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QFI Core Exam Old Advanced Portfolio Management Forum 

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#1




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




V=aX+bY
I assume that you have SDE for X and Y. What you want is the SDE for V..So, apply the twodimensional Ito Lemma as done in MOCK31 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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Bell F. Ouelega FSA CERA MAAA CQF PAK Study Manual Instructor Quantitative Finance & Investment Track Visit our website at: http://www.pakstudymanual.com/ 
#3




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?
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bond, qfic, qficore, total differential 
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