Solve for Q2 and Q3, in terms of Q1!!

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Nazariy

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Hello guys, I have been playing with this for a while, Ive got an exam coming up soon and would appreciate the solution straight away, usually I do everything on my own; Unfortunately there is no time for that right now.

I'll get straight to the point, after differentiating the variance of the change in value of a portfolio of three bonds with respect to bond 2 and bond 3, I obtain following equations, which I equate to zero in order to find minimum value of var:

Q2*(V2^2)*(sigma2^2)+Q1*V1*V2*(correl1&2)*sigma1*sigma2+V2*V3*Q3*(correl2&3)*sigma2*sigma3 = 0 (< With respect to Q2, i.e. par value of bond 2 to be hedged with)

Q3*(V3^2)*(sigma3^2)+Q2*V2*V3*(correl2&3)*sigma2*sigma3+Q1*V1*V3*(correl1&3)*sigma1*sigma3 = 0 (< With respect to Q3, i.e. par value of bond 3 to be hedged with)
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I need to find Q2 in terms of JUST Q1 and Q3 in terms of just Q1. I was thinking about solving these as a system of equations, I don't think that works. So please help me with this one, I would ask for a solution straight away, I mean one that outlines the logical steps, since I already have a solution in my notes(it lacks the logical steps for arriving at it).
 
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Guys need help super fast.. has been working on this for very long now! Mistakenly submitted this into calculus. This is the problem:

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I need to find Q2 and Q3, but only in terms of Q1. Please assist me here, my exam is very soon and I am wasting way too much time on this..
 
That looks to me like a pretty standard "two linear equations in two unknowns" problem:

1) Multiply the first equation by \(\displaystyle V_3^2\sigma_3^2\) and multiply the second equation by
\(\displaystyle V_2^2\sigma_2^2\) so that Q2 has the same coefficient in each equation. Subtract one equation from the other to eliminate Q2 and solve for Q3.

2) Multiply the first equation by \(\displaystyle V_1\rho_{1,3}\sigma_1\) and multiply the second equation by \(\displaystyle V_2\rho_{2,3}\sigma_2\) so that Q3 has the same coefficient in each equation. Subtract one equation from the other to eliminate Q3 and solve for Q2.
 
HallsofIvy said:
That looks to me like a pretty standard "two linear equations in two unknowns" problem:

1) Multiply the first equation by \(\displaystyle V_3^2\sigma_3^2\) and multiply the second equation by
\(\displaystyle V_2^2\sigma_2^2\) so that Q2 has the same coefficient in each equation. Subtract one equation from the other to eliminate Q2 and solve for Q3.

2) Multiply the first equation by \(\displaystyle V_1\rho_{1,3}\sigma_1\) and multiply the second equation by \(\displaystyle V_2\rho_{2,3}\sigma_2\) so that Q3 has the same coefficient in each equation. Subtract one equation from the other to eliminate Q3 and solve for Q2.
Click to expand...

I tried your first suggestion, I get a cube for some sigmas...
 
HallsofIvy said:
That looks to me like a pretty standard "two linear equations in two unknowns" problem:

1) Multiply the first equation by \(\displaystyle V_3^2\sigma_3^2\) and multiply the second equation by
\(\displaystyle V_2^2\sigma_2^2\) so that Q2 has the same coefficient in each equation. Subtract one equation from the other to eliminate Q2 and solve for Q3.

2) Multiply the first equation by \(\displaystyle V_1\rho_{1,3}\sigma_1\) and multiply the second equation by \(\displaystyle V_2\rho_{2,3}\sigma_2\) so that Q3 has the same coefficient in each equation. Subtract one equation from the other to eliminate Q3 and solve for Q2.
Click to expand...

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I'm done with this...
 
I didn't consider it all that strange. Whoever wrote this problem wanted to make it clear that "\(\displaystyle \rho\)" (the Greek letter "rho", not P) had two subscripts, 1 and 3, not the single subscript, 13.
 
HallsofIvy said:
Yes. There is nothing wrong with that.


So you were able to complete the problem?
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I was not. I have a very large volume of material to revise, so I was reading through all of my notes yesterday.

The solving of this is not crucial to my success in the exam because I am not expected to know how to procedurally derive it, but rather to know conceptually the steps. But there are quite a number of other equations that I have to remember, all of them I derived so at least I do not have to memorize them. When I have time later today, I will attempt at solving it again, now I am just too curious and cannot leave it like that.
 
Denis said:
Curious: I think I see P1,2 and P2,3 and P1,3 in those equations; what does that represent?
The "comma" is really what seems strange...
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It is correlation between changes in yield between the bonds in subscript. E.g. correlation of yield change in bond 1 to yield change in bond 2. Used in a method of optimal hedging that overcomes the assumption of conventional hedging whereby the correlation between yield changes of two bonds are assumed to be identical.
 
HallsofIvy said:
Yes. There is nothing wrong with that.


So you were able to complete the problem?
Click to expand...

I have solved it, though not sure how your advised multiplication elements make the part of element to the right of Q2 or Q3 common in both equations. I tried your method yesterday and it didn't work like that.

I have found that in order to solve for Q3, we have to multiply first equation by (correlation2,3)*sigma3*V3 while second by V2*sigma2. Whereas, if you wish to solve for Q2 the first equation has to be multiplied by V3*sigma3 while the second by V2*(correlation2,3)*sigma2.
 
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