (symmetrc Covarace matrx. V(a,a Cov(a,b Cov(a,c Cov(a,d V(b,b Cov(b,c Cov(b,d V(c,c Cov(c,d V(d,d (5 Correlato matrx: ( 同理可建構出 by 除以相應 個 assets 的標準差 r(a,b r(a,c r(a,d r(b,c r(b,d r(c,d ~ 重要運算式, Assume a, b, c are costat ( E[aX+bY+c]aE[X]+bE[Y]+c ( Codtoal expectato: 條件期望值運算. E[X] E[ X Y ]* P[ Y ] where Y s a set of mutually exclusve & exhaustve evets.. E[X Y] (3 COV[aX,b] 0 (4 V[a]0 x * P[ x Y ] (5 COV[aX,bY] ab COV[X,Y] (6 V[aX+bY+c] COV[aX+bY+c, ax+by+c] 此處的 Y, 可為上式的 Y a V[X] + b V[Y] + abcov[x,y] a X + b Y + ab ρ XY X Y (7 V[ a ] a COV [ x, x ] X a j a V[ x ] + j < j j a a jcov[ x, x j ] ex: V[X+4Y-5Z]4V[X]+6V[Y]+5V[Z] +*((4COV[X,Y] +*((-5COV[X,Z] +*(4(-5COV[Y,Z] 4

15 Chapter 5 Commo Probablty Dstrbutos ( Basc deftos: a. A probablty dstrbuto ( 機率分配 specfes the probabltes of all possble outcomes of a radom varable b. The two basc types of radom varables are dscrete radom varables ad cotuous radom varables. Dscrete radom varables mea each varable ca be couted ad probablty ( p(x be assged, whereas each cotuous varable does ot exst (e: P[x]0; we must use probablty desty fucto(pdf ( f(x to geerate the probablty that outcomes le wth a rage of outcomes. c. The cumulatve dstrbuto fucto (CDF, deoted F(x for both cotuous ad dscrete radom varables, gves the probablty that the radom varable s less tha or equal to x. Therefore: CDF has the value betwee [0,] CDF & pdf F ( x x f ( x dx where f(x s pdf" ad F(xP(X<x s CDF" F( 0 F( + CDF & pdf CDF CDF pdf pdf ( Some commo dstrbutos: a. The dscrete uform ad the cotuous uform dstrbutos are the dstrbutos of equally lke outcomes. I dscrete uform dstrbutos, the probablty that the radom varable takes o ay of these possble values s the same for all outcomes. The dstrbuto has a fte umber of specfed outcomes, ad each outcome s equally lkely (e: Dscrete uform RV: gve X {.,.4,.6,.8,.0} the P(.40% I cotuous uform dstrbuto s defed over a rage that spas betwee some lower lmt a, ad upper lmt b, whch serve as the parameter of ths dstrbuto. ~ 可表為 :U(a,b Mathematcally speakg, x x P[X<a]0 ; P[X>b]0 ; P[x<X<x] for ay x, x [a,b] b a (e: Cotuous uform RV: gve X [.~.0] the P(.40%, P(.3<x<.637.5% 5

