- 3028 - INTEREST - The Growth and Decay of Money? Learn the math for the growth and decay of money. Let’s study the growth of money. The math is very interesting and it applies to many situations in nature that experience growth or decay. It involves compound interest in the case of money. In nature it is any continuous rate of change, usually with time.
--------------- 3028 - INTEREST - The Growth and Decay of Money?
- Interest is really the price of money. The higher the interest rate the more valuable the money is. The interest rate is the rate of growth with time. In stable economics the interest rate is normally a point or two above the inflation rate.
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- The inflation rate is how fast prices are rising. Under ideal conditions the inflation rate is tied to the productivity which defines how efficiently the economy is growing.
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- Productivity is the ratio of the value of the output to the cost of the inputs. It all fits together as a natural process, ideally.
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- Let’s start with money and cover other natural processes later:
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- Starting with $100 and letting it grow at a 4% annual interest rate at the end of the first year you would have $104. - $100 is called the Principle and $4 is called the interest on that Principle. For our purposes we will call the $104 the “Future Value of money“.
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---------------------- Future Value = Present Value * ( 1 + interest rate) ^n
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------------------- where: “n” is the number of years at 4% per year.
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--------------------------- FV = PV ( 1+i)^n
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- If we let the money ride for another year, n = 2
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---------------------- FV = 100 ( 1.04)^2 = 100 ( 1.08)
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---------------------- FV = $108
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- If we let it ride for 10 years:
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-------------------- FV = 100 ( 1.04)^10 = 100 ( 1.48)
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--------------------- FV = $148
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- After 10 years the $100 has grown to $148. You have made $48 in interest on your money. If that principle were $100,000 you would have made $48,024.
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- When interest grows on top of interest we call that “compound interest”. Getting interest on your interest. The growth of compound interest is an exponential growth rate. It is the natural growth rate found in nature with most types of growth and decay.
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- In the money example we compounded yearly and got $148 after 10 years. What would you get if you compounded interest hourly instead of yearly? There are 8,760 hours in a year.
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------------------- FV = PV ( 1 + i /n)^n
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-------------------FV = 100 ( 1+ 0.04/8760)^ 87,600
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------------------ FV = $149.18
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- Compounding hourly only gained another $1.16. So, what would happen if you compounded continuously as normally happens in nature?
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- We introduce the natural constant “e” which is the base of natural logarithms. A logarithms is another name for “exponent“, and, we are working with “exponential growth” . e^x is the only function that has a rate of change that is the same e^x.
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-------------------- FV = PV * e^n*I
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-------------------- FV = 100 * 3^10*(0.04) = 100 * ( 1.4918)
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--------------------- FV = $149.18
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- Compounding interest continuously did not get us any more growth in interest over 10 years of compounding hourly. But, note that the exponential curve gave us the same answer, $149.18.
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- Imagine a x-y plot and y = e^x as the exponential curve, y is a function of 2.7 raised to the power of “x”. What is unique about the base “e”, which is a constant that has a value of 2.718, the value of the function is the same as the slope of the curve. The slope is rise over run, which is (delta y) / (delta x).
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----------------------------- y = e^x
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----------------------------- dy / dx = e^x
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- For an exponential curve the value of the function is the same as the change in y to a change in x as you travel up the curve. dy / dx is the tangential slope of the curve. e^x has some other unique properties. It can be expanded into an infinite series.
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---------------- y = e^x = 1 + x / 1 + x^2 / 2! + x^3 / 3! + x^4 / 4! + x^5 / 5! + x^6 / 6! + x^7 / 7! + ………
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- 3! is 3 factorial which is shorthand for 3 * 2 * 1, just as 5! is 5 * 4* 3 * 2 * 1
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- When x + 1 the function y = e which = 1 + 1 / 1 + 1^2 / 2 + 1^3 / 6 + 1^4 / 24 + 1^5 / 120 + 1^6 / 720 + 1^7 / 5040 + ………
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- y = e = 2 + 0.5 + 0.167 + 0.0417 + 0.008 + 0.001389 + 0.0001984 + ………
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---------------- e = 2.718254
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- Note that the series for compound interest is the same type of exponential function:
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-------------- FV = PV ( 1+i) + PV ( 1+i)^2 + PV ( 1+i)^3 + PV ( 1+i)^4 + PV ( 1+i)^5 + ……… PV ( 1+i)^ n
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--------------- FV = PV e ^ n*i
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- This is the equation for all exponential functions. The equation appears everywhere in nature, not just in banking and mortgages. Any situation where the amount (A) of something varies with the time , such that the time rate of change of (A) is proportional to (A) itself.
