Formula Summary for the upcoming FRM Exam

S

ShaktiRathore

Well-Known Member
Subscriber
Hi starting a new thread for the important formulas that can come in handy for your exams.
Let me start with the following(these are relevant for part I) :
1. Holding Period Return=(V1-V0+D1)/V0
2. Standardized Return= (mean return-Target return)/standard deviation of returns
3. Expected Portfolio Return= E(Rp)= w1*E(R1)+w2*E(R2)+.....+wn*E(Rn) for n assets portfolio
4. Standard deviation of portfolio= w1^2*sd1^2+w2^2*sd2^2+..........wn^2*sdn^2, sd=standard deviation
5. CAPM: E(Ri)= Rf+ Betai*[E(Rm)-Rf] where Betai=Cov(Ri,Rm)/var(Rm), Rm is market return and Ri is security return
6. Capital Market Line(CML): E(Rp)= Rf+ [[E(Rm)-Rf] /SDm]SDp
7. CAPM with taxes: E(Ri)= Rf+ Betai*[E(Rm)-Rf]+taxfactor*[(divyield(i)-Rf)-Betai(divyield(m)-Rf)] taxfactor that measures market tax rates, divyield(i) is dividend yield of stock
8. Multi Beta CAPM: E(Ri)= Rf+ Betai,m*[E(Rm)-Rf]+Betai,f1*[E(f1)-Rf]+Betai,f2*[E(f2)-Rf]+........ where Betai,m is sensitivity to market, Betai,f1 is sensitivity to factor 1,....
9. Treynor ratio= [E(Rp)-Rf]/Beta(p) p is subscript for portfolio
10. Sharpe ratio= [E(Rp)-Rf]/StdDev(p)
11.Jensen's alpha= E(Rp)-{Rf+ Betap*[E(Rm)-Rf]}
12. Sortino Ratio=[E(Rp)-Rmin]/sqrt(MSDmin) Rmin is benchmark and MSDmin is standard deviation of portfolio returns below the Rmin
13.APT Model: Rn=Rf+ Xn,1*b1+ Xn,2*b2+ Xn,3*b3+ Xn,4*b4................+un
where Rn is the return for stock n; Xn,k is kth factor exposure for n; bk is return for kth factor, un is the systematic risk
14. Bayes formula: P(A/B)=P(A&B)/P(B) A and b are events
15.Correlation(Ri,Rj)= Covariance(Ri,Rj)/[stdDev(Ri)*stdDev(Rj)]
16. z parameter for ND(mean,stdDev)
z= (observation-mean)/stdDev
17. test of differences between means:
t=[(mean(s1)-mean(s2))-(mean(p1)-mean(p2))]/sqrt[sp^2*(1/n1+1/n2)]
where sp^2= [(n1-1)*s1^2+(n2-1)*s2^2]/(n1+n2-1)
s1 is standard deviation of sample 1, s2 is standard deviation of sample 2
s1,s2 are samples 1 and 2 while p1 and p2 are populations 1, and 2
18. chi-Square test stat= (n-1)*s^2/stdDev^2
19.F test stat. = s1^2/s2^2
20.t stat.= (sample mean- population mean)/(s/sqrt(n))


more later...

