Portfolio Dispersion Statistic
Consider the example of a bond portfolio of just two semiannual coupon bonds, leading to the following cash flows that provide immunization for a three-year investment horizon, as shown by the Macaulay duration of six semiannual periods.
| Period | Cash Flows | PV of Cash Flows | Weight | Time Weight |
|-----|-----|-----|-----|-----|
| 1 | 46,103 | 45,163 | 0.02078 | 0.02078 |
| 2 | 46,103 | 44,241 | 0.02035 | 0.04071 |
| 3 | 46,103 | 43,339 | 0.01994 | 0.05981 |
| 4 | 46,103 | 42,455 | 0.01953 | 0.07812 |
| 5 | 1,235,103 | 1,114,177 | 0.51256 | 2.56280 |
| 6 | 14,000 | 12,372 | 0.00569 | 0.03415 |
| 7 | 14,000 | 12,119 | 0.00558 | 0.03903 |
| 8 | 1,014,000 | 859,883 | 0.39558 | 3.16461 |
| Sum | $$\,$$ | 2,173,749 | 1.00000 | 6.00000 |
The cash flow yield here is 4.1643%. That's the IRR of the cash flows (using the PV of 2,173,749) multiplied by 2, since they represent eight semiannual periods.
What would you say is a main difference between using this portfolio and a three-year zero-coupon bond of the ideal size, if it existed?
Exactly.
Both provide immunization, just with different cash flows. The zero-coupon bond would provide a single cash flow at Year 3 (or Period 6) and be done. No variance. But since tailored zero-coupon bonds aren't often available, a bond portfolio manager needs to put together something like this coupon-bond combination in order to create the Macaulay duration needed. But the varied cash flows provide variance.
No.
It would provide the same return. The "loss" of reinvestment would be compensated by the lower bond price.
Not quite.
This is a true statement, but it doesn't represent a difference. This statement refers to the successful immunization of the investor, which either strategy would provide.
The __dispersion statistic__ measures the weighted variance of cash flows, and here's how it works: it's the squared deviation of the period and the Macaulay duration, multiplied by the weight. Recall that the "weight" here is the PV of that period's cash flow divided by the total. So for example, the dispersion statistic for the first row would use the period of 1, the Macaulay duration of 6, and then the weight of 0.02078.
$$\displaystyle \text{Dispersion} = (1 - 6)^2 \times 0.02078 = 0.51941 $$
What period do you think will have the highest dispersion statistic?
No.
That one will be high, but consider that it's not that far from the Macaulay duration.
No.
This one will be the lowest. Since it's exactly the Macaulay duration, this value will be zero.
Yes.
It's one of the biggest two cash flows, and it's further away from the Macaulay duration of 6 than is the other big cash flow in Period 5. So that will have to be the biggest one. Here they all are:
| Period | Cash Flows | PV of Cash Flows | Weight | Time Weight | Dispersion |
|-----|-----|-----|-----|-----|-----|
| 1 | 46,103 | 45,163 | 0.02078 | 0.02078 | 0.51941 |
| 2 | 46,103 | 44,241 | 0.02035 | 0.04071 | 0.32564 |
| 3 | 46,103 | 43,339 | 0.01994 | 0.05981 | 0.17944 |
| 4 | 46,103 | 42,455 | 0.01953 | 0.07812 | 0.07812 |
| 5 | 1,235,103 | 1,114,177 | 0.51256 | 2.56280 | 0.51256 |
| 6 | 14,000 | 12,372 | 0.00569 | 0.03415 | 0.00000 |
| 7 | 14,000 | 12,119 | 0.00558 | 0.03903 | 0.00558 |
| 8 | 1,014,000 | 859,883 | 0.39558 | 3.16461 | 1.58230 |
| Sum | $$\,$$ | 2,173,749 | 1.00000 | 6.00000 | 3.20305 |
The Macaulay duration is 3, since it's the sum of those weighted average cash flows of 6 divided by the periodicity of 2. But the dispersion statistic is a squared measure, so it is divided by the squared periodicity to arrive at its final value. What is your estimate of this portfolio's dispersion statistic, then?
No.
That's just the sum of the squared terms. Again, you need to divide by something in order to arrive at the statistic as described.
Not quite.
That would be the sum of the dispersion terms divided by periodicity, but not squared periodicity.
That's right!
Squared periodicity is 4, and then 3.2 divided by 4 gives you the dispersion statistic of 0.8. This is obviously higher for longer time horizons, as there is more squared variance. Do you think it would be larger for a bigger investment as well, such as having 10 times the capital invested?
