Estimating Bond Return Covariance for Multi-Asset Portfolios
Summary
The document addresses which bond data to use when estimating a portfolio variance–covariance matrix. Its central recommendation is to model returns rather than covariance of raw prices. For longer-maturity bonds, historical returns from traded prices may be adequate because pull to par is less influential over the relevant horizon.
For short maturities or more detailed modeling, it proposes repricing a bond with its current characteristics, including time to maturity, under historical yield-curve scenarios. Returns can then be computed from those scenario prices relative to the current price. A simpler approximation uses duration multiplied by the change in the relevant yield, with the inverse relationship between bond prices and yields reflected by a negative sign. These are modeling approaches rather than a universal data rule; the response does not discuss curve construction, return horizons, or covariance estimation choices in depth.
Key ideas
- Portfolio covariance is generally estimated from returns rather than price levels.
- Historical traded-price returns can be used for bonds with sufficiently long maturities.
- For shorter maturities, historical yield-curve changes can be applied to today’s curve to reprice bonds.
- Duration gives an approximate bond return from yield changes, with prices moving inversely to yields.
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# Variance covariance matrix for a portfolio containing bonds also with other asset classes # 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? ## Answer by Richi Wa (score 0, accepted) https://quant.stackexchange.com/a/22978 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.
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