Search Results for “%22Problems with Constant Compound Interest%22”

Separate Processes

Posted on 18 December 2021 by David Merkel

Photo Credit: atramos || Inflation isn’t the most organized phenomenon, and investors often all want to be on the same side of the boat…

I have a very irregular series called, “Problems with Constant Compound Interest.” Part of the idea of that series is that it is difficult to assure growth in capital in any sort of constant way. The simple models of the CFPs, and even actuaries that assume constant or near constant growth are ultimately doomed to fail if they try to exceed growth in nominal GDP by more than 2%/year.

Because of the oddities in the current market environment, current interest rates and inflation have decoupled. They are separate processes. We all want to build value in real (inflation-adjusted) terms, but how do you do that in an environment where price to free cash flow multiples are sky high, nominal interest rates are low, and the prices of most commodities are high as well (leaving aside gold as an oddity). Mindless stock bulls talk of TINA [There Is No Alternative (to buying stocks)], as if there is no limit to how high stock prices can go when interest rates are low. I want to tell you about TIN. There Is Nothing (worth buying). This is the nature of financial repression.

If you invest in short bonds, you get gouged by current inflation. If you buy long bonds, you run the risk that the Fed might start monetizing Treasury debt directly, and inflation really runs. With stocks you run the risk of any hiccup in the global economy (when is the omega variant coming so that we can move on to Hebrew letters?) can derail the market, particularly if it leads to higher interest rates.

The Fed has gotten its wish and is forcing an asset bubble on the US to aid growth, however fitfully. All of the relationships of the present to the future are out of whack, because interest rates are too low. But if intermediate interest rates rose to the level of nominal GDP growth, we would see deficits grow even more rapidly as the US government would refinance at higher rates. The Fed is stuck in a doom loop of its own design ever since Alan Greenspan got the great idea to cut short recessions too soon. That has led us into a liquidity trap designed by the Fed.

As I said to one of my clients this week (a bright man), “If you are not bewildered, you are not thinking.” About the only idea I can think of for investing at present is the intersection of high quality and low-ish valuations. As it says in Ecclesiastes 11:2 “Give a serving to seven, and also to eight, For you do not know what evil will be on the earth.” Diversify among safe-ish investments, with a few cyclicals that will do well if things run hot, and stable businesses, if things do not.

That’s all.

The Doubling Rule

Posted on 21 June 2017 by David Merkel

Picture Credit: Vincent Brown || Einstein never said this, either…

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If you are famous and dead, many people will attribute clever sayings to you that you never said. ?As Yogi Berra said:

I really didn?t say everything I said.

Except that Yogi did say that.? Now, if Einstein didn’t do enough for us, he supposedly made many statements praising compound interest, but the articles I have seen haven’t been able to trace it back to an original source. ?Personally, I think compound interest is overrated, because business processes can’t forever compound wealth at a steady rate.

But some also have tried to credit Einstein with the Rule of 72. ?You know, the rule that says the time it takes to double your money in years is equal to 72 divded by the annual compound interest rate expressed as as an integer. ?E.g., at 8% your money doubles in 9 years. ?At 9% you money doubles in 8 years.

Pretty nice, and easy to remember. ?It is an approximation though. ?If Einstein ever did look at the rule of 72, he would have noticed that the approximation is pretty good between 3% and 13%. ?Outside that it gets further away.

One advantage of the rule of 72 is that it is simple. ?The second one is that 72 = 3 x 3 x 2 x 2 x 2. ?That makes it divide more intuitively by many integral interest rates, e.g., 3, 4, 6, 8, 9, 12 — allowing for some intuitive interpolation to aid it.

There is a more complex version of the doubling rule though:

The doubling constant starts (in the limit) at 100 times the natural logarithm of 2 [69.3147], and increases almost linearly from there. ?If you estimate a “best linear fit” line on the observations where the interest rate is between 0 and 30, the R-squared will be over 99.98%. ?The equation would be:

K = 69.3856 + 0.3313 * interest rate ?[Linear Fit K]

To make it a little more memorable rule, it can be turned into:

K = 70 + (interest rate – 2)/3 ?[Rule K]

Thus at 2% the doubling constant would be 70 — money doubles in 35 years. ?At 5%, 71, money doubles in 14.2 years. ?8% is the rule of 72 — nine years to double. ?At 11%, 73, 6.6 years to double. ?At 14%, 74, 5.3 years to double. 17%, 75, 4.4 years to double. 20%, 76, 3.8 years to double. ?I did those in my head.

As you can see from the graph above, the actual doubling constant and its two approximations lie on top of each other. ?Not that I hope we see ultrahigh interest rates, but Rule K does quite well over a long span of rates. ?Here’s how small the deviations are:

Now, almost no one will use “Rule K” because the two advantages of the Rule of 72 are huge, and if interest rates get really high, someone could create an easy smartphone app to calculate the doubling period. (and constant if they wanted)

This is interesting for me, because I ran across what I call the “Rule of K” earlier in my career, and I was able to reproduce it on my own after reading the WSJ article that I cited above. ?Who knows, maybe Einstein took the doubling rule and did a first order Taylor expansion around 2% — that would have produced something very close to the “Rule of K” back when regressions were hard to do.

That’s all, and if you made it this far, thanks for bearing with me.