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----------------------dA / dt = k* A
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- Solving this differential equation gets us to the same formula with different symbols:
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--------------- A = Ao * e^k*t
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- This equation could be the growth of money, the decay of a voltage across a capacitor, the cooling of a voltage regulator in your car. The voltage across a capacitor decays at a rate proportional to the voltage. It the voltage is 20 volts and 2 seconds later is 10 volts it has a decay rate of - 10 volts / second. When will the voltage across the capacitor be 2 volts? ANSWER: 6.64 seconds:
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---------------- A = FV = 10 volts
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---------------- Ao = PV = 20 volts, the initial voltage at time equal zero.
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--------------- k = i = the rate of decay. If it were linear it would be - 10 volts per second. But, it is exponential and we have to solve for it.
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--------------- t = time = 2 seconds
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--------------- FV = PV * e^i*t
---------------- 10 = 20 * e^ 2 i
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- The unknown “i” is in the exponent so we need to take the natural logarithm.
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----------------- 10 / 20 = e^2i
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----------------- e^ 2i = 0.5
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- The natural logarithm to the base “e” of 0.5 = the exponent 2i
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-------------------- log e (0.5) = 2*I
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------------------- -0.693 = 2 * I
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-------------------- - 0.3465 = i
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- In the next interval of time the capacitor discharges from 10 volts to 2 volts at this constant decay rate of - 0.3465 volts per second
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-------------------- 2 = 10 * e^-0.3465 * t
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------------------- log e 0.2 = -0.3465* t
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------------------- - 1.605 = -0.3465 * t
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-------------------- t = 4.64 seconds
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- The voltage across the capacitor decays at an exponential rate. It will decay from 20 volts to 10 volts in only 2 seconds , and, an additional 4.64 seconds for the capacitor to decay down to 2 volts. A total time of 6.64 seconds.
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- How long does it take an amount of money to double if the investment is growing at 10% compounding interest?
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------------------- FV = 2
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------------------- PV = 1
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--------------------- n = ?
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-------------------- i = 10%
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--------------- FV = PV * e^n*I
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--------------- 2 = e ^ n*(0.1)
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--------------- log 2 = 0.1 * n
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-------------- 0.693 = 0.1*n
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--------------- n = 6.93 years
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- It takes 7 years for money to double growing at 10% per year. You will learn i * n = 70 later on:
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--------------- FV = PV * e^n*I
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---------------- FV = $100 * e ^0.7 = 100 * ( 201)
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--------------- FV = $201
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- Objects cool at an exponential rate, temperature decreases at a rate proportional to the difference of the units temperature (u) and the temperature of the surrounding environment (T).
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------------------- du / dT = k (u-T)
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- Integrating to solve the differential equation:
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---------------- (u - T) = (uo - T) * e^ k*t
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---------------- (u - T) = FV = the future temperature difference, then:
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----------------- FV = PV * e^n*i
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- The car’s regulator is at 90 decrees C and the garage is at 20C. Turn off the ignition and 10 minutes later the regulator cools to 60 C. At what temperature will it be after 20 minutes?
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----------- PV = 90 - 20 = 70
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---------- FV = 60 - 20 = 40
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----------- i = rate of decay in degrees per minute. ( - 3 degrees / minute if the decay was linear.)
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------------ n = is in 10 minute intervals and that doubles in time n = 1 for each interval.
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------------ 40 = 70 * e^1* I
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----------- 0.5714 = e^I
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----------- log e (0.5714) = I
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------------ ( - 0.56) = i
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- The next interval drops the temperature from 60 degrees , which is a difference of 40 degrees above ambient to some temperature u = ?.
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--------------- FV = u - T = 40 * e^-.56
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--------------- u - T = 40 / 1.75 = 23
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------------- u = 20 + 23 = 55 C
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- After 20 minutes the regulator cooled 90 - 55 = 35 degrees to get to 55C.
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- “e” has many unique properties: “e” is a “transcendental” number, like pi, it is a series of decimals that goes to infinity and never repeats itself. It is an “irrational” number that can not be written as a ratio of two integers.
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--------------- e^ square root of -1 . pi + 1 = 0 ( think about it )
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----------------- e ^ix = cos x + i sin x
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------------------ i * t = 70
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- By dividing the interest rate into 70 you get how many years it will take for the investment to double. This simple equation has a multitude of every day uses:
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- If we double our national debt in 10 years what is the rate of growth of spending?
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------------------------ 70 / 10 years = 7% per year.
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- If the population is growing at 2% per year when will it double?
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---------------------- 70 / 2% = 35 years
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- If you got a 5% mortgage for 30 years, when will you double the amount of your loan with interest?
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------------- 70 / 5% = 14 years.
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- Put one penny in a kitchen jar on December 1st and double it every day. How much will you have in time for Christmas?
------------------------ $117,772
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- Google’s official revenue target was $2,718,281,828. ( get it? )
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----------------------------- Other reviews on math:
- 1329 - The most important math you will ever learn.
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- 1281 - Math was invented to solve problems.
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February 6, 2021 INTEREST - Growth of Money? 1467 3028
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--------------------- --- Sunday, February 7, 2021 ---------------------------
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