Chapter 2: Time Value of Money & Present Value Calculations

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Every single big financial decision, and I mean every single one, whether you're running a huge company, planning a new factory, or you're just trying to figure out if going to college is worth the debt.

It all comes down to this one core skill.

It's being able to compare money across different points in time.

That is the perfect place to start because the entire world of finance, it just spins around one question.

Why is a dollar today not the same as a dollar next year?

And the answer is what we call the time value of money.

Because you can invest it.

You can invest it.

Money earns a return, it grows.

And that simple fact, that opportunity means getting cash sooner is always, always better than getting it later.

And even if you don't invest it, you've still lost something, right?

You've lost the opportunity to earn that interest.

That's a real cost.

It is a very real cost.

And that idea forces us straight into the most important calculation in finance,

present value or PV.

We're basically asking, what is that dollar I'm getting in the future actually worth to me right now in today's money?

And that leads right to the big one, the ultimate decision tool, net present value,

NPV.

So that's our mission for this deep dive.

That's the mission.

We are building the absolute foundation for how you value, well, anything.

Because at the end of the day, companies create wealth by doing one thing,

accepting projects that are worth more than they cost.

Projects with a positive net present value.

Exactly.

All right.

So let's map this out.

We're going to start with the basic mechanics, right?

Calculating value forward into the future value and then backward to today present value.

And we'll see how risk changes all those numbers.

Then we need the shortcuts because nobody wants to calculate 30 years of mortgage payments one by one.

So that means we're getting into things like perpetuities and annuities.

Those level streams of payments.

And from there, we'll tackle what happens when those payments aren't level, when they actually grow over time.

Which is super important for like dividends or planning for inflation.

It is.

And finally, we have to talk about the tricky part.

The part where a lot of people get tripped up.

How interest rates are quoted.

We'll sort out the difference between the APR you see in an ad and the rate you actually end up earning or paying.

Okay.

Once you have that toolkit, you should be ready to value just about anything.

You really are.

Let's jump in.

Hashtag hack, how to calculate future and present values,

hashtag tag, the time value of money and compound interest.

Let's just really hammer down that first concept.

A dollar today is worth more than a dollar tomorrow.

This isn't just some, you know, abstract economic idea.

No, it's a practical reality.

If someone offers you 100 bucks today or 100 bucks next year, you take it now.

You just do.

Because you can put it to work immediately.

Precisely.

Let's put a number on it.

Let's say we have an investment opportunity and the interest rate, we'll call it R, is 7 % a year.

Okay.

So you invest your initial $100, that's our present value, our PV.

We can calculate what it's worth in the future, its future value, or FV.

So after one year, you've earned $7 in interest.

Your 100 bucks becomes 107.

Simple enough.

For one year, it's just your principal times one plus the interest rate.

Right.

100 times 1 .07.

But the real power, the thing that makes finance so explosive, is what happens in year two.

It totally changes.

Because you're not just earning interest on your original $100 anymore.

You're earning interest on the $7 of interest, too.

You're earning interest on the interest.

So in year two, you get 7 % of the new total, the $107.

Which is $7 .49.

Exactly.

So your investment grows to $114 .49.

And that phenomenon,

interest, earning interest, is what we call compound interest.

It's just the most powerful force in finance.

It really is.

And that's why we see exponents in the general formula.

If we let it be the number of years the money's invested.

The formula is FV equals PV times, in parentheses, one plus R.

All raised to the power of T.

And that little T in the exponent is doing all the work.

Because it shows the base is growing every single year.

And it makes a huge difference.

If you just graph it out, you see this line that starts off kind of flat and then just curves up almost vertically.

It's exponential.

Let's look at the example from the source material.

$100 invested for 20 years.

At 5%, that 100 bucks grows to $265.

Not bad at all.

But then you just bump that rate up to 10%, just 5 percentage points more.

And that same $100,

it explodes.

Becomes $672 .75.

Wow.

OK, just stop there for a second.

Over 20 years, that 5 % difference in the rate gave you an extra $400 on your initial 100.

It's massive.

It just shows that when you're thinking about long term savings, like for retirement, getting even a slightly better rate of return isn't a small thing.

It's a gigantic thing.

It can be the difference between a comfortable retirement and, well, not.

And this isn't some new idea.

There's this great historical example.

Oh, the Ned Kelly story.

Yeah, from Australia in the 1880s.

There was a reward promised to his trackers, $100 Australian dollars.

Which back then was a lot of money.

But today, $100 sounds like nothing.

Right.

