Time value of money | Interest and debt | Finance & Capital Markets | Khan Academy

Summary

This script delves into the concept of the time value of money, illustrating how the timing of receiving money impacts its worth. It uses a hypothetical scenario of a 10% risk-free interest rate to compare the desirability of receiving $100 now versus $109 in a year or $120 in two years. The script explains that, due to the opportunity cost of lost interest, the immediate $100 is more valuable. It also introduces the concept of present and future value, demonstrating how to calculate the present value of a future sum using a simple formula. The script concludes with a problem that calculates the present value of $65 to be received in one year, resulting in $59.09.

Takeaways

• 💰 The concept of the time value of money is crucial in financial decision-making, emphasizing that the timing of money transactions is as important as the amount.
• 🏦 A hypothetical scenario is presented where a bank guarantees a 10% risk-free interest rate, simplifying the math for the discussion of the time value of money.
• 🤔 The script prompts the audience to consider their preference between receiving $100 immediately, $109 in one year, or $120 in two years, given the opportunity to earn interest.
• 📈 It is demonstrated that receiving $100 now and investing it at 10% interest is more advantageous than receiving $109 in a year, due to the power of compounding interest.
• 🔢 The calculation shows that $100 today, with 10% interest, will grow to $110 in one year and $121 in two years, illustrating the advantage of receiving money sooner.
• 💡 The idea of present value (PV) is introduced, which is the current worth of a future sum of money, given a specific interest rate.
• 🔮 The future value (FV) concept is explained as the value of money in the future, calculated by applying the interest rate to the present value over time.
• 🧮 An example is given to calculate the present value of $65 to be received in one year, using the formula and arriving at approximately $59.09.
• 📊 The script uses mathematical formulas to compare the value of money at different times, highlighting the importance of understanding compound interest.
• 💼 The discussion serves as a lesson in financial literacy, teaching the audience how to evaluate different monetary offers over time and make informed decisions.
• 🌐 The script implies that these financial principles are universally applicable, regardless of the specific interest rates or economic conditions.

Q & A

What is the main concept discussed in the script?

The main concept discussed in the script is the time value of money, which emphasizes that the timing of receiving or spending money is as important as the amount due to the potential for interest accumulation.

Why is the 10% risk-free interest rate significant in this context?

The 10% risk-free interest rate is significant as it provides a fixed and guaranteed return on investment, simplifying the calculations and illustrating the concept of the time value of money effectively.

What is the first scenario presented in the script involving money?

The first scenario is a choice between receiving $100 immediately or $109 in one year, and $120 in two years, considering the 10% risk-free interest rate.

Why would someone prefer to receive $100 now rather than $109 in one year?

Someone would prefer to receive $100 now because if they invest it at a 10% risk-free interest rate, it would grow to $110 in a year, which is $1 more than the $109 offered in one year.

How does the future value of $100 compare to the $120 offered in two years?

If you invest $100 at a 10% risk-free interest rate, after two years it would grow to $121, which is $1 more than the $120 offered in two years, demonstrating the time value of money.

What is the concept of 'present value' as introduced in the script?

Present value is the current worth of a sum of money, given a specified rate of return. It is used to determine an amount that, if invested now, would yield a future value at a specified time.

How is the present value of a future sum calculated?

The present value is calculated by dividing the future sum by the sum of 1 plus the interest rate raised to the power of the number of periods (years in this case). For example, the present value of $121 in two years at a 10% interest rate is $100.

What is the future value of $100 if it is invested at a 10% interest rate for one year?

The future value of $100 invested at a 10% interest rate for one year is $110, as 10% of $100 is $10 in interest.

What is the present value of $65 to be received in one year at a 10% interest rate?

The present value of $65 to be received in one year at a 10% interest rate is approximately $59.09, calculated by dividing $65 by 1.10.

What is the relationship between the time value of money and the concept of present value?

The time value of money and the concept of present value are related in that both consider the impact of time on the value of money. The time value of money explains why money available now is worth more than the same amount in the future, while present value is the method to calculate the current worth of a future sum of money.

How does the script illustrate the importance of considering the timing of money in financial decisions?

The script illustrates this by comparing different scenarios where the timing of receiving money affects its value due to potential interest earnings, thus affecting the decision on which option is most financially beneficial.

Outlines

00:00

💰 Time Value of Money

This paragraph introduces the concept of the time value of money, emphasizing that not only the amount of money is important, but also the timing of when it is received or spent. It uses a hypothetical scenario where a bank offers a 10% risk-free interest rate to illustrate the idea. The narrator presents three options for receiving money and explains why receiving $100 immediately and investing it at the bank would be more beneficial than receiving $109 in one year or $120 in two years, due to the compounding interest effect. The paragraph concludes by defining the time value of money and introducing the concept of present and future value.