thanks
 
S

ShaktiRathore

Well-Known Member
Subscriber
21. Poisson distribution: P(X=x)=lambda^x*exp(-lambda)/x!
22.adjusted R^2= 1-(n-1/n-k-1)*(1-R^2) where R^2 is the coefficient of determination, k:no of independent variables
23. Geometric Brownian motion= dSt= mean(t)*St*dt + stdDevt*St*dz where mean(t)*St*dt =constant drift term and stdDevt*St*dz is volatile component
24. Binomial probability function= nCx p^x*(1-p)^n-x where n is no of trials , x is the no of succeses and p is probability of success
25. dollar value of basis point= DV01= Price @YTM0- Price @ YTM1
26. Hedge ratio= HR= DV01(per $ of initial position)/DV01(per $ of hedging instrument)
27.Duration of Bond= BV(y-)-BV(y+)/2*BV0*(total change in y)
28. Convexity of Bond=BV(y-)+BV(y+)-2BV0/2*BV0*(total change in y)^2
29. percent price change in bond= duration effect+ convexity effect= [-duration*(total change in y)] +[.5*convexity*(total change in y)^2]
30. BSM option pricing model: c=S0N(d1)-Xe^(-rT)*N(d2) where d1=[ln(S0/X)+(rf+.5*stdDev^2)T]/stdDev*sqrt(T) and d2=d1-stdDev*sqrt(T)
31. continously compounded return= ln(St/St-1)
32. delta= dc/ds if the rate of change of call option price w.r.t the change in underlying asset price that is stock price
32. gamma= d^2C/dS^2= rate of change of delta w.r.t the underlying asset value
33. theta= dC/dt is the rate of change of option price w.r.t. the time
34. vega= dC/d(volatility)= rate of change of option price w.r.t the volatility of the underlying asset
35. rho= dC/dr is the rate of change of option price w.r.t the interest rate
36. Taylor series: f(x)=f(x0)+f'(x0)*(x-x0)+.5*f''(x0)(x-x0)^2
37. Expected Loss= AE*LGD*PD where AE-Adjusted exposure, LGD: loss given default and PD: probability of default
38.Unexpected loss= AE*sqrt[EDF*(volatility LGD)^2+LGD^2*(volatility EDF)^2] where EDF is expected default frequency; AE=OS(outstanding)+alpha(COMu)
39. Hedge ratio= corr(S,F)*(stdDev of S/stdDev of F)
40.beta(S,F)= co-variance(S,F)/variance (S)
41.No of contracts= beta of portfolio*[portfolio value/value of future contract] after adjustment to beta* No of contracts=(beta*-beta of portfolio)*[portfolio value/value of future contract]
42. Forward price, F0= S0* e^rT
43. Accrued interest= Coupon*[No of days from last coupon to settlement day/No of days in coupon period]
44.T Bill Discount rate= (360/n)*(100-Y)
45. Cheaper to deliver bond= Qouted Bond price- QFP*CF where QFP is quoted futures price and CF is the conversion factor
46. Duration based hedge ratio= N= -(P*Dp/F*Df) where Dp is duration of bond and Df is the duration of futures used to hedge the bond price movement
47.Forward rate(T1,T2)= (R2T2-R1T1)/T2-T1
48. Hedge effectiveness= 1-[var(S-F)/var(S)]
****************************************************
that is it for the part I i may have left out some but above list is the comprehensive one that i can garner.
best wishes

thanks
 
S

ShaktiRathore

Well-Known Member
Subscriber
Ok that was for part I but now lets have a look on part II important formulas from the exam point of view,

1. Arithmetic Return= (Pt+Dt-Pt-1)/Pt-1 where Pt and Pt-1 are prices at time t and t-1 resp. and Dt is the dividends flow between time t-1 and t
2. Geometric Return= ln([Pt+Dt]/Pt-1)
3. delta normal VaR(alpha%)= [-mean(returns)+stdDev(returns)*z(alpha%)]*Pt-1 where Pt-1 is initial portfolio position ...remember this is absolute Var
4. Lognormal VaR(alpha%)= [1-exp[mean(returns)-stdDev(returns)*z(alpha%)]]*Pt-1
5. Standard error of Quantile se(q)= sqrt[p(1-p)/n]/f(q)
6. Generalized Extreme Value Distribution(GEV) :
F(X|E,mean,stdDev)= exp[-[1+E*((x-mean)/stdDev)]^(-1/E)] ;E=shape parameter of tail !=0
F(X|E,mean,stdDev)= exp[-exp((x-mean)/stdDev)] ;E=shape parameter of tail =0
7. Generalized Pareto Distribution(GPD):
1-[1+(E*x/beta)]^(-1/E) ;E=shape parameter of tail !=0
1-[exp(-x/beta)] ;E=shape parameter of tail =0
8. Var Using PoT VaR= u+(beta/E)[[(n/Nu)*(1-CL)]^-E - 1] where u is the upper limit for losses. CL is confidence level, Nu no of losses above u and total no of observations is n
9. Expected Shortfall Using POT parameters
ES= VaR/(1-E) + (Beta-E*u)/(1-E)
10. Yield based DV01= (1/10000)*[(Sum of PV(weighted time) of Bond's Cash flows)/(1+periodic yield)]
11. Modified Duration= 1/P*(1/1+periodic yield)*(Sum of PV(weighted time) of Bond's Cash flows)
12. Macualay duration= (1+periodic yield)* Modified Duration
13.Mortgage payment(monthly)= MB0*[r/1-(1+r)^-T] where r is monthly interest rate and MB0 is original loan balance ,T is loan maturity
14. Loan to Value ratio= Current Mortgage Amount/Current Apprised Value
15. Single monthly Mortality Rate, SMM= 1-(1-CPR)^1/12 where CPR is current prepayment rate
16. Bond Equivalent yield=2*[(1+monthly CF yield)^6-1]
17.option Cost= Zero Volatility Spread- Option Adjusted Spread
18. Put call parity: p+S=c+X*e^(-Rf*T)
19.Information ratio= [Rp-Rb]/std Dev(Rp-b) where Rp is portfolio return, Rb is benchmark return, std Dev(Rp-b) is active risk and Rp-Rb is active return
20. Risk Aversion= Information Ratio/2*Active risk
thats the first half wait for more

