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# algebra in P1.T2. Miller Chapter 2 video

#### hellohi

##### Active Member
hello @David Harper CFA FRM

in the video related to P1.T2. Miller Chapter 2, you mentioned this question:

Assume the probability density function (pdf) of a zero-coupon bond with a notional value of $5.00 is given by f(x) = (3/125)*x^2 on the domain [0,5] where x is the price of the bond: you asked to find the 95% value at risk (VaR)? just I could not follow the following algebra answer steps: f(x) = 3/125*x^2 them turn it to f(x) = 3/125*1/3*x^3 = x^3/125 = p then to x = 1/3√125p = 5p^1/3 then to For p = 5%, x = 5(0.05)^1/3 =$1.8420

David, may you clarify the algebra issues here ?

thanks

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@hellohi I moved this from About FRM to the P1.T2.Quantitative Methods board (where it is now). The reason is not just because it belongs here, but to keep the "About FRM" board clear of posts that are not, you know, about the FRM. We want to make it easy for other visitors to find conversations or information they are looking for. A feature of the forum's intention is to try to minimize clutter. Thanks for understanding!

#### brian.field

##### Well-Known Member
Subscriber
"f(x) = 3/125*x^2

them turn it to
f(x) = 3/125*1/3*x^3 = x^3/125 = p"

This is not algebra per se. Rather, it is calculus. The integral of x^n is (x^(n+1)) / (n+1) .

Do you have any background in undergraduate mathematics? I think that a minimum level of comfort with math is necessary for the exam - but integration isn't necessarily a testable concept. I do think, however, that anyone pursuing the FRM should have a basic understanding of calculus, which means he./she should be able to solve the above easily.

#### ShaktiRathore

##### Well-Known Member
Subscriber
Hi,
@hellohi the basic calculus is involved here not the algebra. Hearing from you that you are from non-quant background i would advise you to clear your calculus basics if required(otherwise calculus is not that stringent/strict requirement for FRM in my opinion) only basic algebra and arithmetic would do.
You could refer to any of these below books if required otherwise not necessary(time permits):
The Calculus Lifesaver: All the Tools You Need to Excel at Calculus (Princeton Lifesaver Study Guide) by Adrian Banner (Author)
Cartooon guide to Calculus by Larry gonick.
thanks

#### hellohi

##### Active Member
"f(x) = 3/125*x^2

them turn it to
f(x) = 3/125*1/3*x^3 = x^3/125 = p"

This is not algebra per se. Rather, it is calculus. The integral of x^n is (x^(n+1)) / (n+1) .

Do you have any background in undergraduate mathematics? I think that a minimum level of comfort with math is necessary for the exam - but integration isn't necessarily a testable concept. I do think, however, that anyone pursuing the FRM should have a basic understanding of calculus, which means he./she should be able to solve the above easily.
thanks dear brian for this information, I am trying to master this concept now

#### mattias@ledunger.com

##### New Member
Hi, I struggled with this one too, but I think in the end the problem for me sat in the solution presented which i think has a typo...
300.2. D. $3.158 As f(x) = 3/125*x^2, F(x) = 3/125*(1/3)*x^3 = p, such that: p = F(x) = (3/125)*(1/3)*x^3 = x^3/125, solving for x: x = (125*p)^(1/3) = 5*p^(1/3). For p = 5%, x = 5*5%^(1/3) =$1.8420. As q(0.05) = $1.8420, 95% VaR =$5.00 - $1.8420 =$3.1580
Solving for x would for me rather be:
p=x^3 / 125
0,05 = x^3 / 125
x^3 = 125 * 0,05 X = 1,842
Right?

#### mattias@ledunger.com

##### New Member
Sorry to be nit picking here but when trying to grasp the concepts it can be tough with mathematical errors in the solutions. In the same Miller Ch2 video there is an example on Bayes Theorem where we are asked to calculate the Probability ( Neutral Economy | Constant Stock Price ) .
Have you not forgotten to add the P(Neutral Economy) 30% in the Numerator there? I.e. Your answer 0,387 is incorrect and the correct answer is 0,11612% (Which is intuitive, given that the probability (Constant Stock Price) is <=0,2 in all states of the economy...

#### ShaktiRathore

##### Well-Known Member
Subscriber
Hi, I struggled with this one too, but I think in the end the problem for me sat in the solution presented which i think has a typo...
300.2. D. $3.158 As f(x) = 3/125*x^2, F(x) = 3/125*(1/3)*x^3 = p, such that: p = F(x) = (3/125)*(1/3)*x^3 = x^3/125, solving for x: x = (125*p)^(1/3) = 5*p^(1/3). For p = 5%, x = 5*5%^(1/3) =$1.8420. As q(0.05) = $1.8420, 95% VaR =$5.00 - $1.8420 =$3.1580
Solving for x would for me rather be:
p=x^3 / 125
0,05 = x^3 / 125
x^3 = 125 * 0,05 X = 1,842
Right?
x = (125*p)^(1/3)
p=.05=> x = (125*.05)^(1/3)= (6.25)^(1/3)=$1.842 is the worst price that the bond can attain at 95% CL,therefore the worst loss that the bond can have is$1.842-$5=$ -3.158 at 95% CL or the Var is \$3.158.

#### mattias@ledunger.com

##### New Member
Thanks Shakti, exactly right. 125 * 0,05 = 6,26. I see now how David had written it. Just looked sooo backwards. Not going via 6,25 there got me.... .-P

#### PortoMarco79

##### New Member
Subscriber
F(X): 3/125 | x^2
= 3x^2/125
= 1/125 * dx 3x^2
= 1/125 * 3/1*x^3/3
3s cancel
= 1/125 * x^3
= x^3/125
x=(0.05*125)*1/3 = 1.842 -5 = 3.158