Good call!
No, actually it wouldn't be.
The weights are all PV of cash flows divided by the total, so they are insensitive to position size, or for that matter, currency units. Dispersion measures have to have this characteristic to be useful.
Then there is one more column to add: __convexity__. Each cash flow has its own convexity measure, which is the time, multiplied by time + 1, multiplied by the weight. It's probably best not to search for reasoning here. Just take a peek at the results.
| Period | Cash Flows | PV of Cash Flows | Weight | Time Weight | Dispersion | Convexity |
|-----|-----|-----|-----|-----|-----|-----|
| 1 | 46,103 | 45,163 | 0.02078 | 0.02078 | 0.51941 | 0.0416 |
| 2 | 46,103 | 44,241 | 0.02035 | 0.04071 | 0.32564 | 0.1221 |
| 3 | 46,103 | 43,339 | 0.01994 | 0.05981 | 0.17944 | 0.2392 |
| 4 | 46,103 | 42,455 | 0.01953 | 0.07812 | 0.07812 | 0.3906 |
| 5 | 1,235,103 | 1,114,177 | 0.51256 | 2.56280 | 0.51256 | 15.3768 |
| 6 | 14,000 | 12,372 | 0.00569 | 0.03415 | 0.00000 | 0.2390 |
| 7 | 14,000 | 12,119 | 0.00558 | 0.03903 | 0.00558 | 0.3122 |
| 8 | 1,014,000 | 859,883 | 0.39558 | 3.16461 | 1.58230 | 28.4815 |
| Sum | $$\,$$ | 2,173,749 | 1.00000 | 6.00000 | 3.20305 | 45.203 |
The sum of these terms is 45.203, and that gets closer to the final measure. From there, just divide by the square of the gross periodic cash flow yield. Recall that the annualized cash flow yield (the IRR of cash flows) is 4.1643%.
$$\displaystyle \text{Convexity}_{\text{semiannual}} = \frac{45.203}{(1 + \frac{0.041643}{2})^2} = 43.3779 $$
And then, as with dispersion, divide by squared periodicity to arrive at the annualized convexity measure.
$$\displaystyle \mbox{Convexity} = \frac{43.3779}{4} = 10.84 $$
Does this calculation seem easy to follow?
No one can blame you.
Wow, you're amazing.
But here's perhaps a simpler relationship:
$$\displaystyle \text{Convexity} = \frac{\text{MacDur}^2 + \text{MacDur} + \text{Dispersion}}{(1 + \text{Cash Flow Yield})^2} $$.
You have the Macaulay Duration (MacDur) of 6, and the sum of the dispersion statistics of approximately 3.20305. With the cash flow yield given, convexity can be calculated as
$$\displaystyle \text{Convexity}_{\text{semiannual}} = \frac{6^2 + 6 + 3.20305}{(1 + \frac{0.041643}{2})^2} \approx 43.3779$$.
Then again, just divide by 4 to get 10.84.
Admittedly, this is quite a complex journey through a spreadsheet of values to arrive at these final numbers. You aren't likely going to be asked to replicate something like this, but focus on the meaning instead. What sign will convexity always have to have?
Absolutely.
No, positive.
Either recall that from past discussions on what convexity is, or just look at the calculations here where there are positive, squared terms dominating the calculation to make sure everything is positive.
Convexity has always been that extra adjustment above and beyond duration to add more accuracy to a price adjustment. So what effect do you think the convexity of a bond portfolio would have on an immunization strategy?
No.
The use of convexity makes estimations better, but its existence doesn't improve the success of an immunization.
Precisely.
The fact that there is convexity means that your Macaulay duration–based immunization strategy has a weakness. Twisting or steepening of the yield curve will cause the strategy to be off by a little, and this is added information about the effectiveness of an immunization strategy. All the numbers and calculations lead to one final conclusion: the disease of interest rate risk still exists, even with proper immunization.
No.
It has an effect. Consider what you're really trying to do with immunization, which is based on Macaulay duration.
To summarize:
[[summary]]
The zero-coupon bond would have no variance of weighted cash flows
The zero-coupon bond would provide less return due to lack of reinvestment
The zero-coupon bond would provide the same return even with a yield curve shift
Period 5
Period 6
Period 8
0.8
1.6
3.2
No
Yes
No
Yes
Positive
Negative
It makes the immunization better
It makes the immunization worse
It doesn't affect immunization, just estimates
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