05:04

📈 Present and Future Value Calculations

The second paragraph delves deeper into the concepts of present and future value, using the previously established 10% risk-free interest rate. It explains how to calculate the present value of a future sum of money by working backwards from the future value to determine the amount that needs to be invested today to achieve that future sum. The narrator provides a step-by-step example to calculate the present value of $65 to be received in one year, which turns out to be $59.09. This paragraph reinforces the importance of understanding how money grows over time and how to make informed decisions based on the time value of money.

Keywords

💡Money

Money, in the context of the video, is the medium of exchange and a measure of value. It's central to the theme of the video as it discusses the concept of the time value of money, which is the idea that a sum of money is worth more now than the same sum in the future due to its potential earning capacity. The script uses scenarios involving different amounts of money to illustrate this concept.

💡Interest

Interest is the cost or reward paid for the use of money. In the video, it's described as a 'risk-free' 10% interest rate offered by a bank, which is used as a benchmark to compare the value of money received at different times. The script explains how interest affects the desirability of receiving money now versus in the future.

💡Time Value of Money

The time value of money is a financial concept that states that money available at the present time is worth more than the same amount in the future due to its earning potential. The video uses this concept to explain why receiving $100 now is more desirable than receiving a larger sum later, given the opportunity to invest and earn interest.

💡Present Value (PV)

Present value is the current worth of a future sum of money or a series of cash flows given a specified rate of return. The video introduces the concept of present value to explain how to determine the amount of money that, if invested now, would yield a certain future value, such as $65 in one year.

💡Future Value (FV)

Future value is the amount of money that a sum of money will grow to in the future, given a specific interest rate. The script uses the concept of future value to demonstrate how an initial investment of $100 can grow to $110 in one year and $121 in two years at a 10% interest rate.

💡Risk-Free

Risk-free refers to an investment that is expected to return a certain rate of interest without the possibility of losing the principal. In the video, the bank's 10% interest rate is described as 'risk-free,' which simplifies the calculations and emphasizes the mathematical aspect of the time value of money.

💡Investment

Investment, in the context of the video, is the act of putting money into financial assets or other accounts with the expectation of earning interest or profit. The script uses the concept of investment to illustrate how money can grow over time, especially when placed in a bank account with a guaranteed interest rate.

💡Opportunity Cost

Opportunity cost is the loss of potential gain from other alternatives when one alternative is chosen. The video implies the concept of opportunity cost when discussing the choice between receiving money now or later, as choosing to receive money later means losing out on the potential interest that could have been earned if the money were received sooner.

💡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. The video uses the example of $100 growing to $121 in two years to demonstrate the effect of compounding, where the interest earned in the first year is added to the principal and earns interest in the second year.

💡Scenarios

Scenarios in the video are hypothetical situations presented to the viewer to help understand the concept of the time value of money. The script uses different scenarios, such as receiving $100 now or $120 in two years, to explore the decision-making process when considering the timing of money receipt.

💡Calculation

Calculation in the video refers to the mathematical process used to determine the present or future value of money. The script provides a step-by-step calculation to find the present value of $65 received in one year, which involves dividing by 1.10 to account for the 10% interest rate.

Highlights

The concept that the timing of money matters as much as the amount, introducing the time value of money.

Assumption of a world with a guaranteed 10% risk-free interest rate for simplifying calculations.

Scenario comparison of receiving $100 now versus $109 in one year or $120 in two years.

The realization that $100 now is more valuable than $109 in a year due to compound interest.

Demonstration of how $100 grows to $121 in two years with 10% interest, outperforming a guaranteed $120 in two years.

The introduction of the term 'present value' in the context of future monetary value.

Explanation of how to calculate the present value of a future sum using the given interest rate.

The formula for calculating present value: dividing future value by (1 + interest rate).

A practical example calculating the present value of $65 received in one year.

The result of the calculation showing $59.09 as the present value of $65 in one year.

The concept of future value and how it relates to the time value of money.

Illustration of how $100 grows to $110 in one year and $121 in two years with compound interest.

The importance of considering both the amount and timing of money in financial decisions.

The impact of opportunity cost in the context of the time value of money.

A deeper understanding of why immediate money is often more desirable than future payments.

The narrative's emphasis on the practical applications of the time value of money in personal finance.

The educational approach used to explain complex financial concepts in an accessible manner.

The use of a hypothetical scenario to illustrate the principles of the time value of money.

The transcript's goal to provide a clear understanding of why the timing of receiving money is crucial.

Transcripts

Narrator: Whenever we talk about money,

the amount of money is not the only thing that matters.

What also matters is when you have to get

or when you have to give the money.

So, to think about this or to make it a little bit

more concrete, let's assume that we live in a world

that if you put money in a bank, you are guaranteed

10% interest, 10% risk free interest in a bank.

This is high by historical standards,

but it will make our math easy.

So, let's just assume that you can always get

10% risk free interest in the bank.

Now, given that, let me throw out scenarios

and have you think about which of these

that you would most want.

