BBA FN415 2016 Formula Sheet.pdf

Formula Sheet for BBA FN415 2016

Sutee Mokkhavesa PhD

1. Probability density function: f(x) = Pr[X=x]

2. (Cumulative) Distribution function: F(x) = _−∞x f(x)dx

3. Mean: E(X) =µ= xf(x)dx

4. Variance: V (X) =σ2 = (x−µ)2f(x)dx 5. Skewness: γ= [ (x−µ)3f(x)dx]/σ3

Positively skewed→mean> median>mode Negatively skewed→mean<median <mode

6. Kurtosis: δ= [ (x−µ)4f(x)dx]/σ4

7. Covariance: σ12= _{X2 X1}(x−µ1) (x−µ2)f1,2(x1, x2)dx1dx2

8. Correlation: ρ12=cov(x1, x2)/ var(x1)var(x2) =σ12/(σ1σ2)

9. Linear Transformation of Random Variables Y =a+bX

Expectation of Y is E[Y] =a+bE[X]

Variance ofY is V[Y] =b2_V[X]

10. Normal Distribution:

68% of observations fall within±1σ 90% of observations fall within±1.65σ 95% of observations fall within±1.96σ 99% of observations fall within±2.58σ

11. Portfolios of random variables: Y =wTX_{, E[Y] =µ}

p =wTµ, σ2p =wTMw w_{={w1, w2, ..., wN,}}T_{, µ={µ}

1, µ2, ..., µN,}T, M =



  

σ11 σ12 ... σ1N σ22

ց

σN1 ... ... σNN



  

12. Treasury Bill: DR= (F−_FP).360_t 13. Macaulay Duration:

D= T_t=1t×wt= _tT₌₁t× TCt/(1+y)t t=1Ct/(1+y)t

14. Wiener Process: ∆z∼N(0,∆t)

15. The Generalised Wiener Process: ∆x=a∆t+b∆z

17. Geometric Brownian Motion: ∆S=µS∆t+σS∆z

18. Forward price, no income on the asset: Fte−rt_=St

19. Forward price, income on the asset:

discrete dividend, Fte−rt_{=St−P V (D)} continuous dividned, Fte−rt_=Ste−r∗

t

20. Valuation of outstanding forward contract: Value of contract: Vt=Ste−r∗

t_−Ke−rt_=Fte−rt_−Ke−rt_{= (F−K)e}−rt

21. Put-call parity: c−p=Se−r∗

t_−Ke−rτ _{= (F−K)e}−rτ

22. Bounds on call value (no dividends): ct Ct St & ct St−Ke−rτ

23. Bounds on put value (no dividends):pt Pt K & pt Ke−rτ _−St

24. Geometric Brownian motion: ln (ST) = ln (S0) + µ−σ

2

2 τ +σ

√

τ ǫ

25. Risk-neutral discounting formula: ft=ERN[e−rτ_F(ST)]

26. Black-Scholes call option pricing: c=SN(d1)−Ke−rτN(d2)

d1 = ln S

Ke−rτ σ√τ +

σ√τ

2 , d2=d1−σ

√

τ

27. Black-Scholes put option pricing: p=S[N(d1)−1]−Ke−rτ[N(d2)−1]

28. VaR: c= ₋∞_{V aR}f(x)dx

29. CVaR: E[X |X < q] =

q

−∞xf(x)dx

q

−∞f(x)dx

30. Square Root of Time Adjustment: V aR(T days) =V aR(1day)×√T

31. Cross-exchange rate: S3(EUR/GBP) = S_S1($/GBP)

2($/EU R)

σ23=σ21+σ22−2ρ12σ1σ2

32. Volatility of the rate of return in the bond price: σ dP_P =|D∗_{| ×σ(dy)}

33. CAPM: Ri =αi+β_iRM +εi

34. Profit on position with unit hedge: Q[(S2−S1)−(F2−F1)] =Q[b2−b1]

35. Optimal Hedge Ratio: N∗ _=−β

sfQfQ.s.f

36. Optimal Hedge Ratio (Unitless): βsf = σsf

σ2

37. Volatility of the Hedge Position: σ∗

V =σs (1−R2) 38. Duration Hedge: N∗ ₌₋DS∗S

D∗

FF

39. Beta Hedge: N∗ _=−βS F

40. Linear VAR, fixed income: V AR(dP) =|−D∗_{P| ×V AR(dy)}

41. Full Valuation VAR, fixed income: dPworst _{=P y}

0+dyworst −P[y0]

