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Important Concepts for the FRM exam

ShaktiRathore

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Hi starting a new thread for discussing various important concepts related to the exam,

C1. Betai= Cov(Ri,Rm)/stdDev(m)^2= correlation*stdDev(i)*stdDev(m)/stdDev(m)^2 =correlation*stdDev(i)/stdDev(m)
stdDev of portfolio= Beta(p)*stdDev(m) ; m stands for market
how to get the above formula see the below derivation-----
We know that CML are most efficient portfolios with maximum sharpe ratio and thus for CAL portfolio to be most efficient,
Sharpe ratio of CAL= Sharpe ratio of CML
E(Rp)-Rf/stdDev(p)= E(Rm)-Rf/stdDev(m)
E(Rp)-Rf=stdDev(p)*[E(Rm)-Rf/stdDev(m)]
E(Rp)=Rf+[stdDev(p)/stdDev(m)]*[E(Rm)-Rf]
compare it with CAPM E(Rp)=Rf+Beta(p)*[E(Rm)-Rf]
we get Beta(p)=[stdDev(p)/stdDev(m)] =>stdDev(p)=Beta(p)*stdDev(m)

C2. stdDev(p) of two Assets A and B
stdDev(p)= sqrt[wA^2*stdDev(A)^2+wB^2*stdDev(B)^2+2*Corr(A,B)*wA*wB*stdDev(A)*stdDev(B)]
for minimum variance portfolio,
wA=[stdDev(B)^2-corr(A,B)*stdDev(A)*stdDev(B)]/[stdDev(A)^2+stdDev(B)^2-2*Corr(A,B)*stdDev(A)*stdDev(B)]
for above formula see the listed below derivation;
wA+wB=1 =>wB=1-wA
stdDev(p)^2=wA^2*stdDev(A)^2+wB^2*stdDev(B)^2+2*Corr(A,B)*wA*wB*stdDev(A)*stdDev(B)

substitute wB for 1-wA =>
stdDev(p)^2= wA^2*stdDev(A)^2+(1-wA)^2*stdDev(B)^2+2*Corr(A,B)*wA*(1-wA)*stdDev(A)*stdDev(B)
for portfolio variance stdDev(p)^2 to be minimum,
first derivative of portfolio variance should be 0,
d(stdDev(p)^2)/dwA=0
=>d/dwA(wA^2*stdDev(A)^2+(1-wA)^2*stdDev(B)^2+2*Corr(A,B)*wA*(1-wA)*stdDev(A)*stdDev(B))=0
=>2*wA*stdDev(A)^2-2(1-wA)*stdDev(B)^2+2*Corr(A,B)*(1-2*wA)*stdDev(A)*stdDev(B))=0
=>wA*stdDev(A)^2+(wA)*stdDev(B)^2-stdDev(B)^2+Corr(A,B)*(1-2*wA)*stdDev(A)*stdDev(B))=0 cancelling out 2 from both sides
=>wA*[stdDev(A)^2+stdDev(B)^2]-stdDev(B)^2+Corr(A,B)*(1-2*wA)*stdDev(A)*stdDev(B))=0
=>wA*[stdDev(A)^2+stdDev(B)^2-2*Corr(A,B)*stdDev(A)*stdDev(B))]-stdDev(B)^2+Corr(A,B)*stdDev(A)*stdDev(B))=0

taking wA common
wA*[stdDev(A)^2+stdDev(B)^2-2*Corr(A,B)*stdDev(A)*stdDev(B))]=stdDev(B)^2-Corr(A,B)*stdDev(A)*stdDev(B))
wA=[stdDev(B)^2-Corr(A,B)*stdDev(A)*stdDev(B))]/[stdDev(A)^2+stdDev(B)^2-2*Corr(A,B)*stdDev(A)*stdDev(B))]
to be certain that the portfolio variance is minimal,
second derivative of variance should be greater than zero,
d^2(stdDev(p)^2)/dwA^2=d/dwA[2*wA*stdDev(A)^2-2(1-wA)*stdDev(B)^2+2*Corr(A,B)*(1-2*wA)*stdDev(A)*stdDev(B))]
=2*stdDev(A)^2+2*stdDev(B)^2+2*Corr(A,B)*(-2)*stdDev(A)*stdDev(B))
=2[stdDev(A)^2+stdDev(B)^2-2*Corr(A,B)*stdDev(A)*stdDev(B))
=2[stdDev(A)^2+stdDev(B)^2-2*stdDev(A)*stdDev(B)+2*stdDev(A)*stdDev(B)-2*Corr(A,B)*stdDev(A)*stdDev(B))]
=2[(stdDev(A)-stdDev(B))^2+2*stdDev(A)*stdDev(B)(1-Corr(A,B))]>=0
as Corr(A,B)<=1
thus proved that variance is minimum for this value of
wA=[stdDev(B)^2-Corr(A,B)*stdDev(A)*stdDev(B))]/[stdDev(A)^2+stdDev(B)^2-2*Corr(A,B)*stdDev(A)*stdDev(B))]
 

