Present Value 2 | Interest and debt | Finance & Capital Markets | Khan Academy

Summary

This video script explores the concept of present value in financial decision-making, offering three hypothetical payment options. It explains the process of discounting future payments to their current value using a 5% risk-free rate. The script illustrates that receiving $100 today is more valuable than a promise of $110 in two years, due to the time value of money. It also demonstrates how to calculate the present value of a series of payments, concluding that option one is the most beneficial based on the given conditions.

Takeaways

• 💰 The script presents a choice between three different payment options, all involving receiving money at different times and amounts.
• 📅 Choice one offers immediate payment of $100 today, highlighted in magenta to indicate the time of payment.
• 🔢 Choice two proposes a payment of $110 in two years, with the payment time also marked in magenta.
• 📈 Choice three is a more complex arrangement, offering $20 today, $50 in one year, and $35 in three years, totaling $105 over the three years.
• 🏦 The speaker assumes a risk-free environment, comparing the payment options to lending money to the federal government at a 5% interest rate.
• 📉 The concept of present value is introduced to determine the worth of future payments in today's money, using a 5% discount rate.
• 🧮 A calculation is provided to show that the present value of $110 received in two years is $99.77, using the formula \( \frac{110}{(1.05)^2} \).
• 🤔 The comparison of choices reveals that taking $100 today and investing it at a 5% interest rate would yield more than the $110 to be received in two years.
• 💡 The importance of the discount rate in financial calculations is emphasized, as it can significantly affect the outcome of present value calculations.
• 📊 The present value of each payment in choice three is calculated individually and then summed up to compare with the other options.
• 📝 The final comparison shows that choice one, receiving $100 today, has the highest present value of approximately $99.37 when compared to the other options.
• 🔑 The takeaway is that understanding present value is crucial for making informed decisions about different payment options and financial investments.

Q & A

What are the two initial payment options presented in the script?

The two initial payment options are: Choice one, receiving $100 today, and Choice two, receiving $110 in 2 years.

Why does the presenter introduce a third payment option?

The third payment option is introduced to provide a more complex scenario and to illustrate the concept of present value more effectively.

What is the significance of circling the payment in magenta in the script?

The magenta circle is used to visually indicate when the payment is received, helping to differentiate between the timing of different payments.

What is the assumed risk-free rate used for discounting future payments in the script?

The assumed risk-free rate used for discounting future payments is 5%.

How does the presenter calculate the present value of $110 received in 2 years?

The presenter calculates the present value by dividing $110 by 1.05 (the factor for 5% growth over one year) twice, which represents discounting back two years.

What is the result of the present value calculation for $110 received in 2 years?

The present value of $110 received in 2 years, using a 5% discount rate, is $99.77.

Why is the present value of $110 in 2 years less than $100 today according to the script?

The present value of $110 in 2 years is less than $100 today because of the time value of money; money received in the future is worth less than the same amount today due to the potential earning capacity of money over time.

What does the script suggest about the comparison between receiving $100 today and $99.77 as the present value of $110 in 2 years?

The script suggests that it is more advantageous to receive $100 today rather than the present value equivalent of $110 in 2 years, which is $99.77.

How does the script handle the third payment option in terms of present value calculation?

The script calculates the present value of each payment in the third option separately: $20 today has a present value of $20, $50 in 1 year is discounted by dividing by 1.05, and $35 in 3 years is discounted by dividing by 1.05 squared.

What is the total present value of the third payment option as calculated in the script?

The total present value of the third payment option, which includes $20 today, $50 in 1 year, and $35 in 3 years, is $99.37.

What insight does the script provide about choosing between different payment options?

The script provides the insight that to make an informed choice between different payment options, one should calculate the present value of each option to determine which is worth more in today's terms.

Outlines

00:00

💰 Understanding Payment Choices and Present Value

The script introduces a financial decision-making scenario with three different payment options. The first option offers immediate payment of $100. The second option proposes a payment of $110 after two years. The third option is a staggered payment plan: $20 today, $50 in one year, and $35 in three years. The video emphasizes the importance of calculating the present value of future payments, especially when comparing different financial offers. It uses a hypothetical risk-free rate of 5% to demonstrate how to calculate the present value of the second option, concluding that receiving $100 today would be more beneficial than a guaranteed $110 in two years, based on the present value calculation.

05:08

📊 Calculating Present Value for Complex Payment Structures

This paragraph delves deeper into the concept of present value, illustrating how to calculate it for more complex payment structures, such as the third option presented. The script explains the process of discounting future payments back to their present value using the risk-free rate of 5%. It breaks down the calculation for each payment in the third option and sums them up to find the total present value. The result, $99.37, is then compared with the present value of the immediate $100 from the first option and the discounted $110 from the second option. The summary highlights the significance of the discount rate in financial calculations and how it can affect the perceived value of different payment options.

Keywords

💡Payment Options

Payment options refer to the different choices one has when receiving money, either immediately or at a future date. In the video, the narrator presents three distinct payment options, each with varying amounts and timelines, to illustrate the concept of present value and time value of money.

