Compound Amount Formula with Unknown Interest Rate and Time
Summary
TLDRThis 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
📚 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.
🔢 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.
⏳ 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.
🎉 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.
Mindmap
Keywords
💡Compound Amount Formula
💡Present Value (P)
💡Future Value (F)
💡Nominal Rate (J)
💡Compounding Periods (N)
💡Interest Rate per Period (i)
💡Logarithms
💡Nth Root
💡Semi-Annually
💡Fractional Exponent
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
[Music]
and this expression will directly
give you this
[Music]
so
[Music]
[Music]
you
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