Compound Amount Formula with Unknown Interest Rate and Time

Summary

This video script teaches the derivation of the compound interest rate formula and its application. It explains how to find the nominal rate 'j', compounded quarterly, using the formula j = i * m, where 'i' is the interest rate per period and 'm' is the number of compounding periods per year. The script also demonstrates solving for time 't' in compounding periods 'n' using logarithms. Practical examples are given, such as calculating the time for an investment to double with a 6% semi-annual interest rate.

Takeaways

• 🧮 The compound amount formula helps solve for present and future values of loans or investments.
• 📈 This video focuses on deriving formulas for the interest rate (i) per compounding period and the number of compounding periods (n).
• 💰 Example problem: A loan of 40,000 pesos accumulates to 100,000 pesos in 10 years with quarterly compounding; the task is to find the nominal interest rate (j).
• 📊 To find the number of compounding periods, use the formula n = m * t, where m is the number of periods per year (4 in this case) and t is the time in years.
• 🔢 The nominal rate (j) is related to the interest rate (i) by the equation j = i * m, where m is the number of compounding periods per year.
• ⚖️ The formula for the interest rate (i) per compounding period is derived as i = (nth root of (F/P)) - 1, where F is the future value and P is the present value.
• 🧮 In the example, substituting values gives an interest rate (i) of 0.0232 (or 2.32%) per compounding period.
• 📉 The nominal rate (j) is found by multiplying the interest rate per period by the number of periods per year: j = 2.32% * 4 = 9.28%.
• 🕰️ For time-related problems, such as finding the duration required for a principal of 60,000 pesos to reach 85,000 pesos with a 6% interest rate compounded semi-annually, the formula for n is used.
• 🔐 To solve for n (the number of periods), logarithms are applied to the compound amount formula, giving n = log(F/P) / log(1 + i). Then, time (t) is calculated as t = n/m.

Q & A

What is the purpose of the compound amount formula?

The compound amount formula is used to find the present value and future value of loans or investments over time.

What unknowns does the video focus on solving within the compound amount formula?

The video focuses on deriving formulas for the interest rate per compounding period (i) and the number of compounding periods (n).

In the example provided, what are the given values to solve for the nominal interest rate (j)?

The given values are a present value of 40,000, a future value of 100,000, and a period of 10 years with quarterly compounding, meaning m = 4.

How is the number of compounding periods (n) calculated in the problem?

n is calculated as the product of the number of compounding periods per year (m) and the time in years (t), which in this case is n = 4 * 10 = 40 quarters.

What is the relationship between the nominal rate (j) and the interest rate per period (i)?

The nominal rate (j) is equal to the interest rate per period (i) multiplied by the number of compounding periods per year (m), i.e., j = i * m.

How is the interest rate per period (i) derived from the compound amount formula?

The interest rate per period (i) is derived by rearranging the formula for the future value (F = P * (1 + i)^n). After dividing both sides by P, taking the nth root, and subtracting 1, the formula becomes i = (nth root of F/P) - 1.

How is the nominal rate (j) calculated once the interest rate per period (i) is found?

The nominal rate (j) is found by multiplying the interest rate per period (i) by the number of compounding periods per year (m). In the example, j = 2.32% * 4 = 9.28%.

How can a scientific calculator or smartphone be used to solve for i when calculating roots with large indices like 40?

On scientific calculators, you can directly input the formula. On smartphones, which may not support high index roots, you can convert the root to a fractional exponent. For example, instead of taking the 40th root, you can raise the fraction to the power of 1/40.

What is another common unknown in the compound amount formula, as shown in the second example?

Another common unknown is time (t), which is related to the number of compounding periods (n). The second example demonstrates how to solve for time.

How is time (t) derived from the number of compounding periods (n) in the second example?

Time (t) is calculated by dividing the number of compounding periods (n) by the number of compounding periods per year (m). In the second example, t = 11.78 / 2 = 5.89 years.

Outlines

00:00

📚 Deriving Interest Rate Formulas

This paragraph introduces the compound amount formula, focusing on finding the present and future values of loans or investments. It explains the derivation of the interest rate per compounding period (i) and the number of compounding periods (n). An example problem is provided where $40,000 grows to $100,000 in 10 years with quarterly compounding, requiring the calculation of the nominal rate (j). The derivation of i is shown using the compound amount formula F = P(1+i)^n, leading to the formula i = (F/P)^{1/n} - 1.

05:02

🔢 Calculating the Nominal Rate

This paragraph continues the example, showing the calculation of the interest rate per compounding period (i), which is found to be 0.0232 or 2.32%. The nominal rate (j) is then calculated by multiplying i by the number of compounding periods per year (m), resulting in a nominal rate of 9.28%. Instructions are provided for calculating this using both a standard and a smartphone calculator, including converting the 40th root into a fractional exponent.