thanks
 
S

ShaktiRathore

Well-Known Member
Subscriber
Here are the rest of the lot for the part II,

21.Marginal Contribution to value added= Alpha of asset- 2*Risk Aversion* Active Risk*Marginal contribution to active risk
22.Average Log Return= ln(1+r1)+ln(1+r2)+ln(1+r3)+................ln(1+rT)/T
23. Alpha and IR test: t(alpha%)= alpha-0/SE(alpha) and t(IR)= IR-0/S.E.(IR)
24. Sharpe Ratio t-Test; t=[MERp/stdDev(p)]-[MERb/stdDev(b)]/sqrt(2/N) where MER is mean excess return
25. Active portfolio return= Rpa= beta(pa)*Rb+[Xpa1*Rf1+Xpa2*Rf2....Xpan*Rfn]+Spar where beta(pa) is sensitivity to benchmark,X are factors sensitivities to portfolio and s is unsystematic risk
26. Fama French three factor model: Ri,t= Rf,t + alphai + Betai,m*[E(Rm)-Rf,t]+ Betai,smb*[E(SMBt)]+ Betai,hml*[E(HMLt)]
27. Total Active Systematic Return=Expected Active beta return+ Active beta surprise+ Active benchmark timing return
28. Liquidity Duration= Q/.1*V where Q is no of shares of security, V is volume of security
29. Diversified VaR= z*stdDev(p)*P ;P is portfolio value
30. Individual VaR=z*stdDev(i)*wi*P ;wi is weight of individual security i
31. VaR of Two Asset Portfolio= z*P*sqrt[w1^2*stdDev1^2+w2^2*stdDev2^2+2w1w2*stdDev1*stdDev2*correlation(1,2)] for securities 1 and 2
32. Covariance(1,2)=stdDev1*stdDev2*correlation(1,2)
33. Undiversified VaRp=VaR1+VaR2
34. std Deviation of equally weighted portfolio of n securities with equal stdDeviation stdDev and correlation rho
stdDeviation of portfolio= stdDev *sqrt[1/n+(1-1/n)*rho]
35.Marginal VaRi= (VaR/P)*Betai
36. Component VaR= VaR*Betai*wi
37.Return on surplus=change in surplus/Assets=(change in Assets- change in Liabilities)/Assets=Rasset-Rliabs.*(Liabs/Assets) where Surplus = Assets- Liabilities
38. Probability of Default, PD= CS/1-RR where CS is spread of corporate bond wrt Rf and RR is expected recovery rate
39. Risk neutral probability of default= 1-[(1+Rf)/(1+y)] where Rf is risk free rate and y is yield on Bond
40. Cumulative probability of default(2yrs)= 1-{(1-PD1)*(1-P(Default in yr2|no default in yr 1))}
41. Merton Model, Payment to Debtholder= Dm- max(Dm-Vm,0) and Payment to Stockholder= max(Vm-Dm,0)
42. Distance to default= [expected Asset return-default threshold]/stdDev(exp asset returns)
43. Distance to Default(lognormal Distribution)= [log(V/defaultThreshold)+[E(ROA)-.5*stdDev^2]*Maturity]/stdDev*sqrt(Maturity) where V is value of firm assests, stdDev is stdDeviation of firm assets, E(ROA) is expected return on assets
44. Portfolio Unexpected Loss of two asset portfolio ULp= sqrt[UL1^2+UL2^2+2*UL1*UL2*correlation(1,2)]
45. Risk Contribution=RC1= UL1*[UL1+UL2*corr(1,2)]/ULp ;RC2= UL2*[UL2+UL1*corr(1,2)]/ULp so that RC1+RC2=ULp
46.Mean Loss Rate=PD(1-RR)= PD*LGD
47. Credit spread= -[(1/T-t)*ln(D/F)]-Rf where T-t is remaining maturity, D is current value of debt, F face value of debt and Rf is risk free rate
48. Vasicek Model: change in interest rate r= speed of reversion of r*(k-r(t))*small change in time t+ stdDev of r* random error term
 