💡Present Value

Present value is the concept of calculating the worth of a future sum of money in today's terms, taking into account a specified rate of return. In the script, the narrator uses present value to compare the value of receiving $100 today versus $110 in two years, using a 5% risk-free rate.

💡Risk-Free Rate

A risk-free rate is the theoretical rate of return of an investment with zero risk of financial loss, often associated with government bonds. The script uses a 5% risk-free rate to demonstrate how to calculate the present value of future payments.

💡Discount Rate

The discount rate is used to determine the present value of future cash flows. It represents the interest rate used to discount future payments back to their present value. In the video, the narrator uses a 5% discount rate to calculate the present value of the future payments in the different options.

💡Time Value of Money

The time value of money is the concept that a dollar today is worth more than a dollar in the future due to its potential earning capacity. The video script discusses this concept by comparing the value of receiving money now versus later under different payment options.

💡Inflation

Inflation is the rate at which the general level of prices for goods and services is rising, and subsequently, the purchasing power of currency is falling. The script briefly touches on inflation as a risk when considering the value of future payments, mentioning the potential for currency devaluation by the federal government.

💡Federal Government

The federal government is the central governing authority of a nation, and in the context of the script, it represents a risk-free entity for borrowing and lending money. The narrator uses the federal government as an example of a safe borrower that would pay back loans with a 5% return.

💡Default on Debt

Defaulting on debt refers to the failure to repay borrowed money in accordance with the terms of the loan agreement. The script mentions that while the U.S. government has never defaulted on its debt, it has inflated its currency, which is a form of indirect default.

💡Retirement Plan

A retirement plan is a savings scheme designed to provide income when an individual retires from the workforce. The video script uses retirement plans as an example of a situation where understanding present value is crucial for evaluating the financial benefits of different payment options.

💡Insurance Company

An insurance company is a business that sells insurance policies to individuals and organizations. In the script, the narrator mentions insurance companies as entities that might offer complex payment schemes, which can be evaluated using the concept of present value.

💡Discounted Cash Flow

Discounted cash flow (DCF) is a valuation method used to estimate the value of an investment based on its future cash flows discounted back to their present value. The script explains the process of calculating present values as a form of DCF analysis, particularly when evaluating the third payment option involving multiple payments over time.

Highlights

Introduction of a complex choice between two payment options, both beneficial as they involve receiving money.

Illustration of choice one: receiving $100 today, with the payment circled in magenta.

Introduction of choice two with a payment of $110 in 2 years, also circled in magenta.

Expansion to three choices to emphasize the comparison and decision-making process.

Presentation of choice three involving payments of $20 today, $50 in 1 year, and $35 in year 3.

Explanation of the concept of present value and its importance in comparing financial options.

Assumption of a risk-free person making guaranteed payments, akin to the federal government's reliability.

Discussion on the risk associated with the federal government's promise to pay, including inflation concerns.

Calculation of the future value of $100 at a 5% risk-free rate over one and two years.

Comparison showing that taking $100 now and lending it at 5% is more beneficial than waiting for $110 in 2 years.

Demonstration of how to calculate the present value of a future payment using discounting.

Calculation of the present value of $110 to be received in 2 years, resulting in $99.77 today.

Introduction of the concept of discount rate and its significance in finance and present value calculations.

Explanation of how to calculate the present value of multiple payments over different time periods.

Calculation of the present value for choice three's payments, resulting in a total of $99.37.

Comparison of the three payment options based on their present values, highlighting the most financially beneficial choice.

Emphasis on the importance of understanding present value to make informed decisions in financial planning.

Anticipation of the next video discussing the impact of varying discount rates on financial outcomes.

Transcripts

Now I'll give you a slightly more complicated choice

between two payment options.

Both of them are good, because in either case

you're getting money.

So choice one.

Today I will give you $100.

I'll circle the payment when you get it in magenta.

So today you get $100.

Choice two.

And I'll try to write this choice a little bit neater.

Choice two is that not in 1 year, but in 2 years.

So let's say this is year 1.

And now this is year 2.

Actually I'm going to give you three choices.

That'll really hopefully hit things home.

So actually let me scoot this choice two over to the left.

Back to green.

So now I'm back in business.

So choice two, I am willing to give you, let's say, oh I

don't know, $110 in 2 years.

So not in 1 year.

In 2 years I'm going to give you $110.

And so I'll circle in magenta when you

actually get your payment.

And then choice three .

And choice three is going to be fascinating.

I've done it in a slightly different shade of green.

Choice three, I am going to pay you-- I'm making this up

on the fly as I go-- I'm going to pay you $20 today.

I'm going to pay you $50 in 1 year.

That's $70.

Let me make this so it's close.

And then I'm going to pay you, I don't know, $35 in year 3.

So all of these are payments.

I want to differentiate between the actual dollar

payments and the present values.

And just for the sake of simplicity, let's assume that

I am guaranteed.

