## PerpetualDiscount Duration Calculation

Very simple.

Given that the yield is “r” per period and that the first payment is received exactly one period hence:

Macaulay Duration = (1+r) / r

Modified Duration = 1 / r

And here’s the proof (TIF file).

Note that, strictly speaking, these are the durations of a perpetual annuity; the assumption of perpetuity gets shakier as redemption becomes more likely.

### 4 Responses to “PerpetualDiscount Duration Calculation”

1. […] The algebra is linked in the post PerpetualDiscount Duration Calculation. […]

2. […] Readers will instantly recognize this as highly suspicious, at best. The duration of a perpetual annuity is the inverse of the interest rate. Long Gilts are currently yielding about 4.5% … assuming that the Gilt Perpetuals are trading […]

3. […] Update, 2009-2-15: Note that the Modified Duration of a PerpetualDiscount is dependent solely upon its yield. […]

4. […] be aware that the Modified Duration of a PerpetualDiscount is simply the inverse of its yield (see http://www.prefblog.com/?p=2582). With yields at about 7%, this means a MD of about 14 years. Many investors will blithely purchase […]