Q- How exactly does convexity work ? Why does a higher dispersion of cash flows magnify positive returns and dampen losses when interest rates fall
Q-Why does a laddered portfolio have less convexity then barbell ? Aren’t its cashfllows more dispersed
Q- Why Does the method( futures, swap , swaption , swaption collar) of adjusting BPV for immunization matter ? How does the interest rate view of the manager impact the hedge result ?
Convexity dampens the effect of rising rates and amplifies the effect of lower rates on bond price, making it desirable.
When dispersion of cash flows is low, the portfolio is more bulleted. So the question is comparing a bulleted portfolio with a barbelled portfolio. When interest rates fall (flatter yield curve), barbell does better. When interest rates rise (steep yield curve), bulletet portfolio does better.
A laddered portfolio lies between a barbell and a bullet when it comes to convexity. Bulleted is very concentrated in the middle, barbell is concentrated in the extremes, while laddered is more spread across maturities. As to your third question, imagine you have a situation where your assets have higher BPV (interest sensitivity) than liabilities. If interest rates fall, you benefit (since assets rise more than liabilities), and if interest rates rise, you lose. In this scenario, you need to hedge in case interest rates rise by going short futures. Here, if interest rates rise, your assets fall more than your liabilities, but you gain from your short futures position to offset the loss. Now, if you expect interest rates to fall, then you’d under hedge (i.e. short fewer futures contracts than needed going by the formula:
(liab BPV - Asset BPV)/ Futures BPV
If you expect interest rates to decline low enough, you’d leave your position completely unhedged. This shows why your interest rate expectations are important in your hedging decisions.
See it for yourself: calculate what happens to a bond with 5y maturity which pays no coupons and currently trades at 5% ytm when 1.1) ytm falls to 4% and 1.2) ytm increases to 6%. See how you ‘lose less’ when rates are up compared to what you win when rates are down? That is exactly the effect of convexity and it is higher for longer maturities (this sort of answers your question about dispersion as well). It makes sense if you know a bit of calculus, where you can interpret convexity as the second derivative of the price of the bond in terms of the yield to maturity
the dispersion is represented by the distance between cash flows. If you look closer, you will see that the distance between cash flows on a laddered portfolios is smaller than the distance on a barbell one
Futures and swaps work essentially as the same instrument, with some differences regarding margin etc, but on swaptions, you pay an upfront premium to enter on a immunization strategy, therefore you pay for the right of being wrong, essentially
hope that helps!
Follow up to the third question…
But as long as the Total BPV ( BPV per contract x total contracts) of the hedge is the same, does it mater what instrument ( futures , swaps or swaption was used ) , Numerically, shouldn’t the result of hedge be same ?