I am the safest person available.

If the world exists, if the sun does not supernova, I will

be paying you this amount of money.

So I'm as risk-free as the federal government.

And I had a post on the previous present value, where

someone talked about, well is the federal

government really that safe?

And this is the point.

The federal government, when it borrows from you $100.

Let's say it borrows $100 and it promises

to pay it in a year.

It'll give you that $100.

The risk is, what is that $100 worth?

Because they might inflate the currency to death.

Anyway, I won't go into that right now.

Let's just go back to this present value problem.

And actually sometimes governments

do default on debt.

But the U.S. government has never defaulted.

It has inflated its currency.

So that's kind of a round about way of defaulting.

But its never actually said, I will not pay you.

Because if that happened, our entire financial system would

blow up and we would all be living off the land again.

Anyway, back to this problem.

Enough commentary from Sal.

So let's just compare choice one and choice two again.

And once again let's say that risk-free, I could put money,

I could lend it to the federal government at 5%.

Risk-free rate is 5%.

And for the sake of simplicity-- in the next video

I will make that assumption less simple-- but for the sake

simplicity, the government will pay you 5% whether you

give them the money for 1 year, whether you give them

the money for 2 years, or whether you give them the

money for 3 years, right?

So if I had $100, what would that be worth in 1 year?

We figured that out already.

It's 100 times 1.05.

So that's $105.

And then if you got another 5%?

So the government is giving you 5% per year.

It would be 105 times 1.05.

And what is that?

So I have 105 times 1.05, which equals $110.25.

So that is the value in 2 years.

So immediately, without even doing any present value, we

see that you'll actually be better off in 2 years if you

were to take the money now and just lend it to the

government.

Because the government, risk-free, will give you

$110.25 in 2 years, while I'm only willing to give you $110.

So that's all fair and good.

But the whole topic, what we're trying to solve, is

present value.

So let's take everything in today's money.

And to take this $110 and say what is that worth today, we

can just discount it backwards by the same method, right?

So $110 in 2 years, what is its 1-year value?

Well, you take $110 and you divide it by 1.05.

You're just doing the reverse.

And then you get some number here.

Well that number you get is 110 divided by 1.05.

And then to get its present value, its value today, you

divide that by 1.05 again.

So you get 110 divided.

If I were to divide by 1.05 again what do I get?

I divide by 1.05, and then I divide by 1.05 again.

I'm dividing by 1.05 squared.

And what does that equal?

And I'm writing this on purpose, because I want to get

you used to this notation.

Because this is what all of our present values and our

discounted cash flow, this type of dividing by 1 plus the

discount rate to the power of however many years out, this

is what all of that's based on.

And that's all we're doing though, we're just dividing by

1.05 twice because we're 2 years out.

So let's do that.

110 divided by 1.05 squared is equal to $99.77.

So once again we have verified, by taking the

present value of $110 in 2 years to today, that its

present value-- if we assume a 5% discount rate.

And this discount rate, this is where all of the fudge

factor occurs in finance.

You can tweak that discount rate and make a few

assumptions in discount rate and

pretty much assume anything.

But right now, for simplification, we're assuming

a risk-free discount rate.

But when the present value is based on that, you get $99.77.

You say, wow, yeah, this really isn't as good as this.

I would rather have $100 today than $99.77 today.

Now this is interesting.

Choice number three.

How do we look at this?

Well what we do is, we present value each of

the payments, right?

So the present value of $20 today, well that's just $20.

What's the present value of $50 in 1 year?

Well the present value of that is going to be-- so plus $50

divided by 1.05, right-- that's the present value of

the $50, because it's 1 year out.

And then I want the present value of the $35.

So that's plus $35 divided by what-- it's 2 years out,

right, so you have to discount it twice--

divided by 1.05 squared.

Just like we did here.

So let's figure out what that present value is.

Notice I'm just adding up the present values of each of

those payments.

Get out my virtual TI-85.

Let's see, so the present value of the $20 payment is

$20, plus the present value of the $50 payment.

Well that's just 50 divided by 1.05, plus the present value

of our $35 payment.

35 divided by-- and it's 2 years out, so we discount by

our discount rate twice-- so it's divided by 1.05 squared.

And then that is equal to-- we'll round it-- $99.37.

So now we can make a very good comparison

between the three options.

This might have been confusing before.

You know, you have this guy coming up to you.

And this guy is usually in the form of some type of

retirement plan or insurance company, where they say, hey,

you pay me this for years a, b, and c, and I'll pay you

that in years b, c, and d.

And you're like, boy, how do I compare if that's really a

good value?

Well this is how you compare it.

You present value all of the payments and you say well what

is that worth to me today.

And here we did that.

We said well actually choice number one is the best deal.

And it just depended on how the mathematics work out.

If I lowered the discount rate, if this discount rate is

lower, it might have changed the outcomes.

And maybe I'll actually do that in the next video, just

to show you how important the discount rate is.

Anyway I'm out of time, and I'll see

you in the next video.