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Ok, so consider an example of a callable bond with market price P. The Z-Spread calculation takes the cashflows for this bond as if it is a non-callable, vanilla bond to arrive at a Z-Spread value which brings the model price equal to P. Obviously, because the bond is callable, there is a...

Nothing you have said is incorrect. OAS is usually derived via a Monte Carlo approach as the flat spread which can be added to underlying curve in order to bring the Monte Carlo expected price equal to the observed market price. For bonds with no optionality the OAS will be very close to the...

Typically used for pricing credit sensitive instruments such as corporate bonds, CDS etc. Usually you would derive a hazard rate curve from observable market data like bond quotes and use it in conjunction with other rates curves to price the instrument or determine sensitivities.

Maybe take a hazard rate type approach? A little bit rusty on this so may be wrong! I'm sure someone will correct me :) If probability of default in a year is 2% then probability of survival must be 98%. Use this to obtain the hazard rate/default intensity: \( P(S)=e^{-Ht} \) \( \therefore...

Hi there, Yes, I work in Fixed Income and have heard of PIMCO ;)

In the vast majority of practical applications of KRDs I've worked on the starting point has always been to shift observable market data rather than the zeros. You may get results which are not particularly intuitive because of the extra step of re-deriving the spot curve but ultimately these...

\[ VAR(A-B)=VAR(A)+VAR(B)-2COV(A,B) \] In this case, since the returns of Fund A and B are independent the covariance of A and B is zero. Leaving: \[ VAR(A-B)=VAR(A)+VAR(B)=0.4^2+0.3^2=0.25=0.5^2 \] Therefore \[\sigma_{A-B}=0.5\] or 50%

This is all about compounding assumptions. By using continuous compounding you are making the assumption that the spot rates are continuously compounded rates, obtaining a valuation of 98.39. If you then calculate a YTM in the same fashion with continuous compounding you will obtain a YTM of...

The difference is the assumed compounding. \( x_n/x_{n-1}-1 \) assumes compounding at the base frequency of your series. i.e. daily, monthly etc. \( \ln(x_n/x_{n-1}) \) assumes continuous compounding. Bearing this in mind, it's a case of how you anticipate the numbers will be used. In your...

Looks fine to me. \[ \sigma^2(X)=E(X^2)-E(X)^2 \] \[ \therefore \sigma(X)=\sqrt{E(X^2)-E(X)^2} \] If your spreadsheet is set up correctly you should just be able to change the probabilities to get back to the example 50/50 numbers. You could also memorise the common quantities for a bernoulli...

Vasicek model is for short rates or instantaneous spot rate. It makes no assumptions about the shape of a curve as it is modelling the spot rate dynamics only.

Rather than trying to memorise these specific formulae, I find it much easier to remember the basic, underlying concept is the same for both in discount factor/growth factor space: \[ GF(0,T_2)=GF(0,T_1)*GF(T_1,T_2) \] Then, depending on the question or data provided you just need to calculate...

The yield-modified duration world is the simplest, text book situation. Calculating KRDs in this flat curve YTM world is not really done or useful. If you REALLY wanted to do this then you could make a flat curve with the YTM at each tenor point and shift each one in turn before repricing. The...

You are correct that the sum of Key Rate Durations should be approximately equal to the Effective Duration. As with all of the these valuation type exercises, consistency is key. In all of the models I've ever seen the KRD and Effective Duration estimation procedure is essentially the same with...

The YTM is generally just used as a measure of quoted price or value. Some bonds quote in the market using YTM, some with actual price. You are correct that the derivation or usage of YTM implies a flat curve at the YTM. YTM and Modified Duration form a sort of self-consistent simplistic...

I think we should first separate the two issues at play here: 1 - Deriving the zero curve 2 - Determining PV01/KRD values for a non-curve instrument Firstly, zero rate curves are derived from observable market prices/yields and these are almost always coupon bearing instruments. The resulting...

This is not correct. It will lower the overall weighted average duration of the portfolio but this is correct as the defaulted bonds are no longer rates sensitive but still have market value. This is exactly what I've said. This is not correct. Including them with zero actually gives the...

I would set the duration to zero but keep them included in the calculation. Remember, (modified) duration is the sensitivity to Yield but defaulted or distressed securities tend to start trading based on Price (estimate of recoverable value) rather than Yield so rates become essentially...