16 b. Bomal Dstrbuto ~ 可表為 :b(,p I. a bomal R.V. may be defed as the umber successes a gve umber of depedet trals ad gve costat probablty of success 可表為 :x~ b(,p II. Uder above codtos, the bomal probablty fucto defes the probablty of x successes trals. Expressed as followg formula: p(x P(Xx (umber of ways to choose x from p x (-p -x C x p x ( p III. The Mea of bomal R.V. s p The Varace of bomal R.V. s p(-p x c. ormal Dstrbuto ~ 可表為 :( μ, I. a ormal dstrbuto s completely descrbed by ts mea, μ, ad varace, stated as X ~ ( μ,. II. Skewess0 whch meas ( μ, s symmetrc about ts mea, thus P(X< μ P(X< μ 0.5, ad meamedamode III. Kurtoss3; excess kurtoss 0 Kurtoss s a measure of how flat the dstrbuto s. IV. The possble outcomes rages from - to + V. A lear combato of ormally dstrbuted radom varable s also ormally dstrbuted. VI. the cofdece terval of a ormally dstrbuted R.V. 粗估 : 約 50% 的觀察值會落在母體平均值上下 0.67 個標準差的範圍內 約 68% 的觀察值會落在母體平均值上下 個標準差的範圍內 約 95.44% 的觀察值會落在母體平均值上下 約 99.74% 的觀察值會落在母體平均值上下 3 細算 :( 要熟記 約 90% 約 95% 約 98% 約 99% 個標準差的範圍內 個標準差的範圍內 的觀察值會落在母體平均數上下 ±.645 個標準差內 的觀察值會落在母體平均數上下 ±.96 個標準差內 的觀察值會落在母體平均數上下 ±.33 個標準差內 的觀察值會落在母體平均數上下 ±.58 個標準差內 VII. Stadard ormal dstrbuto ~ 可表為 :( 0, 不同的常態分配有不同的平均數和變異數, 為了統一, 我們將之轉換為標準常態分 配 標準常態分配的平均數為 0, 標準差為 而標準常態分配下的 Z 值代表的就 是從平均數到某一個觀察值的標準差值 我們可以利用標準常態分配表, 找出一個特 6

17 定的 Z 值, 然後求出對應之機率值 轉換公式 : 在常態分配的題目中, 如果 observato populato _ mea x μ Z ( 要熟記 s ta dard _ devato ( 要求機率, 就先將隨機變數值 (X 換成 Z 值, 再查表即可求出機率 ; 如果知道機率, 給你某數值, 要求機率 X μ 某數值 μ P[X< 某數值 ] P[ < ] P[Z< 某數值 μ ] ( 查表得機率 ( 要我們求 X 值, 則先將機率換成對應的 Z 值, 就可以求出 x 值 給你機率值, 要求 X 值 機率值 查表得 Z 值 Z X 值 μ 得 X 值 VIII. SFRato _(Roy's safety - frst rato SFRato Roy's Safety Frst Rato r portfolo r treshold portfolo _ level ( SFRato ca be stated as: Mmze P( R p < R L Where: R p portfolo retur R L threshold level retur I short, SFRato states that the optmal portfolo mmzes the probablty that the retur of portfolo falls below some mmum acceptable level. Ths m. level s called the threshold level (SFRato s very smlar to SFRato s actually -Z x μ Z. Ideed, f a portfolo retur ~ ( μ,, the the (3the probablty that portfolo s retur wll fall below the threshold retur s r treshold _ level r P(Z< portfolo portfolo or, equally, P(Z<-SFRato. 故若你想從幾個 portfolos 中間, 挑一個 portfolo, 其報酬率小於你的要求報酬率的機會是 最低的, 那你要挑 :SFRato 值最大的那一個 portfolo. 7

18 SFRato: ( SFR0.67 ( SFR %.50% d. Logormal Dstrbuto I. Defto: Ye x where x~ ( μ, the Y follows the logormal dstrbuto. A radom varable Y follows a logormal dstrbuto f ts atural logarthm, ly, s ormally dstrbuted. The reverse s also true: If a radom varable Y follows a logormal dstrbuto, the s atural logarthm (ly follows a ormal dstrbuto II. The logormal dstrbuto s completely descrbed by two parameters. A logormal dstrbuto s defed terms of the parameters of a dfferet dstrbuto. The two parameters of a logormal dstrbuto are the mea ad stadard devato (or varace of ts assocated ormal dstrbuto: the mea ad varace of l Y, gve Y s logormal. III. We must keep track of two sets of meas ad stadard devatos (or varaces: the mea ad stadard devato (or varace of the assocated ormal dstrbuto (these are the parameters, ad the mea ad stadard devato (or varace of the logormal varable tself. Gve Ye x where x~ ( μ, the Y follows the logormal dstrbuto. IV. Logormal V.S. ormal ( Logormal dstrbuto s skew to the rght ( Logormal dstrbuto s bouded from below by zero so that t s useful for modelg asset prces whch ever take egatve values. ormal dstrbuto Logormal dstrbuto 8

