Duration Limit for Perpetual Bonds
A __perpetuity__, or _perpetual bond_, is a bond that pays interest forever—the price the present value of the endless stream of cash flows with no principal.
In reality, perpetual bonds, also called consols, are very rare. However, they have some interesting characteristics and are worth evaluating for that reason.
Which of the following securities could be thought of as a perpetual asset?
Correct!
A common stock has no maturity, so it can be thought of as a perpetual asset.
Incorrect.
Municipal bonds have stated maturity dates.
Incorrect.
US Treasury bills have stated maturity dates.
What happens to the duration of a perpetual bond as the yield increases?
Incorrect.
In general, increases in yields cause durations to go down.
Correct!
As the yield increases, the duration goes down. As an example, at 7.00%, the duration becomes:
$$\displaystyle \text{MacDur} = \frac{(1.07)}{0.07} \approx 15.286$$
Incorrect.
Changes in yields always cause some change in duration.
In the example above, if the three coupon bonds had a maturity of 100 years, what could you conclude about the risk of the four bonds?
Correct!
The fact that durations for the bonds converge as maturity increases means that the risk of all the bonds would be about the same. In essence, all four bonds would be like perpetual bonds.
Incorrect.
Discount bonds tend to have higher durations. However, durations for the bonds converge as maturity increases.
Incorrect.
All the bonds would look like the perpetual bond from a risk perspective.
To summarize:
[[summary]]
Since there is no principal on a perpetual bond, the Macaulay duration reduces to a basic equation. It is simply:
$$\displaystyle \text{MacDur} = \frac{(1+r)}{r}$$
where $$r$$ is the yield on the bond.
As an example, for a perpetual bond with a 6.00% yield, Macaulay duration would be:
$$\displaystyle \text{MacDur} = \frac{(1.06)}{0.06} \approx 17.667$$
The duration on a perpetual bond has an important relationship to the Macaulay duration estimates on other types of bonds. The figure below displays Macaulay duration for a par bond, discount bond, premium bond, and a perpetual bond. All of the bonds have 6.00% yields:
Notice that the duration on par bonds and premium bonds increase as maturity increases. A discount bond's duration also increases initially and then declines slightly. All three of these bonds' durations approach the duration of the perpetual bond as maturity is increased.
Municipal bond
Common stock
US Treasury bill
Increases
Decreases
Stays the same
It would be similar for all four
It would be lowest for the discount bond
It would be highest for the perpetual bond
Continue
Continue
Continue
Continue