S

ShaktiRathore

Well-Known Member
Subscriber
here are rest to

49. Merton Models: PD=N{[ln(F/V)-mean*(T-t)+.5*stdDev^2*(T-t)]/stdDev*sqrt(T-t)}
50. LGD= F*PD-V*exp(mean*(T-t))*N(d)
51. Vulnerable option= (1-PD)*c +PD*RR*c
52. CDS Spread; PV of payoff =s* PV of payments => s=PV of payoff/PV of payments
53. RAROC= [Revenues-Expected Loss-Expenses+Return on Economic capital+/-transfer price]/Economic Capital
54. Economic Capital= Operation VaR-EL=Unexpected Loss
55. Adjusted RAROC= ARAROC= RAROC-Rf/Beta(eqty)
56. spread= (Ask price-Bid Price)/.5*(Ask price+Bid Price)
57. Liquidity Adjusted VaR=LVaR= V*z*stdDev+ .5*V*spread where V is asset value
58. LVaR/VaR= 1+[spread/2*(1-exp(-stdDev*z)]
59. Elasticity=E= (change in P/P)/(change in N/N)
60. LVaR= VaR*(1-change in P/P)=VaR*(1-E*change in N/N)
61. LVaR/VaR|comb.=LVaR/VaR|exog. *LVaR/VaR|endo.
62. Capital Ratio= Total Capital/Total Risk weighted Assets
63. Capital Requirement(K)= [Conditional EL-EL]*maturity Adjustment
64. Liquidity Coverage ratio= Stock of highly Liquid Assets/Total net cash outflow over next 30 days
65. net stable funding ratio= Available amount of stable funding/Required amount of stable funding
66. Stressed VaR= max(SVaRt-1, m*SVaRavg.) where m is a factor
67. EL= PD*LGD

thanks thats sit for part II....
best of luck for your exams..
 