19 ( Uvarate ad multvarate dstrbuto a. Uvarate dstrbuto the dstrbuto of a sgle R.V (e: Up to ow, our dscusso has bee strctly focused o uvarate dstrbuto. b. Multvarate dstrbuto to descrbed the probabltes of outcomes of a group of R.Vs. It could be ether cotuous or dscrete R.V. It s meagful oly whe the behavor of each R.V. s depedet. I the multvarate ormal dstrbuto, Correlatos are the ew factors that are eeded to be add- besdes meas ad varaces to descrbe a gve specfc dstrbuto. (3 Mote Carlo Smulato ad Hstorcal Smulato a. Mote Carlo smulato volves the use of a computer to fd approxmate solutos to complex problems b. Mote Carlo smulato volves detfyg the rsk factors assocated wth a problem ad specfyg probablty dstrbuto for them. Repeated radom samplg from a probablty dstrbuto or dstrbutos s used to smulate the rsk factors c. Mote Carlo smulato s a complemet to aalytcal methods d. Hstorcal smulato s a establshed alteratve to Mote Carlo smulato that volves repeated samplg from a hstorcal data seres e. Hstorcal smulato ca oly reflect rsks represeted the sample. Compared to Mote Carlo smulato, hstorcal smulato does ot led tself to what-f aalyss Mote Carlo Smulato Hstorcal Smulato Methodology The geerato of a large umber of radom samples from specfed Usg a repeated samplg from a hstorcal data seres, you do t eed probablty dstrbutos to represet to assume the dstrbuto of stock the operato of rsk affectg a asset s prce movemet sce you use the actual past data. Applcatos. Value complex securtes. calculate VaR. Complemetary to Mote Carlo smulato 3. Smulate peso fud asset/labltes over tme Lmtatos. Compared wth a aalytcal method, Mote Carlo smulato does ot explctly provde the. Does ot explctly provde the sestvty feedback too. 9

20 sestvty feedback o asset. The rsk dstrbuto the future prce for each varable s chage. may ot be the same that the past ( 註 : 用 M.C 很難看出非線性的關 3. Ifrequetly evet may ot be 系, 到底是指數 對數 次方, reflected. 但線性的可以 4. Lmted past data. 5. Caot ask what f questos Chapter 6. Samplg ad Estmato ( 本章概念 : 我們想要知道母體的參數 (Parameter, 而利用樣本統計數 (Statstc 來推估母體參數, 譬如我們利用樣本平均數 ( X 來推估母體平均數 (μ, X 稱之為點估計 (Pot Estmate 運用中央極限定理(Cetral Lmt Theorem, 我們了解樣本平均數 ( X 分配 (Samplg Dstrbuto 會近似常態分配, 其平均數為母體平均數 (μ, 標準差為樣本標準差 s 除以樣本觀察數的開根號 ( 點估計為最簡單的方法 我們利用樣本平均數分配為常態分配之特性, 給定一個信賴水準, 進而可以決定一個區間 (Cofdece Iterval, 就可說明母體平均數有多少機率會位於這個區間內 ( Radom Samplg A. Parameter: A parameter s a descrptve measure computed from or used to descrbe a populato of data. B. Statstc: A statstc s a descrptve measure computed from or used to descrbe a sample of data. C. Smple radom samplg: A smple radom sample s a sample obtaed such a way that each elemet of the populato has a equal probablty of beg selected. D. Samplg error: Ay dfferece betwee the sample mea ad the populato mea s called samplg error. ( Samplg error of mea X -μ E. Samplg dstrbuto: A sample statstc s a radom varable too. Ths dstrbuto s the statstc s samplg dstrbuto. F. Stratfed radom samplg: I stratfed radom samplg, the populato s subdvded to subpopulatos (strata based o oe or more classfcato crtera. Smple radom samples are the draw from each stratum szes proportoal to the relatve sze of each stratum the populato. These samples are the pooled. (3 The desrable propertes of a estmator: 一個好的點估計值的三個特性 : 0