So, I could give you $100 right now.

That's option 1.

I could, in one year, instead of giving you

the $100 immediately, in one year

I could give you $109 and then in 2 years,

this is kind of option 3, I'd be willing to give you

$120, so your choice is, someone walks up to you

off the street.

I could give you $100 bill now, $109 bill ...

(laughing) $109 bill, $109 in a year, $120, 2 years from now

and you know in the back of your mind

you could get 10% risk free interest.

So, given that you don't have an immediate need for money.

We're assuming that this money, you will save.

That you don't have a bill to pay immediately,

which of these things are the most desirable?

Which of these would you most want to have?

Well, if you just cared about the absolute value

or the absolute amount of the money you would say,

"Hey, look. $120, that's the biggest amount of money."

"I'm going to take that one because that's just the biggest number."

But, you probably have in the back of your mind,

"Well, I'm getting that later, so there's maybe something

I'm losing out there?"

And you'd be right.

You'd be losing out on the opportunity to get

the 10% risk free interest if you were to get the money earlier.

And if you wanted to compare them directly,

the thought process would be,

"Well, let's see. If I took option 1. If I got the $100."

And if you were to put it in the bank,

what would that grow to based on that 10% risk free interest?

Well, after 1 year 10% of $100 is $10.

So, you would get $10 in interest.

So, after one year, you're entire savings

in the bank will now be $110.

So, just doing that little exercise

we actually see that $100 given now,

put it in the bank at 10% risk free,

will actually turn into $110 in a year from now,

which is better than the $109 one year from now.

So, given this scenario, or given this kind of situation

or this option, you would rather do this

than do this.

A year from now you're better off by $1.

What about 2 years from now?

Well, if you take that $100 after 1 year

it becomes $110, then 10% of $110 is $11.

You want to add $11 to it, so it becomes $121.

So, once again you're better off taking the $100,

investing it in the bank risk free, 10% per year.

It turns into $121. That is a better situation

than just someone guaranteeing you to give

the $120 in 2 years.

Once again, you are better off by $1.

So, this idea that not just the amount matters,

but when you get it, this idea is called

the time value of money.

Time value of money.

Or another way to think about it is,

think about what the value of this money is over time.

Given some expected interest rate

and when you do that you can compare this money

to equal amounts of money at some future date.

Now, another way of thinking about the time value

or, I guess, another related concept to the time value of money

is the idea of present value, present value.

Maybe I'll talk about present and future value.

So, present and future value, future value.

So, given this assumption, this 10% assumption,

if someone were to ask you, "What is the present value

of $121 2 years in the future?"

They're essentially asking you,

so what is the present value?

PV stands for present value.

So, what is the present value of $121 2 years in the future?

That's equivalent to asking what type of money

or what amount of money would you have to put into the bank

risk free for the next 2 years to get $121?

We know that. If you put $100 in the bank

for 2 years at 10% risk free, you would get $121.

So, the present value here,

the present value of $121 is the $100.

Or another way to think about present and future value

if someone were to ask what is the future value?

So, what is the future value of this $100 in 1 year?

So, in 1 year. Well, if you get 10% in the bank

that's guaranteed, it's future value is $110.

After 2 years, it's 2 year future value is $121.

So, with that in mind let me give you

one slightly more interesting problem.

So, let's say that I have ... let's say,

we're going to assume this the whole time

that makes our math easy at 10% risk free interest.

And let's say that someone says they're willing

to give us $65 in 1 year

and we were to ask ourselves,

"What is the present value of this?"

So, what is the present value of this.

Remember, the present value is just asking you

what amount of money, that if you were to

put it in the bank at this risk free interest,

would be equivalent to this $65?

Which of these 2 are equivalent to you?

You would say, "Well, look. Whatever amount of money that is?"

Let's call that X.

Whatever amount of money that is, times,

if I grow it by 10%, that's literally,

I'm taking X+10%X+ ... let me write it this way.

+10%xX ... Let me write it ... Let me make it clear this way.

X+10%X should be equal to our $65.

If I take the amount I get 10% of that amount

over the year, that should be equal to $65.

This is the same thing as 1X

or we can say that 1X+10% is the same thing

as 0.10X is equal to 65, or you add these 2.

1.10X = 65, and if you want to solve

for the actual amount of the present value here,

you would just divide both sides by the 1.10.

You get X is equal to ... let me do it this way.

It will be a little bit more clear about it.

So, let's divide both sides by 1.0

and really that trailing zero doesn't matter.

We're not really too worried about the precision here

because this actually exactly 10%.

So, this is going to be ... these cancel out

and X is going to be equal to,

let me get the calculator out,

X is going to be equal to 65 divided by 1.1,

$59.09, rounding it.

So, X=59.09, which was the present value

of $65 in one year,

or another way to think about it is

if you wanted to know what the future value

of $59.09 is in 1 year, assuming the 10% interest,

you would get the $65.