10:05

⏳ Determining Time in Compound Interest

This paragraph introduces another example problem where $60,000 grows to $85,000 with a 6% interest rate compounded semi-annually, focusing on finding the time required. The relationship between time (t) and the number of compounding periods (n) is explained, leading to the formula n = log(F/P) / log(1+i). Substituting the given values results in n = 11.78, which, when divided by the number of compounding periods per year (m = 2), gives a time of 5.89 years.

15:10

🎉 Conclusion and Final Notes

The final paragraph wraps up the discussion, summarizing the process of deriving and calculating various components of the compound amount formula. It reinforces the steps taken to solve for the nominal rate and the time required for investments to grow under given conditions, emphasizing the practical application of these formulas using different types of calculators.

Keywords

💡Compound Amount Formula

The compound amount formula is used to calculate the future value of an investment or loan based on compound interest. It connects the present value (initial amount), the interest rate per period, and the number of compounding periods. In the script, this formula is central to understanding how to derive the interest rate per period and nominal rate, given a present and future value.

💡Present Value (P)

Present value represents the initial amount of money invested or loaned. In the video, the problem given uses a present value of 40,000 pesos, which is the starting amount before interest is applied. It plays a crucial role in determining how much this investment grows over time under compound interest.

💡Future Value (F)

The future value is the amount accumulated over time, including interest. For example, in the given problem, the future value is 100,000 pesos after 10 years of compound interest. This value helps to illustrate how investments grow and is essential when deriving the interest rate or determining the growth period.

💡Nominal Rate (J)

The nominal rate is the annual interest rate before taking compounding into account. It is often denoted by 'J' and needs to be adjusted based on how frequently interest is compounded. The video shows how to derive the nominal rate from the interest rate per period by multiplying it by the number of compounding periods (m).

💡Compounding Periods (N)

Compounding periods refer to the total number of times interest is added to the principal over the investment or loan duration. It’s calculated as N = m × t, where 'm' is the number of times interest is compounded per year, and 't' is the total number of years. In the video, an example of 40 quarters (or compounding periods) is calculated for a 10-year period with quarterly compounding.

💡Interest Rate per Period (i)

The interest rate per period represents the rate applied at each compounding interval. It’s derived from dividing the nominal rate (J) by the number of compounding periods per year (m). The script demonstrates how to calculate 'i' using the formula and how it affects the accumulation of the present value into the future value over time.

💡Logarithms

Logarithms are mathematical tools used to solve equations involving exponents. In the context of the video, logarithms help isolate the exponent (N) in the compound amount formula when solving for the number of compounding periods. This is crucial for understanding the time it takes for an investment to reach a certain future value.

💡Nth Root

The nth root is used to remove an exponent in equations involving powers. It’s applied when deriving the formula for the interest rate per compounding period (i) from the compound amount formula. In the video, extracting the 40th root helps find 'i' for the problem where interest is compounded quarterly over 10 years.

💡Semi-Annually

Semi-annually refers to interest being compounded twice a year. In the second example from the video, the interest rate is compounded semi-annually, leading to an m-value of 2. Understanding semi-annual compounding is crucial when adjusting the nominal rate to find the actual interest rate per period.

💡Fractional Exponent

A fractional exponent is an alternative way to express a root, such as the nth root being equivalent to raising a number to the power of 1/n. The video explains how to convert the 40th root into a fractional exponent (1/40) for calculation purposes, particularly when using a standard calculator that doesn’t directly support roots beyond the square root.

Highlights

The compound amount formula is used to find the present and future value of loans or investments.

The formula for the interest rate per compounding period (i) is derived from the compound amount formula.

The nominal rate (j) is the product of the interest rate per period (i) and the number of compounding periods per year (m).

A step-by-step guide on deriving the formula for i is provided.

The formula for i is i = (nth root of (F/P)) - 1, where F is the future value, P is the present value, and n is the number of compounding periods.

The nominal rate (j) can be calculated by multiplying i by m, where m is the number of compounding periods per year.

A practical problem is solved to find the nominal rate when given the present value, future value, and time period.

Instructions on how to use a calculator to find the value of i are provided.

The concept of converting the index of a radical into a fractional exponent is explained for ease of calculation.

Another problem is presented to find the time it takes for a principal to reach a certain amount with compound interest.

The relationship between time (t), the number of compounding periods (n), and the number of compounding periods per year (m) is explained.

A formula for determining the number of compounding periods (n) is derived using logarithms.

The formula for n is n = log(F/P) / log(1 + i), where F is the future value, P is the present value, and i is the interest rate per period.

The time (t) in years is calculated by dividing n by m.

Instructions on how to use a scientific calculator or smartphone calculator to find the value of n are given.

The importance of understanding the compound amount formula for financial calculations is emphasized.