42. Quadratic VAR, fixed income:

V AR(dP) =|−D∗_{P| ×V AR(dy)−} 1

2CP ×V AR(dy) 2

43. Delta VAR: V AR(df) =|∆|V AR(dS)

44. Delta-normal VAR: V AR=ασ(Rp,t+1), σ2(Rp,t+1) =xTtΣt+1xt

45. Historical-simulation VAR: ∆fk

i ={∆fi,1,∆fi,2, ...,∆fi,t}

46. Monte Carlo simulation VAR: ∆fk ∼g(θ)

47. Absolute Risk: σ(∆P) =σ(∆P/P)×P =σ(RP)×P

48. RelativeRisk: σ(e)P = [σ(RP −RB)]×P = [σ(∆P/P −∆B/B)]×P =ω×P

49. Tracking error volatility (TEV): ω=σ(∆P/P −∆B/B)

ω= σ2

P +σ2B−2ρσPσB 50. Sharpe Ratio (SR): SR= µ(Rp_σ(RP)−RF₎

51. Information Ratio (IR): IR= µ(Rp)−_ωµ(RB)

52. Alpha, from: RP,t−RF,t=α+βP[Rm,t−RF,t] +εP,t

53. Credit Loss: CL= N_i=1bi×CEi×(1−fi)

54. Joint probability with independence: p(A andB) =p(A)×p(B)

55. Joint probability: p(A and B) =p(A)p(B)+Corr(A, B) p(A) [1−p(A)] p(B) [1−p(B)], using E[bA×bB] =Cov[bA, bB] +E[bA]E[bB]

56. Binomial expansion:

1 =pN ₊ N

1 pN−1(1−p) 1₊ N

2 pN−2(1−p)

2_+...+ N

N−1 p1(1−p)

N−1_{+ (1}

−p)N

where N_k = _k!(NN₋!_k)!

57. Default rateX mean and variance: E X =p ; V X = p(1_N−p)

58. Marginal default rate for firm initially ratedRduring yearT =t+N :

dN(R) = m[t+n]

59. Survival rate forN years: SN(R) = N

i=1(1−di(R))

60. Marginal default rate from start to year T:kN(R) =SN₋1(R)dN(R)

61. Cumulative default rate: CN(R) =k1(R) +k2(R) +....+kN(R) = 1−SN(R)

62. Average default rate,d: CN = 1− N_i=1(1−di) = 1−(1−d)N

63. Implied default probability, 1 period: (1 +y) = (1 +y∗_{) [1−π(1−f)]}

64. Approximation of implied default probability: y∗ _{≈y+π(1−f)}

65. Implied default probability, T period: (1 +y)T = (1 +y∗₎T ₍₁

−π)T +f 1−(1−π)T

66. Approximation of physical default probability: y∗_≈y+π′_{(1−f) +rp}

67. Merton model for stock price: ST = max (VT −K,0)

68. Merton model for bond price: BT =VT −ST = min (VT, K)

69. Stock valuation: S =Call=V N(d1)−Ke−rτN(d2)

70. Firm value and stock volatility: σV = ∆σS(S/V)

71. Bond valution: B/Ke−rτ _{= [N(d}

2) + (V /Ke−rτ)N(−d1)]

72. Risk-neutral PD: 1−N(d2) =N(−d2)

73. ECL at maturity: ECLT =N(−d2) [K−V erτN(−d1)/N(−d2)] =p×[Exposure×LGD]

74. Credit default swap, or put option: P ut=Ke−rτ _{− {Ke}−rτ_N(d

2) +V [1−N(d1)]}=−V N[(−d1)] +Ke−rτ[N(−d2)]

75. Credit loss: N_i=1bi×CEi×LGDi

76. Expected credit loss: ECL= Pr [default]×E[Credit exposure]×E[LGD]

77. A credit portfolio of sizeSmade up ofN independent credits with equal probability of default P.

The expectation of the credit loss:E[X]_NS =P N._NS =P S

The variance of the credit loss: V[X] _NS 2 =P(1−P)N _NS 2 78. Present value of expected credit losses (PVECL):

P V ECL= _tE[CLt]×P Vt= _t[kt×ECEt×(1−f)]×P Vt

79. Approximation to PVECL: PVECLF =cT ×ECE×(1−f)×P VT

80. Credit VAR: CVAR = WCL - ECL

81. Net Beta: (β_LVL−β_SVS) =β_EVE

83. Gross leverage: VL_VE+VS=´Long P ositions +Absolute value of short positions_Equity

84. Net leverage: VL−VS VE =´