Portfolio duration with derivatives

CFA Level III Fixed Income
#1

Hi all, a broad question on calculating portfolio duration when swaps, treasury futures, options on futures, or CDS are held.

The general formula for portfolio duration is sum(duration * market value) / sum(market value). The numerator is the dollar duration of the portfolio and the denominator is the total market value of the portfolio.

In the case of futures we would add (duration * Notional value * price) to the numerator of the formula. This quantity represents the dollar duration and therefore the economic impact of the of the futures contract to the portfolio. Futures contracts have no market value and therefore the denominator does not change. I understand this to be correct

Things get a bit interesting with swaps. Again, we add (duration * notional value * price) to the numerator. This is the dollar duration of the swap. Generally, at the initiation of the swap, the market value is zero which means the market value of the portfolio does not change. However, as rates move, the market value and the price of the contract does change. Does this market value get added to the market value of the portfolio? What is the generally accepted way to roll up swap exposure in the portfolio duration statistic? Can I assume that because swaps are netted at regular intervals that the price remains at par and the market value is effectively 0? In this scenario we simply add (duration * notional value) to the numerator.

Treasury futures options. Again, I’m not sure that these have a market value. They cost nothing, they are exposed to a contract that has no market value. For these instruments, adding the dollar duration to the numerator should suffice.

CDS- no interest rate exposure, should not be rolled up into portfolio duration.

Any comments appreciated

#2

The curriculum does not seem to give a general formula for portfolio duration with derivatives. For such a generalization I would look at two possibilities.

  1. The market value approach can be used to compute the effective duration numerically. I.e. you need to calculate total portfolio values given small + and - shifts in rates.

  2. For a more analytical approach, I would model derivatives in terms of the leverage they provide. The curriculum has a formula for leveraged portfolio duration, it is basically using the leverage factor as an amplifier for the asset duration (when liabilities are short term floaters, i.e. near zero-duration).

The concept of leverage is reflected in the delta of a derivative. So, a long futures contract will give you delta +1*CF for the duration of the CTD bond, a receiver swap will give delta +1 for the duration of fixed leg (less D of floating), and for options you need to calculate the actual delta (and mind the gamma). The delta measures the exposure as a fraction of the notional principal of the underlying.

Or, in your formula, in both numerator and denominator, where you sum up the market values for derivative positions use the exposure given by that derivative position in lieu of its market value.

Hope this makes some sense.

#3

Curriculum deals with the Target Duration of the Portfolio after you add derivatives to it

and based on that - how many contracts of derivatives you would need to achieve that target duration.