Transcripts

the usual problem that we solve under

the compound amount formula is finding

the present value and the future value

of any loan or any investment

in this video i'm going to show you how

to derive the formula

for i which is the interest rate

per compounding period and n

which is the number of compounding

periods

and also we're going to solve some

problems relating to these

two other parts of the compound amount

formula

so for instance given this problem if 40

000

accumulates 2 100 000 in 10 years

find the nominal rate if the interest

rate is compounded

quarterly so in this problem the unknown

is

the nominal rate which is

denoted by j and this is compounded

quarterly therefore m is equal to

4 also we are given

a present value of 40 000

and the future value of this 40 000

is equivalent to 100 000

pesos also we are given time

which is equivalent to 10 years

since this is compounded quarterly

therefore the number of compounding

periods

given by the formula n equals m times t

where m is 4 and

t is equal to 10 therefore we have

40 quarters we all know that

in the compound amount formula j is not

present

but we know that j is related to i

which is part of the compound amount

formula

where i is equal to the nominal rate

divided by the number of compounding

periods

cross multiplying m to i therefore

j is equal to i times m or the nominal

rate is

equal to the interest rate per period

times the number of compounding periods

in a year

now we're going to derive the formula

for

the interest rate or the formula for

i we know that the compound amount

formula

is given by f is equal to p

times 1 plus i

raised to exponent n

so we're going to solve for i or

determine the formula of i so first

we divide both sides by p

this will cancel out on the right side

we're going to have 1 plus i raised to n

is equal to the ratio of the future

value and

the present value then we're going to

extract the n root

to cancel this exponent so nth root of

the left side also the nth root of

the right hand side

this will cancel out what is left

on the left side is 1 plus

i and on the right side is nth root of

f over p

and finally we're going to transpose 1

on the other side of the equation

therefore

i is equal to the nth root of

f over p minus

1. this will be our formula in

determining

the interest rate per compounding period

so in the problem so f is the future

value

p is the present value and

n is equal to 40. so therefore

so substituting those values

ends equal to 40 so that's the 40th root

of the future value over p that's

one hundred thousand divided by

forty thousand

minus one take note that

this minus 1 is not part of the

radical part of the formula

so this value is equal to

so the value of i is equal to taking

four decimal places

this is zero point zero

two three two

we're going to multiply this by 100

percent

this is equal to

two point thirty two percent

and finally the nominal rate is j is

equal to i

times m therefore we're going to

multiply

m or for this case

we're going to multiply m is equal to 4

as indicated in the problem since it is

compounded quarterly

to determine the value of j

so that's two point thirty two

percent times four which is equal to

nine point twenty eight percent

if you're having difficulty in

determining this value using your

calculator so particularly if you're

using

cash calculator the input should be

40 shift

card sign

open parenthesis

100 000 divided by

forty thousand minus one and you should

get

this value and if you're using the

calculator

the standard calculator on your

smartphone which is typically cannot

solve

radicals or higher than an

index of two we can

we can use this by converting

the index of 40 into a

fractional exponent the index of 40 in

this

radical into a fractional exponent

and this part is equivalent to an

exponent

of 1 over 40.

so on your smartphone you can enter

100 000 divided by

40 000 raised to

exponentiation usually that's

y raised to x or x raised to y on your

smartphone on the calculator of your

smartphone

then the radical part index of 40

is 1 divided by 40

and then we subtract 1.

on your calculator on the smartphone

will give you

this value

another possible unknown in the compound

the one formula

is the time and here's an example

problem

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so how long will the principal of 60 000

reach to an amount of 85 000 if it earns

six percent compounded semi-annually

so the question how long pertains to

the time

and under compound the mod formula time

is related to

n where n is equal to

m times p therefore

p is equal to n over

m

and the other given in the problem given

principle or the present value

equal to 60 000

and a future value of

85 000

are also given the nominal rate

equal to six percent compounded

therefore the value of i is equal to

j over m is equal to 6

divided by 2 is equal to

3 or

0.03 in decimal and after

identifying the given we're now going to

derive the formula for

n so again the compound amount formula

is f is equal to p one plus

raised to n we're going to determine a

formula for

n which is an exponent of this part

so first we're going to divide both

sides by e

on the right side and what is left is

one plus i is equal to

f over t so we're going to solve for the

value of

n since n is an exponent we're going to

apply the concept of logarithms

we're going to apply this property

log of x raised to n is equal to

n log of x

so applying logarithms on both sides

so this will give us log of

1 raised to n is equal to

log of f

over p

like this property will give us n

log of 1 plus i

is equal to log of

f over b then we can solve for

x dividing both sides by

log off one plus i

this part will cancel out therefore

n is equal to

log of f over p

divided by log of

one plus r

so in the problem we are given the value

of f

which is equal to eighty eighty-five

thousand the value of p is sixty

thousand

and the value of i is three percent

substituting those values will give us

log of 85 000

divided by 60 000

divided by log

one plus 0.0

and this is equal to 11.78

pertaining only two decimal places

which is equal to n the number of

compounding periods

but we're interested in determining time

which is

usually expressed in the number of years

so to determine the value of t we're

going to divide this

by m which

is equal to two t

is equal to 11.78 divided by

2

which is equal to 5.89

units

this expression on your scientific

calculator

and on your android phones calculator is

pretty

straight forward so directly you input

log of 85 000

divided by 60 000

divided by log

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and this expression will directly

give you this

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so

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you