21 A. Ubasesdess 不偏 (the expected value of the estmate equals the populato parameter, B. Effcecy 有效 (the ubasedess estmator has the smallest varace C. Cosstecy 一致 (the estmator gets better as use more data (4 Cetral Lmt Theorem a. Defto: Gve a populato descrbed by ay probablty dstrbuto wth mea μ ad fte varace, the samplg dstrbuto of sample mea X computed from samples of sze from ths populato wll be approxmately ormal wth mea μ ad varace whe the sample sze s large. b. 中文 : 不管 X I 的母體如何分配, 只要 X I 間獨立, 且每次取樣組的 組內樣本數 則各 組間的 平均值 的分配, 會是常態分配 X ~ ( μ, x μ [ 當然, 你也可以說 :Z ~ ( 0, ] c.i practce, we take 30 ( stead of as the acceptable level to apply C.L.T. d.the stadard devato of a sample statstc s kow as the stadard error of the statstc, so the stadard error of the sample mea: wth kow populato varace:, where x s wth ukow populato varace: s x, where s ( X μ ( X X (5 Cofdece Itervals(C.I. A. Defto: A cofdece terval s a terval for whch we ca assert wth a gve probablty -α, called the degree of cofdece, that t wll cota the parameter t s teded to estmate. It s ofte referred to as the (-α% cofdece terval for the parameter. The resultg terval gves you a rage of values that the mea value wll be betwee wth a certa probablty, say 90, 95 or 99 percet. B. Costructg the Cofdece Itervals:

22 Where A (-α% cofdece terval for a parameter has the followg structure: Pot estmate ± (Relablty Factor x Stadard Error Pot Estmate a pot estmate of the parameter (a value of a sample statstc Relablty factor a umber based o the assumed dstrbuto of the pot estmate ad the degree of cofdece (-α% for the cofdece terval Stadard Error the stadard error of the sample statstc provdg the pot estmate C. Wth a kow varace, the formula for a cofdece terval of populato mea s x ± Z α * ( 要熟記 平均值 μ 的信賴區間是左右對稱 ( 又稱雙尾, 因此信賴水準 (-α% 所對 應之 z 臨界值是以 Z 表示 以下為 3 個常用的信賴水準所對應的 Z 值 : α Z α.645 for 90% cofdece tervals Z α.960 for 95% cofdece tervals Z α.575 for 99% cofdece tervals (6 Studet s t-dstrbuto~ 可表為 :t(υ A. Propertes: I. t-dstrbuto looks lke ormal dstrbuto, fact, Whe df The t-dstrbuto ormal dstrbuto II. Symmetrcal. III.K>3, fat tal IV. Defed by a sgle parameter, the degree of freedom (df, orυ, where the (df for sample mea s equal to the umber of sample observato mus, -. B. Applcato: I. For the ormally dstrbuted populato, whe sample sze s small ad populato varace s ot kow (ad, the use of a relablty factor based o the t-dstrbuto s essetal II. Whe workg wth cofdece tervals, due to the relatvely fatter tals of the t-dstrbuto, cofdece tervals costructed usg t relablty factors (t (-,α wll be more coservatve (wder tha those costructed usg z relablty factors (z α. (e: t( whle Z

23 III. Summary: Test Statstc Whe samplg from a Small Sample (<30 Large Sample ( 30 ormal dstrbuto wth kow varace z-statstc z-statstc ormal dstrbuto wth ukow varace t-statstc t-statstc * ( 因 CLT 用 z 勉強可以 o-ormal dstrbuto wth kow varace ot avalable z-statstc o-ormal dstrbuto wth ukow varace ot avalable t-statstc * ( 因 CLT 用 z 勉強可以 The wdth of the cofdece terval decreases as we crease the sample sze. Ths decrease s a fucto of the stadard error becomg smaller as crease. Ths relablty factor also becomes smaller as the df creases. (7 Samplg Selectg Bas a. All of the followg are mportat for pot ad terval estmato ad hypothess testg. If the sample s based as descrbed these ssues, the pot ad terval estmates ad ay other cocluso we from the sample well be error Types of Bas Descrpto Example Soluto Data-mg Bas Data mg s the practce of fdg forecastg models by extesve 技術分析 lack of ecoomc theory ad repeatedly test all Usg out-of-sample test searchg through databases for patters or tradg rules. possble patters, most of whch are fal, utl sgfcat oe was foud. Sample Selecto Bas Whe some data s systematcally excluded from the aalyss, usually because of the lack of 之前的例子 : 你都只取到 ~5 年級學生的 IQ 值 ( 因為大部分 6 年級生今天校外教學 avalablty. Survvorshp Bas Data avalablty leads to certa assets beg excluded from the aalyss The sample mea of the performace of Hedge fud looks better the t should be. ( 差的早就不見了 Look-ahead Bas Use formato that was ot avalable o the test data I prce to book rato, book value per share s ot kow ahead 3

