Managing the Interest Rate Risk of Multiple Liabilities: Duration Matching
Recall duration matching as an immunization strategy for a single liability. Just choose a bond (or bonds) that give you the same duration as the liability, and maintain it as closely as possible with minimum variance. Done.
With multiple liabilities, it's a little different. There's still a single duration to aim for, but it represents cash flows not at that time, just centered there. So start with this: if the yield curve shifts or twists, do you think it's more important what happens to average timing or to the money in the portfolio?
No.
Money is definitely more important here.
Yes.
You have a portfolio of semi-annual coupon bonds valued at 2,173,749 to target liabilities with a Macaulay duration of three years (six periods), perhaps. The cash flow yield is 4.1643%. The modified duration is the Macaulay duration divided by the gross cash flow yield. To do this, what do you think needs to be adjusted?
Ultimately, the goal of immunization is allowing you to use the portfolio to pay the liabilities. So money duration is more useful than Macaulay duration when it comes to multiple liabilities. If it's been a while, here's an example:
No.
The duration is six years, and there's no adjustment required in order to divide it by something.
No.
The bond value is what it is, no adjustment needed.
Right.
The cash flow yield should be cut in half for the periodic cash flow yield. Then, money duration is annual modified duration multiplied by the PV of the portfolio. So here you go.
$$\displaystyle \text{Money duration }= \left[ \frac{3.0}{1 + \frac{0.041643}{2}} \right] \times 2{,}173{,}749 \approx 6{,}388{,}234$$
This is where basis point value (BPV) is useful. These are some big numbers, even for this relatively small portfolio. Just multiply the money duration by a basis point to get the BPV, so you can work with smaller numbers.
$$\displaystyle \text{BPV }= 12{,}776{,}470 \times 0.0001 \approx 1{,}278 $$
Do you think the use of BPV is going to affect what needs to be done in order to perform duration matching?
Of course not.
It just adjusts the values for easier use.
No, it won't.
It just adjusts the values for easier use.
Now, this money duration just provides an approximate value change to a basis point shift in the yield curve. What do you suppose is the main cause of the difference?
No.
The duration estimate just leads to an approximate change and does a good job for small movements.
Exactly.
Convexity makes the approximations a bit off. In fact, for single liabilities, convexity is really the enemy of immunization since it does cause a disturbance to an otherwise perfect setup.
No.
The bond can be at a discount or a premium, but there's still another underlying reason for this approximation error.
But consider what convexity really does. If you have a portfolio, and the yield curve changes, the portfolio value changes according to duration and the shift, approximately. How does convexity affect the portfolio's actual price, compared to the estimate?
No.
That can't be said with any certainty. It's convexity, not concavity.
Absolutely.
Convexity gives rate decreases a little more power to increase price and gives rate increases a little less power to decrease price. The convexity adjustment is always positive. So when immunizing against multiple liabilities, the bond portfolio has to have a little more convexity and dispersion than the liabilities. That leaves the portfolio with a bit more power in responding to yield curve changes.
No.
It moves in a single direction, actually.
The portfolio itself is often constructed of very high-quality bonds. Nice, safe issues. These offer lower yields, of course, which means they grow at a slower rate than typical corporate liabilities. What does that tell you about the starting balances of assets and liabilities in an immunization strategy of multiple liabilities?
No.
Consider the implications of assets having lower cash flow yields. That changes the reinvestment income.
Precisely.
If assets are going to grow more slowly, then you'll need a higher starting balance.
Not quite.
If you started with a higher liability balance, you wouldn't be able to pay them off with these assets.
And once time starts to pass, convexity might not be the only issue. The curve can twist and "flutter" and all sorts of things. Even steepness is an issue with this sort of immunization. Go back to the basis point value (BPV) calculation in your mind. This is the monetary change of a single basis point change. What time characteristic would make the BPV the highest?
Exactly.
The further in the future, the greater the effect from a basis point change. Also consider that the immunizing bond portfolio most likely has cash flows on both sides of the liabilities: as before, more dispersion and greater convexity. So which would probably move the portfolio toward being underfunded, flattening or steepening?
No.
These have the smallest BPVs.
No.
Proximity to duration isn't an issue. It's more of an extreme value thing.
No, steepening is the threat.
That's right!
A steeper curve means higher long-term rates. Down go the longest-term values, which affects some of your assets the most. So assets fall faster than liabilities out there, and that leaves an imbalance. Hopefully things will flatten out in the future to even out.
When you're immunizing against multiple liabilities, hope for flatter paths ahead.
To summarize:
[[summary]]
Timing
The money
Duration
Bond value
Cash flow yield
No
Yes
Duration
Convexity
Discount pricing
It makes it lower
It makes it higher
It depends on the shift
They should be equal
Assets should be greater
Liabilities should be greater
Long term
Short term
Close to duration
Flattening
Steepening
Continue
Continue
Continue