K

kolkhi

New Member
Subscriber
ShaktiRathore said:
Hi starting a new thread for the important formulas that can come in handy for your exams.
Let me start with the following(these are relevant for part I) :
1. Holding Period Return=(V1-V0+D1)/V0
2. Standardized Return= (mean return-Target return)/standard deviation of returns
3. Expected Portfolio Return= E(Rp)= w1*E(R1)+w2*E(R2)+.....+wn*E(Rn) for n assets portfolio
4. Standard deviation of portfolio= w1^2*sd1^2+w2^2*sd2^2+..........wn^2*sdn^2, sd=standard deviation
5. CAPM: E(Ri)= Rf+ Betai*[E(Rm)-Rf] where Betai=Cov(Ri,Rm)/var(Rm), Rm is market return and Ri is security return
6. Capital Market Line(CML): E(Rp)= Rf+ [[E(Rm)-Rf] /SDm]SDp
7. CAPM with taxes: E(Ri)= Rf+ Betai*[E(Rm)-Rf]+taxfactor*[(divyield(i)-Rf)-Betai(divyield(m)-Rf)] taxfactor that measures market tax rates, divyield(i) is dividend yield of stock
8. Multi Beta CAPM: E(Ri)= Rf+ Betai,m*[E(Rm)-Rf]+Betai,f1*[E(f1)-Rf]+Betai,f2*[E(f2)-Rf]+........ where Betai,m is sensitivity to market, Betai,f1 is sensitivity to factor 1,....
9. Treynor ratio= [E(Rp)-Rf]/Beta(p) p is subscript for portfolio
10. Sharpe ratio= [E(Rp)-Rf]/StdDev(p)
11.Jensen's alpha= E(Rp)-{Rf+ Betap*[E(Rm)-Rf]}
12. Sortino Ratio=[E(Rp)-Rmin]/sqrt(MSDmin) Rmin is benchmark and MSDmin is standard deviation of portfolio returns below the Rmin
13.APT Model: Rn=Rf+ Xn,1*b1+ Xn,2*b2+ Xn,3*b3+ Xn,4*b4................+un
where Rn is the return for stock n; Xn,k is kth factor exposure for n; bk is return for kth factor, un is the systematic risk
14. Bayes formula: P(A/B)=P(A&B)/P(B) A and b are events
15.Correlation(Ri,Rj)= Covariance(Ri,Rj)/[stdDev(Ri)*stdDev(Rj)]
16. z parameter for ND(mean,stdDev)
z= (observation-mean)/stdDev
17. test of differences between means:
t=[(mean(s1)-mean(s2))-(mean(p1)-mean(p2))]/sqrt[sp^2*(1/n1+1/n2)]
where sp^2= [(n1-1)*s1^2+(n2-1)*s2^2]/(n1+n2-1)
s1 is standard deviation of sample 1, s2 is standard deviation of sample 2
s1,s2 are samples 1 and 2 while p1 and p2 are populations 1, and 2
18. chi-Square test stat= (n-1)*s^2/stdDev^2
19.F test stat. = s1^2/s2^2
20.t stat.= (sample mean- population mean)/(s/sqrt(n))


more later...

thanks
Click to expand...

ShaktiRathore said:
Hi starting a new thread for the important formulas that can come in handy for your exams.
Let me start with the following(these are relevant for part I) :
1. Holding Period Return=(V1-V0+D1)/V0
2. Standardized Return= (mean return-Target return)/standard deviation of returns
3. Expected Portfolio Return= E(Rp)= w1*E(R1)+w2*E(R2)+.....+wn*E(Rn) for n assets portfolio
4. Standard deviation of portfolio= w1^2*sd1^2+w2^2*sd2^2+..........wn^2*sdn^2, sd=standard deviation
5. CAPM: E(Ri)= Rf+ Betai*[E(Rm)-Rf] where Betai=Cov(Ri,Rm)/var(Rm), Rm is market return and Ri is security return
6. Capital Market Line(CML): E(Rp)= Rf+ [[E(Rm)-Rf] /SDm]SDp
7. CAPM with taxes: E(Ri)= Rf+ Betai*[E(Rm)-Rf]+taxfactor*[(divyield(i)-Rf)-Betai(divyield(m)-Rf)] taxfactor that measures market tax rates, divyield(i) is dividend yield of stock
8. Multi Beta CAPM: E(Ri)= Rf+ Betai,m*[E(Rm)-Rf]+Betai,f1*[E(f1)-Rf]+Betai,f2*[E(f2)-Rf]+........ where Betai,m is sensitivity to market, Betai,f1 is sensitivity to factor 1,....
9. Treynor ratio= [E(Rp)-Rf]/Beta(p) p is subscript for portfolio
10. Sharpe ratio= [E(Rp)-Rf]/StdDev(p)
11.Jensen's alpha= E(Rp)-{Rf+ Betap*[E(Rm)-Rf]}
12. Sortino Ratio=[E(Rp)-Rmin]/sqrt(MSDmin) Rmin is benchmark and MSDmin is standard deviation of portfolio returns below the Rmin
13.APT Model: Rn=Rf+ Xn,1*b1+ Xn,2*b2+ Xn,3*b3+ Xn,4*b4................+un
where Rn is the return for stock n; Xn,k is kth factor exposure for n; bk is return for kth factor, un is the systematic risk
14. Bayes formula: P(A/B)=P(A&B)/P(B) A and b are events
15.Correlation(Ri,Rj)= Covariance(Ri,Rj)/[stdDev(Ri)*stdDev(Rj)]
16. z parameter for ND(mean,stdDev)
z= (observation-mean)/stdDev
17. test of differences between means:
t=[(mean(s1)-mean(s2))-(mean(p1)-mean(p2))]/sqrt[sp^2*(1/n1+1/n2)]
where sp^2= [(n1-1)*s1^2+(n2-1)*s2^2]/(n1+n2-1)
s1 is standard deviation of sample 1, s2 is standard deviation of sample 2
s1,s2 are samples 1 and 2 while p1 and p2 are populations 1, and 2
18. chi-Square test stat= (n-1)*s^2/stdDev^2
19.F test stat. = s1^2/s2^2
20.t stat.= (sample mean- population mean)/(s/sqrt(n))