24 Tme-perod Based o a tme perod Use a adequate tme Cover a Bas that may make the results perods to justfy value loger perod tme-perod specfc stocks outperformg growth stocks Chapter 7. Hypothess Testg (Basc cocept of hypothess testg 假設檢定 (Hypothess Testg, 是要我們取一組樣本平均數, 然後看這個樣本平均數所對應的 Z 值或 t 值 ( 或其他分配值, 是不是超過一個顯著水準 (Sgfcace Level, 以致於讓我們足以拒絕母體平均數是某個值的統計推論 假設顯訂有兩個對立的假設, 前者稱之為基本假設或虛無假設 (ull Hypothess, 後者稱之為對立假設 (Alteratve Hypothess a. ull Hypothess (H 0 V.S. Alteratve Hypothess (H or H a ull Hypothess (H 0 : a hypothess that you actually wat to reject. Alteratve Hypothess (H : a hypothess that you are really tryg to access. It ca be cocluded f there s suffcet evdece to reject the ull. b. Hypothess Testg Formula H 0 : θ θ 0 versus H a: θ θ 0 ( 雙尾檢定 two-taled testg H 0 : θ θ 0 versus H a: θ>θ 0 ( 單尾檢定 (upper oe-taled testg 3 H 0 : θ θ 0 versus H a: θ<θ 0 ( 單尾檢定 (lower oe-taled testg Where θ 0 s a hypotheszed value of the populato parameter ad θ s the true value of the populato parameter (e: μ or c. Sgfcace Level Lke the cofdece terval secto, hypothess testg, you also have to assg the sgfcace Level. d. Type I error ad Type II error I. Type I error >α Sgfcace level P[ReH 0 H 0 ] II. Type II error > β cosumer error P[AcH 0 H ] III. Power of test β P[ReH 0 H ] 4

25 Fact H 0 s true H s true Decso Fal to reject H 0 Correct (-α Reject H 0 Type I error α P(Type I error Type II error β P(Type II error Correct Power of the test β 當 Sgfcace levelα 從 α β Power-of-test( β 當 Sgfcace levelα 從 α β Power-of-test( β α β e. Test Statstc Defto: A test statstc s a quatty, calculated o the bass of a sample, whose value s the bass for decdg whether to reject or ot to reject the ull hypothess. I other words, the value of test-statstc s the used to compare the crtcal value of gve sgfcace level.( 要注意單 雙尾的不同 Test statstc Sample statstc - Value of the populato parameter uder Ho Stadard error of the sample statstc 附註 : 臨界 Z 值表 ( 要熟記 Crtcal Z-values Level of Two-taled test Oe-taled test sgfcace 0.0 0% ± or % ± or % ± or α0.05 Two-taled α0.05 Oe-taled α.5% α.5% α5%

