Compound Interest

00:10:52
https://www.youtube.com/watch?v=Hn0eLcOSQGw

الملخص

TLDRThe video provides a comprehensive look at compound interest through practical examples. It begins with Luke's investment of $1,000 at a 9% annual interest rate, showing how his investment grows to $2,367.36 after 10 years and $5,604.41 after 20 years. It emphasizes the significance of time in growing investments. Next, Sam invests $5,000 at a 10% rate with quarterly and monthly compounding, demonstrating how frequency influences returns. Finally, Megan invests $10,000 at a 6% fixed rate, calculating the time needed for her investment to double using logarithms and the Rule of 72 for approximation. Overall, it highlights the impact of interest rates, compounding frequency, and time on investment growth.

الوجبات الجاهزة

  • 📈 Understanding compound interest is crucial for investing effectively.
  • 💵 Time significantly enhances investment returns through compound interest.
  • 🕒 Doubling time can be estimated using the Rule of 72.
  • 🔍 Wealth accumulation is impacted by how often interest is compounded.
  • 🔢 Logarithms help calculate time needed for investments to grow.
  • 👍 Regularly investing can result in substantial future gains.
  • 💡 More frequent compounding yields greater returns.
  • 💰 Principal amount is the starting investment before interest.
  • ✔️ Realizing the importance of long-term investments is key.
  • 📊 Comparing different compounding methods helps in strategic planning.

الجدول الزمني

  • 00:00:00 - 00:10:52

    Moving forward, Sam invests $5,000 in an account with a 10% annual interest rate, compounded quarterly initially. After calculations, it reveals that the account would grow to $36,047.84 over 20 years. By shifting to a monthly compounding frequency, the amount slightly increases to $36,640.37, demonstrating that more frequent compounding can yield higher returns. Lastly, Megan aims to learn how long it will take for her $10,000 investment at a 6% annual interest rate to double. By applying a logarithmic approach, it is calculated that it will take approximately 11.896 years, aligning closely with the Rule of 72 estimate of 12 years.

الخريطة الذهنية

فيديو أسئلة وأجوبة

  • What is compound interest?

    Compound interest is the interest on a loan or deposit calculated based on both the initial principal and the accumulated interest from previous periods.

  • How is compound interest calculated?

    Compound interest can be calculated using the formula A = P(1 + r/n)^(nt), where A is the amount of money accumulated after n years, including interest, P is the principal amount, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years.

  • What is the Rule of 72?

    The Rule of 72 is a formula that estimates the number of years required to double the investment at a fixed annual rate of return by dividing 72 by the annual interest rate.

  • How does compounding frequency affect investment returns?

    The more frequently interest is compounded, the more interest you earn on your investment. For example, quarterly compounding yields more than annual compounding.

  • How can I estimate the time it takes to double an investment?

    You can estimate the doubling time by using the Rule of 72, which states that the time to double is approximately 72 divided by the annual interest rate in percentage.

  • What does it mean if an account is credited with interest annually?

    If an account is credited yearly, it means that the interest is calculated once a year and added to the principal.

  • What is the importance of time in investing?

    Time allows compound interest to accumulate, leading to greater returns. The longer an investment is held, the more significant the growth.

  • Can logarithms be used in calculating investment growth?

    Yes, logarithms can be used to find the time required for an investment to grow to a certain amount when dealing with compound interest.

  • What does 'n' represent in the compound interest formula?

    In the compound interest formula, 'n' represents the number of times interest is compounded per year.

  • What is meant by principal in investment terms?

    Principal refers to the initial amount of money invested or loaned, not including any interest.