more later...

thanks
Click to expand...
Formula 4 is for variance, but true only when the assets are mutually uncorrelated.
 
A

Argiope

New Member
Are they going to provide formula sheet during the exam or are we supposed to memorize all of them? I think it is the latter but i hope i am wrong cos even if you need to apply the formulas at work you need not dig them out from your memory. :D
 
S

ShaktiRathore

Well-Known Member
Subscriber
Hi
No they are not going to provide you with the Formula sheet in the exam,you are expected to know all the important formulas for the exam.
David has provided you with the Formula sheet so that you could memorize and know all the important formulas for the exam.
Thanks
 
K

kolkhi

New Member
Subscriber
No formula sheet will be allowed; moreover, my calculator cover had a famous formula from physics E=mc^2 written by sharpie on it, which is totally irrelevant to the exam, yet they took the cover away. The bottom line is - memorize!
 
B

bpdulog

Active Member
I have uploaded my formula sheet from Part 1, I hope someone can provide something similar for Part 2
 

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B

bpdulog

Active Member
ShaktiRathore said:
here are rest to

49. Merton Models: PD=N{[ln(F/V)-mean*(T-t)+.5*stdDev^2*(T-t)]/stdDev*sqrt(T-t)}
50. LGD= F*PD-V*exp(mean*(T-t))*N(d)
51. Vulnerable option= (1-PD)*c +PD*RR*c
52. CDS Spread; PV of payoff =s* PV of payments => s=PV of payoff/PV of payments
53. RAROC= [Revenues-Expected Loss-Expenses+Return on Economic capital+/-transfer price]/Economic Capital
54. Economic Capital= Operation VaR-EL=Unexpected Loss
55. Adjusted RAROC= ARAROC= RAROC-Rf/Beta(eqty)
56. spread= (Ask price-Bid Price)/.5*(Ask price+Bid Price)
57. Liquidity Adjusted VaR=LVaR= V*z*stdDev+ .5*V*spread where V is asset value
58. LVaR/VaR= 1+[spread/2*(1-exp(-stdDev*z)]
59. Elasticity=E= (change in P/P)/(change in N/N)
60. LVaR= VaR*(1-change in P/P)=VaR*(1-E*change in N/N)
61. LVaR/VaR|comb.=LVaR/VaR|exog. *LVaR/VaR|endo.
62. Capital Ratio= Total Capital/Total Risk weighted Assets
63. Capital Requirement(K)= [Conditional EL-EL]*maturity Adjustment
64. Liquidity Coverage ratio= Stock of highly Liquid Assets/Total net cash outflow over next 30 days
65. net stable funding ratio= Available amount of stable funding/Required amount of stable funding
66. Stressed VaR= max(SVaRt-1, m*SVaRavg.) where m is a factor
67. EL= PD*LGD

thanks thats sit for part II....
best of luck for your exams..
Click to expand...

I am just wondering if there is a condensed formula of this list? Understandbly, the goal is to remember all 67 of these but in reality that may not be possible. Is there some kind of list of 15-20 "No Excuses" formulas floating around? For instance I would include Merton model and LVaR.
 
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