ShaktiRathore

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C3. standard deviation of portfolio of a risky asset R and a risk free asset F
stdDev(p)= wR*stdDev(R)
stdDev(p)^2=wR^2*stdDev(R)^2+wF^2*stdDev(F)^2+2*wR*stdDev(R)*wF*stdDev(F)*Corr(R,F)
now for risk free asset there is no volatility or risk stdDev(F)=0 and there is no correlation b/w risk free and risky asset thus, Corr(R,F)=0
the above implies,
stdDev(p)^2=wR^2*stdDev(R)^2+wF^2*0^2+2*wR*stdDev(R)*wF*0*0
stdDev(p)^2=wR^2*stdDev(R)^2+0+0
stdDev(p)^2=wR^2*stdDev(R)^2
stdDev(p)=wR*stdDev(R)


C4. CAPM return is directly proportional to systematic risk and the unsystematic risk is diversified away,
E(Ri)= a+beta(i)*Rm from the security characteristic line by regressing the stock return with the market returns we get this regression line,
E(Ri) the expected return of the security depends on its systematic risk beta(i)
beta(i)=0,E(Ri)=Rf=>Rf= a+0*Rm=a=Rf
beta(i)=1,E(Ri)=E(Rm)=>E(Rm)= a+Rm=>E(Rm)- Rf=Rm from above a=Rf
substituing the value of Rm and a in the equation of security characteristic line ,
E(Ri)= a+beta(i)*Rm
E(Ri)= Rf+beta(i)*(E(Rm)- Rf) which is nothing but the CAPM equation

C5. Treynor Measure= E(Rp)-Rf/beta(p) ..1
Sharpe Ratio= E(Rp)-Rf/stdDev(p) ..2
Dividing 1 by 2
=> Treynor Measure/Sharpe Ratio= [E(Rp)-Rf/beta(p)]/[ E(Rp)-Rf/stdDev(p)]
=> Treynor Measure/Sharpe Ratio= [stdDev(p)]/[beta(p)] cancelling E(Rp)-Rf from denominator and numerator
noe we have already seen that stdDev(p)=Beta(p)*stdDev(m) =>Beta(p)=stdDev(p)/stdDev(m) threfore substituting relation above,
Treynor Measure/Sharpe Ratio=[stdDev(p)]/ [stdDev(p)/stdDev(m)] =stdDev(m) cancelling stdDev(p) from denominator and numerator
=> Treynor Measure/Sharpe Ratio=stdDev(m)
=>Treynor Measure=Sharpe Ratio*stdDev(m)
 

ShaktiRathore

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C6. Bayes Theorem
P(AB)= P(A/B)*P(B)
P(BA)= P(B/A)*P(A)
P(AB)=P(BA)
=> P(A/B)*P(B)=P(B/A)*P(A)
=> P(A)= P(A/B)*P(B)/P(B/A) similarly P(B)= P(B/A)*P(A)/P(A/B)

Suppose only two events occurs in the experiment,
probability that both A and B occurs is,
P(AB)= P(A/B)*P(B)
now probability that A occurs provided B does not occurs,
P(AB')= P(A/B')*P(B')
Now event A occurs when either B occurs or B does not occur as there are only two events in the experiment therefore whenever A occurs there are two cases that either B occurs or it does not occur so total probability of occurance of A is sum of these two cases which happens independently w/o one depending on the other,
P(A)=probability that both A and B occurs+probability that A occurs provided B does not occurs
P(A)=P(A/B)*P(B)+P(A/B')*P(B') which is the total probability rule
similarly P(B)=P(B/A)*P(A)+P(B/A')*P(A')

C7. let r be the Holding period return over period n compounded every m periods within the period n,
FV= PV(1+r/m)^m*n
FV/ PV= (1+r/m)^m*n
(FV/ PV)^1/mn= (1+r/m)^(m*n*1/mn)
(FV/ PV)^1/mn= (1+r/m)
(FV/ PV)^1/mn -1= (r/m)
r=m*[(FV/ PV)^(1/mn) -1]


C8. Discount of a t day T bill, D=F*(t/360)*rBD where rBD is discount rate of T-Bill
so price= F- Discount=F-D=F-F*(t/360)*rBD=F*[1-(t/360)*rBD]
price of t day T bill is Price(T-bill)=F*[1-(t/360)*rBD] where F is the face value which is usually 100.

