# Variance covariance matrix for a portfolio containing bonds also with other asset classes

What should we take for a bond or a zero coupon bond in order to make a variance covariance matrix? For example:- Equities - we take the market price Cash - we take the spot rates Bonds - Do we take yield points from bloomberg or cash flows or simply the traded price of the bond ?????

P.S - I had initially taken the yield points for which I got the feedback as "yield points are taken as proxy mapping while not considering the inverse relationship with the bond prices". What does this statement means??? Please help me out in this regard as I am very much confused what to do with bonds in order to make variance covariance matrix?

You most probably don't want to estimate the covariance of prices but rather the covariance of returns. Thus for equities you can take the return of the traded price.

For bonds:

• if the maturity is long enough (say bigger than 2 years), then you can take the returns of traded prices. The pull to par should not be too relevant here.
• if the maturity is short or in any case for better modelling: if you do historical simulation then you reprice the bond (with current characteristic, especially current time to maturity) in historical scenarios of the yield curve (applying historical deltas to today's curve). Then you can calculate returns of these scenario prices to the current price. This will give you returns.
• as approximation you can calculte each bonds duration and approximate returns by $$r_i = -D \Delta y_i$$ where $D$ is the duration and $\Delta y_i$ is the change at time $i$ of the corresponding yield.

One could (and some people did) fill books with this. You could have a look at Meucci's The quest for invariance.

• Yeah, you got that right. We take returns for covariance matrix. But if I calculate returns according to the last point of yours which is 'duration * change in yield', then what if I have a zero coupon bond in my portfolio. Do I have to follow the same logic as for normal bonds? – Manish Jan 28 '16 at 9:52
• Zero coupon or coupon-bearing - it does not matter, the approximation using duration applies (as an approximation). Take into account that credit risk is ignored in the considerations above. e.g. in the case of a floater you have little market (interest rate) risk but credit/spread risk. – Ric Jan 28 '16 at 10:01
• So if we want to calculate returns of a 3 month TBILL, then we only have to take duration as we cannot find out the current yield of a TBILL as there are no coupon payments. So can I take change in duration for everyday from bloomberg? Am I going on the right track? – Manish Jan 29 '16 at 6:44
• No, please check out the term duration. This measures the sensitivity to changes in interest rates. Thus Duration is fixed everyday and the bond price is affected by changes in yields. By how much is measures by duration (approximately). – Ric Jan 29 '16 at 7:01
• But then how come when I change the settlement date in bloomberg, it changes the duration also???? – Manish Jan 29 '16 at 7:11