MoneyMarathon wrote: ↑
Tue Aug 13, 2019 10:00 pm
MotoTrojan wrote: ↑
Tue Aug 13, 2019 8:41 pm
These are not two portfolios held separately then added together, they are means of simulating each assets contribution to the portfolio as a whole. If one of them goes up 10x and the other 3x, then the total growth was 30x, not 13x.
Let me try to explain the hand-wavy math I used (but... be warned it might only be more confusing than the original hand waving).
Weighted average rate of returns is a weighted sum of the rate of growth of each investment. Wikipedia page:
https://en.wikipedia.org/wiki/Rate_of_r ... _portfolio
When I set up my little experiment, I calculated a total return (over time) that had an associated CAGR that was tied to a larger amount of capital than what was relevant (e.g., the whole 100% instead of the 40% in UPRO). To get the weighted sum of the rate of growth of each investment, we'd first want to get the CAGR based on each individual smaller amount of capital allocated to the investment. But, if we use these weird, unadjusted whole-portfolio CAGR to get a weighted sum, then the weighted sum: 0.4 * (x / 0.4) + 0.6 * (y / 0.6) = x + y, where x and y are the respective CAGRs of the 40% UPRO/60% CASHZERO and 60% TMF/40% CASHZERO rebalanced portfolios. So the weighted sum for the rate of growth is just x + y of the weird (unadjusted to the actual amount of principal for the individual investment, and instead looking at contribution to the whole portfolio) CAGRs. So expected portfolio CAGR = x + y, with steady growth (and without including any bonus effects from rebalancing negatively correlated investments).
A = P * e ^ (rt) is one expression of the exponential growth formula. And if r = x + y, then:
A = P * e ^ ( (x + y) t) = P * e ^ (x t) * e ^ (y t)
Now if we compute B = P * e ^ (x t) / P = e ^ (x t) and compute C = P * e ^ (y t) / P = e ^ (y t), where B is the total return on cash over the whole time period for the 40% UPRO/60% CASHZERO portfolio and where C is the same thing for 60% TMF/40% CASHZERO, and where P is the original principal, then:
P * B * C = P * e ^ (x t) * e ^ (y t) = P * e ^ ( (x + y) t) = P * e ^ (rt) = A
And to the extent that the observed result is different from P * B * C, that looks like the effect of the 'rebalancing bonus' from negative correlation. Note again that it's larger for the quarterly rebalancing, so the 0.5% CAGR figure that was thrown around does not
necessarily apply to quarterly rebalancing. The approximate 0.5% CAGR figure was based on looking at annual rebalancing over the last 32 years. So anyone using quarterly rebalancing may want to include an appropriately larger bonus from rebalancing based on that method.
This is pretty cool, but there's a slight problem, which is mixing CAGR (compound annual
growth rate) and CCGR (compound continuous
Start with this:
But, if we use these weird, unadjusted whole-portfolio CAGR to get a weighted sum, then the weighted sum: 0.4 * (x / 0.4) + 0.6 * (y / 0.6) = x + y, where x and y are the respective CAGRs of the 40% UPRO/60% CASHZERO and 60% TMF/40% CASHZERO rebalanced portfolios.
Using numbers from the original example, x = 7.77% and y = 8.07%, so x + y = 15.84%. Using the CAGR formula for 32 years:
Cumulative return factor = (1 + 15.84%)^32 = 110.5.
But the individual cumulative growth factors for 40% upro and 60% tmf are 10.96 and 11.98 respectively, and 10.96 * 11.98 = 131.4 (slight differences from original example because I start with the CAGR numbers rounded to 2 decimal places, rather than the return factors).
I had already worked out the portfolio weighting math, and calculated the 110.5 value, so I PM'd MoneyMarathon about the discrepancy with the calculated value of 131.X, and MM kindly PM'd me the derivation before posting it in the thread.
To resolve this discrepancy, let's calculate CCGR for growth factors 11.98 and 10.96 for 32 years:
CCGRu = LN(11.98)/32 = 7.48%
CCGRt = LN(10.96)/32 = 7.76%
These are of course slightly smaller than CAGR values, since continuous compounding results in larger cumulative growth for a given growth rate. Using the derived approach, we add these to get 15.24% as the weighted CCGR, and:
e^(rt) = e^(15.24% * 32) = 131.4
This verifies the derivation; i.e., e^(CCGR * t) is the product of the cumulative growth factors, 10.96 * 11.98 = 131.4.
But again, multiplying the individual growth factors does not give the correct growth factor for annual compounding. To get this, simply calculate:
(1 + x + y)^t
Which as shown above, gives a cumulative growth factor of 110.5.