Money market return rmm= (D/price)*360/t in annual terms
rmm= (D/F*[1-(t/360)*rBD])*360/t substitute for price from above formula,
rmm= (D/F*[1-(t/360)*rBD])*360/t since D=F*(t/360)*rBD=>D/F=(t/360)*rBD=>D/F*(360/t)=rBD
=> rmm= (D/F)*360/t*{1/[1-(t/360)*rBD]}=rBD*{1/[1-(t/360)*rBD]}
multiply denominator and numerator by 360,
rmm= 360*rBD/[360-(t)*rBD]

thats it for now
thanks
 

ShaktiRathore

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C9.
Duration= %change in Bond Price/ % change in yield
Duration= Average %change in Bond Price/ Average % change in yield
Duration= [(BV- - BV0)/BV0+(BV0-BV+)/BV0]*.5/.5(chg in y+ chg in y)
Duration= [(BV- - BV0)+(BV0-BV+)]*.5/.5*BV0(chg in y+ chg in y)
Duration= [(BV- - BV0+BV0-BV+)]*.5/.5*BV0(chg in y+ chg in y)
Duration= [(BV- - BV0+BV0-BV+)]*.5/.5*BV0(2*chg in y)
Duration= [(BV- -BV+)]*.5/(chg in y)= [(BV- -BV+)]/2*BV0(chg in y)
Duration=[(BV- -BV+)]/2*BV0(chg in y)
Convexity is the rate of change of duration w.r.t the yield
Convexity= change in duration/change in yield
Convexity=[(BV- -BV0)]/BV0(chg in y)-[(BV0-BV+)]/BV0(chg in y)/(chg in y+ chg in y)
Convexity=[(BV- -BV0)]/BV0(chg in y)-[(BV0-BV+)]/BV0(chg in y)/2*chg in y
Convexity=[(BV- -BV0)]/BV0-[(BV0-BV+)]/BV0/2chg in y^2
Convexity=[(BV- -BV0)]-[(BV0-BV+)]/2BV0*chg in y^2
Convexity=[(BV- -BV0-(BV0-BV+)]/2BV0*chg in y^2
Convexity=[(BV-+BV+ -BV0-BV0)]/2BV0*chg in y^2
Convexity=[(BV-+BV+ -2BV0)]/2BV0*chg in y^2

DV01= change in bond price with 1 basis point change in yield
DV01=Duration*.01%=.0001*Duration*Bo

thanks
 

ShaktiRathore

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C10.
Monthly mortgage payments= MP, MB0= original Mortgage Outstanding, T is maturity of mortgage and rm is monthly mortgage rate
MB0=PV of all future Monthly payments
MB0= MP/(1+rm)^1+MP/(1+rm)^2+MP/(1+rm)^3+................+MP/(1+rm)^T
MB0= Geometric series with first term a=MP/(1+rm)^1 and common ratio r=1/(1+rm), no of terms=n=T
MB0= a*[r^n-1]/[r-1]=[MP/(1+rm)^1]*[(1/(1+rm))^T-1]/[(1/(1+rm))-1]
MB0=[MP/(1+rm)]*[1-(1/(1+rm))^T]/[(rm/(1+rm))]
MB0=[MP/(1+rm)]*[1-(1/(1+rm))^T]*[(1+rm)/rm]
MB0=MP*[1-(1/(1+rm))^T]/rm
MB0*rm=MP*[1-(1/(1+rm))^T] multiply both sides by rm
MP=MB0*rm/[1-(1/(1+rm))^T] divide both sides by [1-(1/(1+rm))^T]

SMM=1-(1-CPR)^1/12
1-SMM=(1-CPR)^1/12
raise to the power 12 both sides, (1-SMM)^12=(1-CPR)^1/12*12
(1-SMM)^12=(1-CPR)
CPR=1-(1-SMM)^12

