In the fixed income world, duration is defined as the percentage change in price of the bond when there is a change in one percentage change in interest rate. What is the duration of a perpetual bond?

I shall not go into differentiation equation but would demonstrate it with just a very simple example. Initially when the bond is issued, it is issued at par with a coupon (assuming annual coupon for simplicity) of C. Let's say the coupon rate is 5%. So C = $5 for par $100. Over time, the value of the bond is P = $5 / R where R is the discount rate. If R = 5%, P = $100 which is the par value. If the interest rate goes up by 100 bps (assuming paralleled shift in the yield curve), P = $5/6% = $83.33. The % change in price of the bond is 83.33/100 -1 = -16.7%. The duration is 16.7 for this case. As it can be seen that a perpetual bond is highly sensitive to interest rate.

A traditional bond will mature at par (assuming no default) and thus any previous losses due to interest increase is not relevant if held to maturity.

But for the perpetual bond, there is no maturity date. Thus, there is no definite date which the bond will mature at par. It is possible to be sitting at a lost for a very long time. But it appears that these perpetual bond is callable after X years. When it is callable, the bond issuer redeems the bond because it can reissue new bonds at a lower discount rate. This happens when the interest rate is lower than that of initially. But this is bad news for holders of these called bonds as they supposed to be sitting at a profit only to be redeemed at par. It is for this reason, the callable feature is a call option held by the bond issuer. For bond holder is effectively buying a bond with infinite maturity and writing a call option. Thus, the price of the bond is roughly given by:

P = B - c

Where B is the price of a normal bond with infinite maturity and c the value of the call option. When interest rates drop, the value of 'c' gets more valuable putting a cap on the value of P. This means, the upside to the bond is limited.

The conclusion is this: The perpetual bond has limited upside (if it is callable), extremely sensitive to interest rate risk and a bond holder could be sitting at a lost for a very long time.

Currently, interest rate in the world is at all time low with the developed countries in its expansionary monetary policy. Imagine what happens when interest rate goes up. Good luck for perpetual bond holders! Make sure you got enough fire power otherwise your retirement age could be extended perpetually!

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