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Kindle Notes & Highlights
by
Kalid Azad
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December 3 - December 19, 2019
Negative exponential growth is just another way to change: you shrink, instead of grow.
Negative exponential growth
e raised to “r · t” gives you the growth impact of rate r and time t.
The natural log gives you the time needed to reach a certain level of growth.
Intuitive definition of ln
If you want 10x growth, assuming continuous compounding, you’d wait only ln(10) or 2.302 years.
ex is the amount of continuous growth after a certain amount of time The natural log (ln) is the amount of time needed to reach a certain level of continuous growth
e vs ln
By converting to a rate of 100%, we only have time to think about: ex = erate · time = e1.0 · time = etime
ex means: How much growth do I get after after x units of time (and 100% continuous growth)
Latin name is logarithmus naturali, giving the abbreviation ln.
ex lets us plug in time and get growth. ln(x) lets us plug in growth and get the time it would take.
e vs ln
e3 is 20.08. After 3 units of time, we end up with 20.08 times what we started with. ln(20.08) is about 3.
Example of e vs ln
What is ln(1)? Intuitively, the question is: How long do I wait to get 1x my current amount?
ln(1)
ln(.5) = –ln(2) = –.693
In general, you can flip the fraction and take the negative: ln(1/3) = –ln(3) = -1.09.
ln(<1)
ln(negative number) = undefined
Time to grow 4x = ln(4) = Time to double and double again = ln(2) + ln(2)
ln(a · b) = ln(a) + ln(b)
Well, growing 5 times is ln(5). Growing 1/3 is –ln(3) units of time. So ln(5/3) = ln(5) – ln(3)
ln(a/b) = ln(a) – ln(b)
Don’t memorize the rules, understand them.
Indeed!
“100% return for 3.4 years is 30x growth”. We can consider the equation to be: ex = erate · time e100% · 3.4years = 30
The natural log can be used with any interest rate or time as long as their product is the same.
The Rule of 72 is a mental math shortcut to estimate the time needed to double your money.
Rule of 72
69.3/5 or 13.86 years. However, 69.3 isn’t the most divisible number. Let’s pick a close neighbor, 72, which can be divided by 2, 3, 4, 6, 8 and many more numbers. time to double = 72/rate which is the rule of 72!
Rule of 72 eases calculation
The Rule of 72 is useful for interest rates, population growth, bacteria cultures, and anything that grows exponentially.
I consider it “natural” because e is the universal rate of growth, so ln could be considered the “universal” way to figure out how long things take to grow.
Why "natural" log?
e is the amount of growth after 1 unit of time, so ln(e) = 1.
APR (annual percentage rate): The rate someone tells you (“12% per year!”). You’ll see this as “r” in the formula.
APR
APY (annual percentage yield): The rate you actually get after a year, after all compounding is taken into account. You can consider this “total return” in the formula. The APY is greater than or equal to the APR.
APY
Getting a credit card or car loan? They’ll show the “low APR” you’re paying, to hide the higher APY. But opening a savings account? Well, of course they’d tout the “high APY” they’re paying to look generous.
Depending on the case, bank will show APR or APY
APY (actual yield) is what you care about, and the way to compare competing offers.
Simple interest pays a fixed amount over time.
Simple interest rate
Simple interest has a simple formula: Every period you earn P · r (principal · interest rate). After n periods you have: interest = P · r · n This formula works as long as “r” and “n” refer to the same time period. It could be years, months, or days
In practice, simple interest is fairly rare because most types of earnings can be reinvested.
Compound growth means your interest earns interest.
Compound growth
In general, we have (1 + r) times more “stuff” each year. After n years, this becomes: total = P · (1 + r)n
Compound growth
Yearly GDP growth of 3% over 10 years is really (1.03)10 = 1.344, or a 34.4% increase over that decade.
Annual payouts are man-made artifacts, used to keep things simple. But in reality, money should be earned all the time.
For 1 year, the impact of rate r compounded t times is: (1 + r/t)t In our case, we had (1 + 50%/2)2. Repeating this for n years (multiplying n times) gives: total = P · (1 + r/t)tn
Continuous growth is compound interest on steroids: you shrink the gap into oblivion, by dividing the year into more and more time periods:
If we have rate r and time t (in years), the result is: total = P · ert If you have a 50% APR, it would be an APY of e.50 = 64.9% if compounded continuously.
ert is the adjustable, one-size-fits-all exponential. It sounds strange, but e can even model the jumpy, staircase-like growth we’ve seen with compound interest.
e^rt is flexible
Most interest discussions leave e out, as continuous interest is not often used in financial calculations. (Daily compounding, (1 + r/365)365, is generous enough for your bank account, thank you very much. But seriously, daily compounding is a pretty good approximation of continuous growth.)
Reality of bank interests
Should I pay my mortgage at the end of the month, or the beginning? The beginning, for sure. This way you knock out a chunk of debt early, preventing that “debt factory” from earning interest for 30 days.
Paying mortgage
prepayment will save you money.
It's better to pay every week than every month
The general principle: When investing, get interest paid early, so it can compound. When borrowing, pay debt early to prevent that interest from compounding.
General principle
Man-made growth uses (1 + r)n, or some variant. We like our loans to line up with years. Nature uses ert. The universe doesn’t particularly care for our solar calendar.
Summarising
Addition is sliding along the number line (+3 means slide 3 to the right) and multiplication is scaling (×3 means scale it up 3x).
New concept of addition and multiplication
original · growthduration = new or growthduration = new / original For example, 32 = 9/1.
Exponent
“Start with 1 and double 3 times” means 1 · 23= 1 · 2 · 2 · 2 = 8 “Start with 3 and double 3 times” means 3 · 23 = 3 · 2 · 2 · 2 = 24
Most things in nature don’t know where they’ll end up! Do you think bacteria plans on doubling every 14 hours? No — it just eats the moldy bread you forgot about in the fridge as fast as it can, and as it gets bigger the blob grows even faster