But their descendants finally claimed the reward 113 years later.

So now you can plug that into our future value formula.

The PV is 100.

The time, t, is 113 years.

And if we assume a pretty conservative interest rate for that period, say 4 .5 %… That $100 becomes $14 ,459.

It's a perfect demonstration of just the raw, relentless power of compounding over a really, really long timeline.

Hashtag, tag, calculating present value, PV, and the discount factor.

OK, so that's future value looking forward.

Now let's run the whole thing in reverse, which for making business decisions is usually what you're actually doing.

Exactly.

If future value asks what will my money be worth later, present value asks a different question.

It asks, how much do I need to invest today to get a specific amount of money in the future?

We're just running the calculation backward.

We call the process discounting.

So instead of multiplying by 1 plus r to the t… You divide.

You divide.

The formula for present value is just the future cash flow, which we'll call c sub t divided by 1 plus r to the power of t.

And in this context, when we're discounting, we don't just call r the interest rate.

It has a specific name.

We call it the discount rate.

And the PV is the discounted value of that future cash flow.

There's also a shortcut term for the division part of the formula, right?

The discount factor?

Yeah, the discount factor, or DF, is just 1 divided by 1 plus r to the t.

It's a handy little number.

What does it represent, really?

You can think of it as the price today for $1 to be delivered to you at some point in the future.

So if I'm getting $100 in 2 years and the discount factor for year 2 is .87, I just multiply 100 by .87.

Then you get $87.

It tells you that the market today values a promise of $1 in 2 years at only $0 .87.

The intuition here feels really important, especially how sensitive those future values are to the discount rate.

That is the critical insight.

The further out in time you go, the more powerful discounting becomes.

If you graph the discount factor over time, it's this curve that just plunges downward.

So the longer you wait for money and the higher the interest rate?

The less it's worth today, the present value gets smaller and smaller.

Let's use the numbers.

$100, 20 years from now, if the discount rate is 5%, what's its present value?

It's worth $37 .69 today.

OK.

But what if risk goes up or inflation kicks in and that discount rate jumps to 10 %?

The value plummets.

It falls all the way down to just $14 .86.

Wow, it lost over 60 % of its present value just from that rate change.

And that shows you a fundamental rule of finance.

Long -term cash flows are incredibly sensitive to changes in discount rates.

If you're managing a pension fund, for example, and you're dealing with promises 30 or 40

A small, sustained increase in interest rates can have a massive impact on the value of your assets and liabilities.

Hashtag, tag, tag, tag, comparing cash flows at different dates.

OK, so this brings us to a really common problem.

In the real world, you don't usually get one single payment.

You get a stream of payments at different times.

Right.

And you can't just add them up.

$1 in year one is not the same as $1 in year two.

This is where you always say you have to draw a timeline.

You have to.

It's not optional.

You map out exactly when each cash flow happens.

It prevents so many errors.

The car example, Gamma versus Delta Motors, is perfect for this.

Delta's deal is simple, $19 ,000 right now.

Easy.

The PV is $19 ,000.

But Gamma offers free credit.

You pay $8 ,000 today and then the rest, $12 ,000 in two years.

And we'll assume the market interest rate is 10%.

To compare these, we have to get everything into today's dollars.

In two years, it's $0.

So Gamma's $8 ,000 payment is already in today's dollars.

Its PV is $8 ,000.

Simple.

But that $12 ,000 payment is two years away.

We have to discount it.

So we take $12 ,000 and divide it by 1 .10 squared.

Which gives you a present value of $9 ,917 .36.

That's what that future payment is worth today.

And now we can use what you call the Additivity Principle.

Since both numbers are now in today's dollars, we can just add them together.

Exactly.

$8 ,000 plus $9 ,917 give you a total present cost for the Gamma deal of $17 ,917 .36.

And now you can compare.

Delta's price is $19 ,000.

Gamma's true cost in today's money is just under $18 ,000.

Gamma is the better deal by more than $1 ,000.

Which is so counterintuitive because the total cash you pay to Gamma is $20 ,000 and to Delta it's only $19 ,000.

But the time value of that delayed payment is so significant that it makes the whole deal cheaper.

That's the lesson.

You must bring everything back to present value before you can make a valid comparison.

Hashtag tag dialing investments and net present value.

NPV.

OK, this is it.

This is the leap into the core decision rule for all of corporate finance.

Net present value.

The NPV rule.

But to use it, we first need the right interest rate.

And that rate is called the opportunity cost of capital.

Can you define that really clearly?