C11. Binomial Tree

S0 the initial stock price can go in two possible states D(down) and U(Up) the probability of up state is pU and the probability of down state is pD. The magnitude of Up jump is U and down move is D
So the So after time t is S0*U in Up state and S0*D in Down state .
Hence value after time t the value of stock is pU*S0*U+pD*S0*D
PV of this value e^-rt*[pU*S0*U+pD*S0*D]
For no arbitrage, S0=e^-rt*[pU*S0*U+pD*S0*D]=e^-rt*S0[pU*U+pD*D]
eliminate S0 from both the sides, e^rt=pU*U+pD*D
pU+pD=1=>pD=1-pU put this in above equation to get,
e^rt=pU*U+(1-pU)*D=pU(U-D)+D
e^rt-D=pU(U-D)+D-D
e^rt-D=pU(U-D)=> pU=e^rt-D/U-D also pD=1-pU=1-[e^rt-D/U-D]=U-D-(e^rt-D)/U-D=U-e^rt/U-D
pU=e^rt-D/U-D ; pD=U-e^rt/U-D

C12. Put Call parity
from BSM; call value=c= S0*N(d1)-X*e^-rT*N(d2)
from put call parity: p+S0=c+X*e^-rT=> p=c+X*e^-rT-S0
put c from BSM,
p=S0*N(d1)-X*e^-rT*N(d2)+X*e^-rT-S0
p=S0*N(d1)+X*e^-rT*(1-N(d2))-S0
p=X*e^-rT*(1-N(d2))-S0+S0*N(d1)=X*e^-rT*(1-N(d2))-S0(1-*N(d1))
p=X*e^-rT*N(-d2)-S0(N(-d1)) derived value of put option from BSM model and put call parity
d1=ln(S0/X)+[r+.5*stdDev^2]T/stdDev*sqrt(T)
d2=d1-stdDev*sqrt(T)=[ln(S0/X)+[r+.5*stdDev^2]T/stdDev*sqrt(T)]-stdDev*sqrt(T)
d2=[ln(S0/X)+[r+.5*stdDev^2]T-stdDev^2*T]/stdDev*sqrt(T)
d2=[ln(S0/X)+[r-.5*stdDev^2]T/stdDev*sqrt(T)

thanks
 

ShaktiRathore

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C13. Delta of call= first derivative of call price w.r.t the underlying asset S
from BSM: c=SN(d1)-X*e^-RfTN(d2)
partially differentiating above bsn equation w.r.t the S the underlying asset price,
dc/dS=d/dS[SN(d1)-X*e^-RfTN(d2)]
dc/dS=d/dS[SN(d1)]-d/dS[X*e^-RfTN(d2)]
dc/dS=N(d1)d/dS-d/dS[X*e^-RfTN(d2)]
dc/dS=N(d1)*1-0
dc/dS=N(d1) which is nothing but the delta of call so for small change in S dS the value of call changes by N(d1)*dS

delta of call=N(d1)
from put call parity,
p+S=c+X*e^-RfT
differentiating above equation w.r.t the underlying asset price S
d/dS[p+S]=d/dS[c+X*e^-RfT]
dp/dS+dS/dS=dc/dS+d[X*e^-RfT]/dS
dp/dS+1=delta of call+0
dp/dS+1=delta of call
dp/dS=delta of call-1=N(d1)-1
delta of put= N(d1)-1
Delta neutral Portfolio:
It comprises of long shares+short call options
portfolio value= Ns*S+Nc*c
with change in S the stock price the change in portfolio value =d/dS[Ns*S+Nc*c]=0 for portfolio to be perfectly hedged
d/dS[Ns*S+Nc*c]=0
d/dS[Ns*S]+d/dS[Nc*c]=0
Ns*dS/dS+Nc*dc/dS=0
Ns+Nc*dc/dS=0
Ns=-Nc*delta so long Nc*delta shares and short Nc calls for perfectly hedging the portfolio against he price movement of assets.
delta=c2-c1/S2-S1
Initially portfolio value=Ns*S1-Nc*c1 as we short the calls and long the shares
for portfolio to remain unchanged in value Ns=delta*Nc
Initially portfolio value=delta*Nc*S1-Nc*c1
After time the stock price changes to S2 and the call price to c2,
Finally portfolio value=delta*Nc*S2-Nc*c2
change in portfolio value=Finally portfolio value-Initially portfolio value=[delta*Nc*S2-Nc*c2]-[delta*Nc*S1-Nc*c1]
change in portfolio value=delta*Nc*(S2-S1)-Nc*(c2-c1)
change in portfolio value=[c2-c1/S2-S1]*Nc*(S2-S1)-Nc*(c2-c1)
change in portfolio value=[c2-c1]*Nc-Nc*(c2-c1)=0 so we have perfectely hedged portfolio with a long position in delta*Nc shares and short Nc calls.

thanks for now..
 

