Actuarial Outpost > SoA Total Differential of Bond Portfolio
 User Name Remember Me? Password
 Register Blogs Wiki FAQ Calendar Search Today's Posts Mark Forums Read
 FlashChat Actuarial Discussion Preliminary Exams CAS/SOA Exams Cyberchat Around the World Suggestions

 Not looking for a job? Tell us about your ideal job, and we'll only contact you when it opens up. https://www.dwsimpson.com/register

 QFI Core Exam Old Advanced Portfolio Management Forum

 Thread Tools Search this Thread Display Modes
#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?
__________________
P, FM, MFE, MLC, C, VEE, FAP, APC
ERM Mod, FR Mod, FM Mod, DMAC,
ERM Exam, QFI Core, QFI Adv, FAC
#2
10-11-2017, 11:56 AM
 Bell Member Join Date: Jun 2005 Posts: 397

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!!!
__________________
Bell F. Ouelega FSA CERA MAAA CQF
PAK Study Manual Instructor
Quantitative Finance & Investment Track

http://www.pakstudymanual.com/
#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

 Thread Tools Search this Thread Search this Thread: Advanced Search Display Modes Linear Mode

 Posting Rules You may not post new threads You may not post replies You may not post attachments You may not edit your posts BB code is On Smilies are On [IMG] code is On HTML code is Off

All times are GMT -4. The time now is 10:38 AM.

 -- Default Style - Fluid Width ---- Default Style - Fixed Width ---- Old Default Style ---- Easy on the eyes ---- Smooth Darkness ---- Chestnut ---- Apple-ish Style ---- If Apples were blue ---- If Apples were green ---- If Apples were purple ---- Halloween 2007 ---- B&W ---- Halloween ---- AO Christmas Theme ---- Turkey Day Theme ---- AO 2007 beta ---- 4th Of July Contact Us - Actuarial Outpost - Archive - Privacy Statement - Top