Duration of a levered portfolio

Reading 23, practice problem 13 says: Palme asks Ibahn: “How can we adjust the portfolio’s duration without contributing significant funds to purchase additional bonds in the portfolio?” Ibahn responds: "We could lever the portfolio by entering into either an overnight of 2-year term repurchase agreement and use the repo funds to purchase additional bonds that have the same duration as the current portfolio. E.g. if we use funds from a $25m over-night repo agreement to purchase bonds in addition to the current $100m, the levered portfolio’s change in value for a 1% change in interest rates would equal $5,125,000 while giving you the portfolio duration you require. Answer says: 2 year term leverage would shorten the total duration of the levered portfolio relative to overnight repo by the dollar duration of the 2-year liability. The levered portfolio duration would be longer using overnight repo because its proceeds are being invested in bonds to have the same duration as the unlevered portfolio- thus net effect is longer duration because overnight repo duration is zero. Can someone please explain the answer, i’m lost on it?

Think of portfolio duration of asset duration - liability duration.

Using a repo and investing in bonds is adding leverage (an asset and an equal liability).The (dollar) duration of the new liability (repo) will be subtracted from the portfolio duration, and the duration of the new assets will be added to the portfolio’s duration.

The duration of the assets in this case is greater than the duration of the liabilities, increasing overall duration of the portfolio.

Where in the text does it say that portfolio duration is = asset duration - liability duration? I am trying to understand this concept.

Thanks mattmania.

it all relates back to - asset is yours - so PLUS. Liability belongs to someone else - so MINUS.

Applies to positions, durations, etc. Liability is a level of leverage.

Is the repo rate’s duration 0 because it’s technically an overnight loan? How does the 2-year length of the agreement come into play when analyzing the duration of the portfolio?

Is this a case of where interest rate products that reset on specificed date have their par values reset to 1xx? Seems fairly confusing…

bump on repo quesiton

This is how I like to look at it:

Using the weighted average return formula

Ra = (D/A)*Rd + (E/A)*Re

where Ra = return on asset; Rd = return on debt (interest rate); Re = return on equity. The respective returns can also be viewed as durations since duration represents % change in value

Rearranging the formula we get

Re = (A/E)*Ra - (D/E)*Rd ; OR equivalent

[Levered Portfolio Duration] = (A/E) * [Asset Duration] - (D/E)*[Liability Duration]

Clearly a longer liability duration results in a lower levered portfolio duration for the same leverage ratio

Yes, this seems to be a confusing question with not a lot of discussion in CFAI text.

There was a similar question asked on this forum in the past.


"Dp = (Di x I - Db x B) / E

where Dp = duration of levered portfolio

Di = duration of invested fund

I = invested funds

Db = duration of borrowed funds (which is repo in this case)

B = Borrowed funds & E = equity

So Duration of levered portfolio is indirectly proportional to duration of borrowed fund"