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الترجمات
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التمرير التلقائي:
  • 00:00:01
    in this video we're going to work on
  • 00:00:02
    some problems associated with compound
  • 00:00:04
    interest
  • 00:00:05
    like this one
  • 00:00:07
    luke puts a thousand dollars in an
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    account
  • 00:00:10
    that pays an annual interest of nine
  • 00:00:12
    percent
  • 00:00:13
    what will be the value of the account
  • 00:00:15
    after 10 years
  • 00:00:16
    so the formula that we need to use is
  • 00:00:18
    this one a
  • 00:00:20
    which is the final account value equals
  • 00:00:22
    p the principal or the amount that he
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    invests
  • 00:00:26
    times 1 plus r over n
  • 00:00:29
    raised to the nt
  • 00:00:31
    r is the interest rate as a decimal
  • 00:00:34
    n is the number of times
  • 00:00:36
    interest is credited to an account on an
  • 00:00:39
    annual basis or per year rather
  • 00:00:42
    so let's say if
  • 00:00:44
    the account is credited with interest
  • 00:00:46
    only once per year and is one
  • 00:00:50
    if it's compounded quarterly that is if
  • 00:00:52
    the account receives interest four times
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    a year and it's four
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    if it's compounded weekly if the account
  • 00:00:59
    receives interest every week
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    then n is 52 because there's 52 weeks
  • 00:01:04
    per year
  • 00:01:05
    so n
  • 00:01:06
    is the number of times
  • 00:01:08
    the account is credited with interest
  • 00:01:10
    per year and t is the time in years
  • 00:01:14
    so in this problem the amount invested
  • 00:01:17
    is a thousand dollars
  • 00:01:19
    the interest rate is nine percent
  • 00:01:21
    nine percent as a decimal
  • 00:01:24
    is point zero nine to get that value
  • 00:01:27
    take nine divided by a hundred
  • 00:01:31
    now
  • 00:01:33
    how often is interest credited
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    to this account
  • 00:01:38
    well we see that the count pays an
  • 00:01:40
    annual interest of nine percent
  • 00:01:42
    and there's nothing else to tell us that
  • 00:01:44
    it's compounded monthly or weekly
  • 00:01:47
    so n is one for this problem
  • 00:01:50
    and we want to find the value of the
  • 00:01:51
    account ten years later
  • 00:01:54
    so a is going to be a thousand
  • 00:01:56
    times one plus point zero nine or one
  • 00:01:59
    point zero nine
  • 00:02:01
    raised to the tenth power
  • 00:02:09
    and so this is going to be
  • 00:02:11
    two thousand three hundred
  • 00:02:13
    sixty seven dollars
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    and thirty six cents
  • 00:02:17
    now let's move on to part b
  • 00:02:19
    so what will be the value of the account
  • 00:02:21
    after 20 years
  • 00:02:23
    the only thing we need to change is the
  • 00:02:24
    time from 10 years
  • 00:02:26
    to 20 years
  • 00:02:28
    so this is going to be a thousand
  • 00:02:31
    times 1.09 raised to the 20th power
  • 00:02:39
    and the answer
  • 00:02:41
    is going to be five thousand
  • 00:02:43
    six hundred and four dollars and forty
  • 00:02:45
    one cents
  • 00:02:49
    so this example really illustrates
  • 00:02:52
    how time is important when investing
  • 00:02:56
    so let's analyze this
  • 00:02:58
    in 10 years
  • 00:03:01
    his account went from a thousand to two
  • 00:03:04
    thousand three hundred sixty seven
  • 00:03:06
    dollars and thirty six cents
  • 00:03:08
    so basically
  • 00:03:10
    his gain
  • 00:03:12
    was
  • 00:03:13
    thirteen hundred dollars
  • 00:03:15
    1367.36
  • 00:03:18
    after 10 years
  • 00:03:22
    but when he keeps it for 20 years
  • 00:03:26
    his investment multiplies by a factor of
  • 00:03:29
    5.6
  • 00:03:32
    so his gain
  • 00:03:35
    is more than double
  • 00:03:37
    it's more than thirteen hundred dollars
  • 00:03:39
    in this case it's four thousand
  • 00:03:43
    six hundred and four dollars and forty
  • 00:03:44
    one cents
  • 00:03:47
    so by leaving it for ten years he made
  • 00:03:49
    about thirteen hundred dollars
  • 00:03:51
    but by leaving it for twenty years he
  • 00:03:53
    made forty six hundred dollars
  • 00:03:54
    from his investment
  • 00:03:57
    so the lesson is simple
  • 00:03:59
    the more time that an investment has to
  • 00:04:01
    work
  • 00:04:02
    the greater the return will be
  • 00:04:05
    number two
  • 00:04:07
    sam invests 5 000
  • 00:04:09
    in an account that pays an annual
  • 00:04:11
    interest of 10 percent
  • 00:04:13
    the funds are invested for 20 years
  • 00:04:15
    what will be the value of the account if
  • 00:04:17
    interest is credited to the account on a
  • 00:04:19
    quarterly basis
  • 00:04:21
    so on a quarterly basis
  • 00:04:23
    n is for there's four quarters in a year
  • 00:04:26
    each quarter represents three months
  • 00:04:29
    so now let's use the same formula
  • 00:04:37
    so p the principal or the amount
  • 00:04:39
    invested
  • 00:04:40
    is thousand
  • 00:04:42
    the interest rate is ten percent
  • 00:04:45
    ten divided by a hundred is point ten
  • 00:04:49
    n is four
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    and we wanna determine the value of the
  • 00:04:53
    account twenty years later
  • 00:05:01
    so let's do this one step at a time
  • 00:05:03
    0.10
  • 00:05:05
    divided by 4
  • 00:05:06
    is 0.025
  • 00:05:08
    if we add 1 to that we're going to get
  • 00:05:10
    1.025
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    and 4 times 20 is 80.
  • 00:05:16
    so it's 5 000 times
  • 00:05:18
    1.025 raised to the 80th power
  • 00:05:22
    and so the value of the investment
  • 00:05:24
