So they’ve told us that at a bond’s Macaulay duration, reinvestment income gains exactly offset present value losses.
That is, gains from reinvestment income of cash flows already received fully offset losses from discounting future cash flows to PV.
So I decided to check this.
I typed in all the cash flows from the " Cash Flow" column into a spreadsheet.
Then I typed in all the cash flows from the " Total Return at 3.7608%" column.
Then for each date, I subtracted the entry in the Cash Flow column from the entry in the Total Return column. This gave me the reinvestment income gains from all cash flows BEFORE the Macaulay duration date (15 Feb 2023), and the PV discount losses on all cash flows AFTER the Macaulay duration date.
I then added up all the reinvestment income gains, and all the PV discount losses. I expected they would be equal.
BUT THEY’RE NOT!
There is 10,797,665 in total reinvestment income, but a whopping 12,228,914 in PV discount losses.
WHY ARE THEY NOT EQUAL?
Shouldn’t they exactly offset each other? If not, why not? Am I misunderstanding something here?
The difference you find (12,228,914 minus 10,797,665) is the same as the difference between the sums of cash flows and total returns… (251,598,250 minus 250,167,000). I know this doesn’t answer but thought it might help.
Thanks. It’s the same because it is basically another way of adding up the same numbers.
Instead of finding the difference in cash flows for each date and adding up those differences, you’re adding up all the actual cash flows into one number and all the cash flows adjusted for reinvestment income and PV discounts into another number, and then finding the difference between those two sums.
I guess that helps me rephrase the question, though: Why isn’t the sum of the discounted cash flows and reinvested cash flows equal to the sum of the actual cash flows? I mean, the whole point is that the PV discount on future cash flows exactly offsets the gain on reinvesting past cash flows, right?
I must be missing something here, but I can’t figure it out and it’s driving me nuts!
I’m going to keep exploring this (thanks… by the way, for getting me into this problem it is 1am now)
You are comparing the value of the undiscounted cash flows to the value of the cash flows discounted to the date corresponding to the macaulay duration years past today. Is there a reason why these should match? I’ve thought about it for a while and I don’t see why they should.
If you change the date you are discounting to to any day other than the macaulay duration date, that difference you calculated balloons out. (eg: discount to 15-Aug-23 the difference is -3,272,891;15-feb-24, -8,065,488; 15-aug-22 +6,048,566… and if you discount to today… can you guess what difference you get? yes: the difference between market value and the undiscounted cash flows)
However, if you keep the discount date the same (macaulay duration) but play around with the IRR you generally get a similar discounted value, which is what the original exhibit in the curriculum is trying to demonstrate: no matter what IRR you end up with, so long as you match time horizon to macaulay duration, you don’t lose out from changes in interest rates.
I’m comparing the value of the undiscounted future cash flows, and un-reinvested past cash flows, as of the Macaulay Duration time, to the value of the discounted future cash flows and reinvested past cash flows.
The CFAI book says on page 53, that at the Macaulay duration, the reinvestment income from past cash flows exactly offsets the discount to PV of future cash flows. Yet in this big example they give us, that doesn’t seem to hold true. Either they made a mistake or I’m just not understanding the concept.
Could it be that the table doesn’t account for convexity?
If you compare the different total return columns you get very little change in value for a 1% move in IRR, but if you compare the 3rd and 4th columns as you did, you get a large difference when it should be zero - because the change in IRR is from 3.7608% to 0%. That’s a large move.
Edit: no matter what new IRR you choose, the new IRR results in a gain compared to the original IRR of 1.8804%. This sounds a lot like the convexity effect.