ShaktiRathore

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C14. Rc be the continously compounded return an R is the annual return
suppose we invest $1 today at continously compounded return Rc and also invest $1 today at annual return R
Now This two positions should yield the same amount after time T,
Amount at continously compounded return Rc=Amount at return R
1*e^Rc*T=1*(1+R/m)^m*T where $1 today at annual return R is compounded m times per yr.
e^Rc*T=(1+R/m)^m*T
taking logs on both sides,
ln[e^Rc*T]=ln[(1+R/m)^m*T]
Rc*T=m*T*ln(1+R/m)
Rc=m*T*ln(1+R/m)/T=m*ln(1+R/m)

Rc=m*ln(1+R/m) ....this converts discrete rate R to continous rate Rc
if compounded annually than m=1 Rc=ln(1+R)
if compounded semiannually than m=2 Rc=2*ln(1+R/2)
if compounded quarterly than m=4 Rc=4*ln(1+R/4)

Conversion from continuous rate Rc to discrete rate R,
from our earlier equation
e^Rc*T=(1+R/m)^m*T
raise both sides by power 1/mT,
e^Rc*T*(1/mT)=(1+R/m)^m*T*(1/mT)
e^(Rc*1/m)=(1+R/m)
e^(Rc*1/m)-1=(1+R/m)-1
e^(Rc/m)-1=(R/m)
R=m*[e^(Rc/m)-1]

C15. Number of futures contracts= Nf to hedge bond portfolio=P and duration of bond portfolio=Dp
Duration of futures contract =Df and F = futures position value
change in bond portfolio+change in futures contracts value=0
Nf*Df*F+P*Dp=0
Nf*Df*F=-P*Dp
Nf=-P*Dp/Df*F


thanks
 

ShaktiRathore

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C16. For a equally weighted portfolio with n assets with equal weights of 1/n each with variance of sigma^2 and average covariance of Cov with each other and avg correlation of Corr
we have, stdDev(p)^2= [wA^2*stdDev(A)^2+wB^2*stdDev(B)^2+......nterms]+[Corr*wA*wB*stdDev(A)*stdDev(B)+...n(n-1)terms]
wA=wB=...1/n and stdDev(A)=stdDev(B)=....=sigma
thus , stdDev(p)^2= [(1/n)^2*sigma^2+(1/n)^2*sigma^2+......nterms]+[Corr*(1/n)*(1/n)*sigma*sigma+...n(n-1)terms]
stdDev(p)^2=n [(1/n)^2*sigma^2]+n(n-1)[Corr*(1/n)*(1/n)*sigma*sigma]
stdDev(p)^2=[(1/n)*sigma^2]+(n-1)[Corr*(1/n)*sigma^2]
stdDev(p)^2=[(1/n)*sigma^2]+(n-1)*(1/n)*[Corr*sigma^2]
stdDev(p)^2=[(1/n)*sigma^2]*(1-Corr)+(n)*(1/n)*[Corr*sigma^2]
stdDev(p)^2=[(1/n)*sigma^2]*(1-Corr)+[Corr*sigma^2]=sigma^2*[(1/n)*(1-Corr)+Corr]
stdDev(p)^2=sigma^2*[(1/n)+Corr*(1-1/n)] in terms of correlation
Corr*sigma^2=Cov =>stdDev(p)^2=[sigma^2*(1/n)*1+sigma^2*Corr*(1-1/n)]=stdDev(p)^2=[sigma^2*(1/n)*1+Cov *(1-1/n)]
stdDev(p)^2=sigma^2*(1/n)+Cov *(1-1/n) in terms of covariance
as can be seen as n tends to infinity, stdDev(p)^2=[sigma^2*(0)*1+Cov *(1-0)] =Cov variance f equally weighted portfolio tends to Average Covariance
 
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