26 假設檢定和信賴區間的判斷是一體兩面 : 殊途同歸 如果利用假設檢定計算出來的 z 值 (or t 值 etc 在臨界值內, 則從樣本平均數所訂出來的信賴區間範圍必有包括其所假設的母體平均數 同樣地, 如果 x 所形成之信賴區間包含 μ 0, 則假設檢定之結果就不會拒絕 H 0 的虛無假設 (μ μ 0 f. P-value( 要熟記 I. Defto: P-value explas the portos of outcome probabltes that beyod the calculated test statstc. Whe applyg the p-value to make decsos, you do t have to care about whether the test s oe-taled or two-taled sce P-value tself already adjust t for you. II. Decso Rule: III. If P-value < α Reject H 0 If P-value > α Fal to reject H 0 Why P-value? The smaller (larger the p-value, the more (less lkely you are to reject the ull hypothess. The same decso wll be reached regardless of whether the p-value approach s used or the computed value of the test statstc s compared to the crtcal value. However, the p-value provdes more formato o the stregth of the evdece tha does the rejecto pots approach. (3 Test Statstcs: a. Test μ kow Z b. Test μ - μ ukow t( υ x μ 單一母體平均數 ( 變異數已知 x μ s where df ( 自由度 - (>30 時勉強可用 Z 單一母體平均數 ( 變異數未知 I. Two ormal populatos are Idepedet. We ofte wat to kow whether a mea value dffers betwee two groups.. H 0 : μ -μ 0 versus H a : μ -μ 0 (the alteratve s thatμ μ. H 0 : μ -μ 0 versus Ha: μ -μ >0 (the alteratve s thatμ>μ 3. H 0 : μ -μ 0 versus Ha: μ -μ <0 (the alteratve s thatμ<μ 6

27 (, kow Z ( x x μ μ + ( (, ukow (a t ( υ ( ( S S x x p p + μ μ, df +, ˆ ( ˆ ( + + s s S p (b t ( υ ( ( S S x x + μ μ, df ˆ ˆ ˆ ˆ ( s s s s + + II. Two ormal populatos are Depedet. Defe:, ~ ( 共抽 對 y x d We wat to kow how much the mea of par dffereces, whch s commoly set to zero.. H 0 : μ d μ do vs u d u do. H 0 : μ d μ do vs u d > u do 3. H 0 : μ d μ do vs u d < u do ( μ, μ pared comparsos t ( υ ( d d S d s d d μ d, where df - ad c. Test 單一母體變異數 I tests cocerg the varace of a sgle ormally dstrbuted populato. H 0 : 0 versus H : a 0. H 0 : 0 versus H : a > 0 3. H 0 : 0 versus H : a < 0 we make use of a ch-square test statstc, doated by X. Test statstc for tests cocerg the value of a ormal populato varace 7

28 s χ( υ (, where df - d. Test, 兩母體變異數 For tests cocerg dffereces betwee the varaces of two ormally dstrbuted populatos based o two radom, depedet samples, the approprate test statstc s based o a F-test (the rato of the sample varaces. H 0 : versus H a :. H 0 : versus H a : > 3. H 0 : versus H a : < s ( υ, υ, where, s F df df ( 記得, 分子用大的那個 s ( 做假設 H 0 ; H ( 做抽樣檢定 (3 算出 test statstc (4 決定 sgfcat level (5 查表得 ( 上下 or 上 or 下 臨界值, 或用電腦程式得 p-value (6 判斷是否此檢定有統計學上的意義 ( 接受 拒絕 (7 判斷是否此檢定有經濟學上的意義 Chapter 8. Correlato ad Regresso ( Covarace ad Correlatos: a. Sample VS Populato Varace Covarace Correlato Coeffcet Populato parameters ( X μ Cov X, Y] E[( X μ ( Y μ ] [ X Y ( X μ X ( Y μy ρ xy Cov[ X, Y] x y Sample estmators s ( X X Cov[ X, Y ] E[( X X ( Y Y ] r xy ( X x X ( Y Cov[ X, Y] S S y Y b. Correlato Testg I. Hypothess testg assumpto: 我們只關心一件事, 就是這兩個變數到底相不相關?? H 0 : ρ 0 : H : ρ 0 (Two-taled 8

29 II. Test Statstc r t( υ where df - r c. Problem regardg the Correlato Aalyss I. Correlato caot represet the olear relatoshp. Eve t s very strog. II. Correlato may ot be a relable measure whe outler are preseted oe or both of seres. III. Correlato does ot mply ay ecoomc causato. Spurous correlato ca make two seres appear closely assocated whe o causal relato exsts. ( The Lear Regresso Equatos: a. Assumptos of the Lear Regresso: I. The depedet varable (X s ad depedet varable (Y s really has lear relatoshp. II. ρ 0 x, ε III. all error terms ε s d (0, d depedet ad detcally cost dstrbuted IV. Addtoal assumpto for Mult-lear regresso(k: all K depet varables (X k, s are depedet to each other Regresso model ( 真實 : Y b0 + b X + ε Y ad X are the th observato of the decedet ad depedet varables, respectvely. b 0 tercept term (represets the value of Y f X s zero. b slope coeffcet (measures the chage Y for a oe ut chage X. ε resdual error of the th observato; ts expected value s 0, therefore: Regresso equato ( 估計 : Gve X, estmate Y ˆ ˆ + ˆ Y b0 b X 9