    twenty years later will be
  • 00:05:26
    thirty six thousand
  • 00:05:29
    forty seven dollars
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    and eighty four cents
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    so to round it to the nearest set
  • 00:05:36
    now let's move on to part b let's see
  • 00:05:38
    what's going to happen
  • 00:05:40
    if
  • 00:05:41
    the account is credited
  • 00:05:43
    with interest on a monthly basis
  • 00:05:50
    so everything is going to be the same
  • 00:05:51
    but we're going to change n
  • 00:05:53
    from 4
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    to 12.
  • 00:06:06
    10 divided by 12 is
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    0.0083 repeating
  • 00:06:10
    if we add 1 to that this is going to be
  • 00:06:15
    1.0083 repeating
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    so there's many threes after that and
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    then 12 times 20
  • 00:06:23
    is 240.
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    so if we type in 5000 times
  • 00:06:28
    1.008 333333
  • 00:06:31
    raised to the 240th power
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    we're going to get thirty six thousand
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    six hundred forty
  • 00:06:43
    and thirty seven cents
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    so notice that
  • 00:06:47
    this amount
  • 00:06:48
    is about six hundred dollars more
  • 00:06:51
    than this amount
  • 00:06:53
    so what does this tell us
  • 00:06:56
    when interest is credited
  • 00:06:59
    to an account more frequently
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    the return will be slightly more
  • 00:07:04
    so in this example when interest is
  • 00:07:05
    credited 12 times a year
  • 00:07:08
    the amount earned over 20 years is 600
  • 00:07:10
    more than if interest is credited four
  • 00:07:14
    times a year
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    now the difference is not that great but
  • 00:07:19
    over many years it could be significant
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    now let's move on to our last problem
  • 00:07:25
    megan
  • 00:07:26
    invests ten thousand dollars in an
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    annuity that pays a fixed interest of
  • 00:07:31
    six percent on an annual basis
  • 00:07:34
    how many years will it take for
  • 00:07:36
    investment to double in value
  • 00:07:41
    so let's start with this equation once
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    more
  • 00:07:52
    now interest is credited to the account
  • 00:07:54
    on an annual basis
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    so n is 1.
  • 00:07:59
    her original investment is 10 000.
  • 00:08:03
    we want to find the time it takes for
  • 00:08:05
    her to double the investment so we're
  • 00:08:07
    looking for t
  • 00:08:10
    so when her investment doubles it's
  • 00:08:12
    going to be worth twenty thousand
  • 00:08:13
    dollars
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    so let's replace a with twenty thousand
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    and let's replace p with ten thousand
  • 00:08:22
    i'm not going to plug in the value for r
  • 00:08:24
    yet
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    n is one
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    if we divide both sides by ten thousand
  • 00:08:34
    we'll get that two is equal to one plus
  • 00:08:37
    r
  • 00:08:38
    raised to the t
  • 00:08:41
    now what i'm going to do in the next
  • 00:08:42
    step is i'm going to take the natural
  • 00:08:44
    log
  • 00:08:44
    of both sides
  • 00:08:50
    a property of logs and natural logs
  • 00:08:52
    allows us to move the exponent to the
  • 00:08:54
    front
  • 00:08:55
    so we're going to get ln 2 is equal to t
  • 00:08:59
    times ln
  • 00:09:00
    1 plus r
  • 00:09:02
    so dividing both sides by ln
  • 00:09:05
    1 plus r
  • 00:09:09
    we're going to get a formula that tells
  • 00:09:11
    us
  • 00:09:12
    the time it takes to double
  • 00:09:15
    when we receive interest on an annual
  • 00:09:17
    basis
  • 00:09:18
    and here it is
  • 00:09:19
    the time it takes to double is equal to
  • 00:09:21
    the natural log of two
  • 00:09:23
    divided by the natural log
  • 00:09:25
    of one plus r
  • 00:09:36
    so now let's plug in r
  • 00:09:38
    r is six percent
  • 00:09:40
    so that's point zero six as a decimal
  • 00:09:43
    so this is going to be the natural log
  • 00:09:45
    of two
  • 00:09:46
    divided by the natural log
  • 00:09:48
    of one point zero six so you could use
  • 00:09:50
    log or natural log
  • 00:09:52
    both will give you the same answer
  • 00:09:55
    so let's type in ln 2 divided by ln
  • 00:09:58
    1.06
  • 00:10:00
    and the answer is 11.896
  • 00:10:04
    years
  • 00:10:06
    so that's how long it's going to take
  • 00:10:08
    for
  • 00:10:09
    megan's investment to double in value
  • 00:10:13
    now you can get the answer quickly
  • 00:10:16
    or at least an estimate of the answer
  • 00:10:18
    using what is known as the rule of 72.
  • 00:10:22
    according to the rule of 72 the time it
  • 00:10:24
    takes to double is equal to 72
  • 00:10:28
    divided by
  • 00:10:29
    the interest rate as a percentage not as
  • 00:10:31
    a decimal
  • 00:10:33
    so we're going to take 72 and divide it
  • 00:10:35
    by 6 not by 0.06
  • 00:10:38
    72 divided by 6 is 12.
  • 00:10:42
    and so we can see that this formula
  • 00:10:44
    gives us a good approximation
  • 00:10:46
    of the time it takes to double
  • 00:10:48
    11.896 years is approximately 12 years
الوسوم
  • compound interest
  • investment
  • interest rate
  • Rule of 72
  • logarithms
  • financial literacy
  • investment growth
  • compounding frequency
  • principal
  • annual interest