Actuarial Outpost > SoA Total Differential of Bond Portfolio
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#1
10-03-2017, 09:09 PM
 charli175 Member SOA Join Date: Jul 2010 Location: NJ College: Georgetown University Posts: 118
Total Differential of Bond Portfolio

Let's say you have a portfolio consisting of two bonds of different maturities. The weight of the first bond maturing at time T is "a" and the weight of the second bond maturing at time S is "b." For ease of notation, let's just say the value of the two bonds is X and Y respectively.

Then our portfolio value, V = aX + bY where a + b = 1.

How do we take the total differential here?

It seems like it should be dV = a*dX + b*dY where dX and dY are the total differentials of bonds X and Y using Ito's lemma.

The solution I'm looking at though is dV = V * (a*dX/X + b*dY/Y). How do I get there?
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#2
10-11-2017, 11:56 AM
 Bell Member Join Date: Jun 2005 Posts: 400

V=aX+bY

I assume that you have SDE for X and Y. What you want is the SDE for V..So, apply the two-dimensional Ito Lemma as done in MOCK-31 for V=XY and V=X/Y..it shall be straightforward.

Let me know if you do not see how....still!!!
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#3
10-13-2017, 09:14 AM
 charli175 Member SOA Join Date: Jul 2010 Location: NJ College: Georgetown University Posts: 118

To avoid confusing each of the functions, let's call:
F = X*Y
G = X/Y

So we then have:
dF = Y*dX + X*dY + dX*dY
(dF/F) = (1/X)*dX + (1/Y)*dY + (1 / X*Y)*dX*dY

dG = (1/Y)*dX - (X / Y^2)*dY + (X / Y^3)*dY^2 - (1 / Y^2)*dX*dY
(dG/G) = (1/X)*dX - (1/Y)*dY + (1 / Y^2)*dY^2 - (1 / X*Y)*dX*dY

How do we connect these for V = X + Y?
__________________
P, FM, MFE, MLC, C, VEE, FAP, APC
ERM Mod, FR Mod, FM Mod, DMAC,
ERM Exam, QFI Core, QFI Adv, FAC

 Tags bond, qfic, qficore, total differential