30 where ˆb Cov ( X, Y V [ X ] bˆ 0 Y bˆ X ( X X ( Y ( X X Y ( X ( X X X ( Y Y The least squares method uses the sample data to provde the values of b 0 ad b that mmze the sum of the squares of the error terms, amely, the devatos betwee the observed values of the depedet varable y ad the estmated values of the depedet varable ŷ m (ε Y ˆ Y ( 名詞群 0: SSE: Uexplaed Sum of Square Error ˆ (ε ( Y Y SSR: Explaed Sum of Square Regresso ( ˆ Y Y ( Y SST: Uexp+Exp Sum of Square Total ( SSE + SSR Y R : Coeffcet of determato SST SSE SSR SST SST R : Defto: the percetage of total varato the depedet varable explaed by the depedet varable. 名詞群 0: SSE: Sum of Square Error (ε Y Yˆ ( 同上 MSE: Mea of Square Error SSE SEE: Stadard Error of Estmate MSE ( 稱 S e SSE 稱 S e The stadard error of estmate (SEE measures how well the regresso model fts the data. If SEE s small, the model fts well. 觀念 : 在單一變數迴歸中, 判定係數 (R-Squared 直接等於相關係數的平方值 30

31 (3Hypothess Tests ad Cofdece Itervals for Lear Regresso: A. Cofdece Iterval for the slope ( ˆb C.I. for b ˆb + t c S where ˆb υ, S ˆb MSE ( X X B. Hypothess Test for the slope ( ˆb ( Hypothess testg assumpto: H 0 : b cost b H: b cost b ( 常見是假設 b 0 ( Test Statstc t bˆ b ( υ where S b ˆ υ S ˆb MSE ( X X C. Cofdece Iterval for forecasted value ( depedet varable: Yˆ C.I. for Y Yˆ + t c S f where υ ( X X S f Se* + + ( The predcto terval for a regresso equato for a partcular predcted value of the depedet varable s Yˆ + t c S f, where S f s the square root the estmated varace of S x ( X X the predcto error(s f Se* + +, whle t c s the crtcal level for the ( S x t-statstc at the chose sgfcace level at - degree of freedom. Ths computato specfes a (-α percet cofdece terval. D. Issues related to the Lear Regresso: I. The relatoshp of X ad Y chages over tme. II. Samples go agast the assumpto of lear regresso (3 AOVA table: a. What s AOVA table: The AOVA table s the table to streamle the process developg the F statstc ad use the F dstrbuto to measure how well the system ( our case, the regresso equato explas the varato the depedet varable. I lear regresso aalyss, we use F-test to determe f all the slope coeffcets are equal to 0. 3

32 Although here t looks lke the F-dstrbuto s two-taled test, t s deed the OE-TAILED test. For oly OE depedet varable, ths s a test of: Ho: b 0 V.S. H : b 0. For TWO or MORE depedet varables, ths s a test of: Ho: ALL b b b 3 b - b 0 V.S. H : Ay of b,b,b 3,,b -,b 0. A typcal AOVA table looks lke ths: (Yb 0 + b X + b X + + b k X k Source of Varace Degrees of Freedom (df Sum of Squares (SS Mea Sum of Squares (MSS Regresso k SSR MSRSSR/K Resdual -k- SSE MSESSE/ -k- Total - SST F MSR F MSE υ k ; υ k For oly oe depedet varable the AOVA table looks lke ths: (Yb 0 + b X Source of Varace Degrees of Freedom (df Sum of Squares (SS Mea Sum of Squares (MSS Regresso SSR MSRSSR/ Resdual - SSE MSESSE/ - Total - SST F MSR F MSE υ ; υ 3