Mechanics of Immunization
Once you have the idea in mind of how a coupon bond can provide interest rate immunization simply by choosing a bond where the Macaulay duration is the investment horizon, you might want to see how that can be managed.
In fact, it works with a single bond or a portfolio of bonds, and that's really nice. What can you easily adjust if you are using a portfolio of bonds for immunization?
Exactly!
You can choose bonds in order to create the Macaulay duration needed.
No.
That's up to the client. But with the portfolio, you can control the Macaulay duration by choosing bonds in order to create the Macaulay duration needed.
To demonstrate as simply as possible, consider two semiannual coupon bonds:
| $$\,$$ | Bond A | Bond B |
|-----|-----|-----|
| Coupon | 5.4% | 2.8% |
| Term | 2.5 years | 4.0 years |
| Price | 104.1 | 93.6 |
| Face Value | 1,189,000 | 1,000,000 |
| YTM | 3.6687% | 4.5688% |
About how many years is the investment horizon if this small portfolio of bonds is to immunize interest rate risk?
No.
Both bonds are offering face values at periods past just two years. It will have to be longer than this.
That's right.
In fact, it is precisely three years, or six periods from now. You can start to see this once you look at the cash flows.
| Period | Bond A | Bond B | Portfolio |
|-----|-----|-----|-----|
| 1 | 32,104 | 14,000 | 46,103 |
| 2 | 32,104 | 14,000 | 46,103 |
| 3 | 32,104 | 14,000 | 46,103 |
| 4 | 32,104 | 14,000 | 46,103 |
| 5 | 1,221,103 | 14,000 | 1,235,103 |
| 6 | $$\,$$ | 14,000 | 14,000 |
| 7 | $$\,$$ | 14,000 | 14,000 |
| 8 | $$\,$$ | 1,014,000 | 1,014,000 |
No.
All cash flows will have ended four years from now. It will have to be shorter than this.
But you need more. The Macaulay duration is the weighted average term to maturity of cash flows, which requires the present value of cash flows so that these can be weighted by dividing by the total portfolio value and then can be time weighted by multiplying each weight by the period.
The discount rate is available from that last table. The IRR of the portfolio cash flows is 2.0822%, suggesting a discount rate of 4.1643%. Which one would you think should be used for discounting?
Perfect!
These are semiannual cash flows, requiring a semiannual discount rate:
No.
It's 2.0822%, actually. These are semiannual cash flows, requiring a semiannual discount rate:
Again, these cash flows were discounted and then divided by the total portfolio value, given as the sum of present values at the bottom. That gets you the weights. Then the time weights are the weights multiplied by the period. Why do you suppose the time weights add up to exactly six?
You're right.
In fact, both responses were. It works out that way since the relative weighting of these two bonds was carefully chosen by the portfolio manager to be that way, and that value of six was chosen since it is the Macaulay duration of six periods, or three years.
You're right.
In fact, both responses were. It's the Macaulay duration of six periods, or three years, but it works out that way since the relative weighting of these two bonds was carefully chosen by the portfolio manager to be that way.
The next step in showing what this all means is to look at all cash flows compounded (or discounted) to Period 6, and to do this for different yields, representing potential yield-curve shifts. That's going to show what each early cash flow gets the investor as far as reinvestment, as these are compounded to Period 6. But what would the remaining cash flows discounted to Period 6 represent?
No.
That really doesn't make much sense. Think about what discounted cash flows usually represent.
Correct!
That's what discounted cash flows usually represent. Here are those cash flows adjusted to Period 6, using the portfolio's periodic IRR, and then -2% and +2%, to show shocks to the yield curve.
| Period | Cash Flows | Return at 4.1643% | Return at 2.1643% | Return at 6.1643% |
|-----|-----|-----|-----|-----|
| 1 | 46,103 | 51,107 | 48,652 | 53,659 |
| 2 | 46,103 | 50,064 | 48,131 | 52,055 |
| 3 | 46,103 | 49,043 | 47,616 | 50,499 |
| 4 | 46,103 | 48,043 | 47,106 | 48,989 |
| 5 | 1,235,103 | 1,260,820 | 1,248,469 | 1,273,171 |
| 6 | 14,000 | 14,000 | 14,000 | 14,000 |
| 7 | 14,000 | 13,714 | 13,850 | 13,581 |
| 8 | 1,014,000 | 973,057 | 992,405 | 954,269 |
| Sum | $$\,$$ | 2,459,848 | 2,460,229 | 2,460,223 |
No.
They won't sum to zero. Think about what discounted cash flows usually represent.
You're just about to the end now. You have the bond portfolio designed around a six-period investment horizon, and now all of the cash flows compounded or discounted to that investment horizon using three different discount rates. To show the immunization at this point, you just need the returns of each of those three scenarios. That is, divide the sum of the Period 6 values in each scenario by the 2,173,749 PV of cash flows determined earlier, and then annualize. For example, the 4.1643% return column would provide
$$\displaystyle 2 \left[ \left( \frac{2,459,848}{2,173,749} \right)^{\frac{1}{6}} - 1 \right] = 4.1643 \%$$.
It's perfect. Now if this strategy worked, what would the return column representing the lower yield curve show for a total return?
Exactly.
That's the whole idea. The investor was immunized against this move by having an investment horizon equal to the Macaulay duration of three years. So the return should be about the same as 4.1643%. A little positive error due to convexity, but nothing big.
In fact, the large yield curve drop would have made the return 4.1696%, and the large increase in the yield curve would have made the return 4.1695%. Just about the same. It worked!
Hopefully not.
A smaller total return would mean that the immunization effort failed.
Not really.
If it was bigger, then the immunization didn't work very well.
To summarize:
[[summary]]
| 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 Macaulay duration
The investment horizon
2
3
4
2.0822%
4.1643%
It's the duration
It was designed by the portfolio manager
Negative reinvestment
The bond price at that time
How all cash flows sum to zero
Something bigger
Something smaller
About the same thing
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