Because I think this is where a lot of people get tripped up.

The opportunity cost of capital is the return that the company shareholders could get for themselves if they invested their own money in other financial assets with the same level of risk.

So it's a return they're giving up by letting the company invest their money for them.

Precisely.

It's the opportunity they've foregone.

So if a project is super safe, the opportunity cost might be the rate on a government bond, let's say 7%.

OK.

Let's use the office building example.

We're thinking of buying it.

The cost today, C0, is a $700 ,000 outflow.

And we expect to sell it in one year for $800 ,000.

We'll assume for now that this payoff is certain, totally risk free.

So the opportunity cost of capital is that 7 % government bond rate.

Right.

So first we find the present value of that $800 ,000 payoff.

We divide it by 1 .07.

Which gives us $747 ,664.

And that number, that PV,

is effectively the market price of the building today.

In a perfect market, that's what it would sell for because it gives investors the 7 % return they demand on a risk free asset.

But that's just its value.

That's not the net value.

Correct.

To get the net present value, the NPV, you take the present value and you subtract the initial investment.

So $747 ,664 minus the $700 ,000 we paid.

Which equals $47 ,664.

And that positive number is the holy grail.

That's the value created.

That's the net contribution to shareholder wealth.

And it gives us the single most important rule in finance,

the NPV rule.

You accept investments that have a positive NPV.

Period.

Because they make the firm, and its owners, richer.

The simple formula for a one year project is just NPV equals C0, which is negative,

plus C1 divided by 1 plus R, hashtag, hashtag, hashtag adjusting for risk and alternative rules.

Okay, but let's get real.

That $800 ,000 payoff was never a certainty.

Almost nothing in business is.

Which brings us to the second big financial principle.

A safe dollar is worth more than a risky dollar.

And what does that mean in practice?

It means investors hate risk.

They're risk averse.

So to get them to invest in something uncertain, you have to offer them a higher expected return.

Which means for a risky project, our discount rate has to go up.

It has to.

The discount rate must be the expected return you could get on other investments of similar risk.

So let's make our office building risky.

Let's say it's as risky as the stock market in general, and the expected return on the stock market is 12%.

So now our opportunity cost of capital.

Our discount rate jumps from 7 % to 12%.

And we rerun the numbers, $800 ,000 divided by 1 .12.

The PV drops to $714 ,286.

And the NPV?

$714 ,286 minus the $700 ,000 cost is only $14 ,286.

It's still positive, so we'd still do it, but it's much, much smaller.

And that's the point.

The riskiness of the project directly reduced its present value.

We're now discounting for two things.

The time delay and the uncertainty of the payoff.

Okay, I have to ask the question that I know students always ask here.

I think I know what's coming.

Let's say our company is so successful, we can go to the bank and get a loan for the $700 ,000 at, say, 8%.

Why aren't we using 8 % as our discount rate?

That's what the money actually costs us.

That is the single most common and most dangerous mistake in this field.

The 8 % bank loan is the cost of financing.

It reflects how creditworthy our company is.

The 12 % is the opportunity cost of the project.

It reflects the project's risk.

The shareholders are the owners.

They could take their money and invest it in the stock market to get a 12 % return.

So if we only earn 10 % on this project, even though our loan is 8%, we've actually failed our shareholders.

You've destroyed value for them because they could have gotten 12 % on their own.

The discount rate must always reflect the risk of the cash flows being valued, not the source of the funds.

That's a critical distinction.

Now, what about the other rule people sometimes use, the rate of return rule?

It's another way of looking at the same problem.

You calculate the project's return as a percentage.

So profit divided by investment,

our profit is $100 ,000.

Divided by the $700 ,000 investment gives you a rate of return of 14 .3%.

And the rule is,

accept the project if its rate of return is higher than the opportunity cost of capital.

14 .3 % is greater than our 12 % hurdle rate, so we accept.

For simple one -period projects,

the NPV rule and the rate of return rule will always give you the same answer, hashtag, hashtag, multiple cash flows and the DCF formula.

Okay, so we've established that because present values are all in today's dollars, we can just add them up.

That's the additivity principle.

Which is incredibly convenient.

It means we can value a project with a whole stream of cash flows over many years.

And that's where we get the full discounted cash flow, or DCF formula.

Right.

Which looks a bit intimidating with the summation sign and all, but all it's saying is...

The total PV is just the sum of each individual future cash flow, each one discounted back by the right number of years.

That's all it is.

And to get the NPV, you just add your initial investment, C0, to that sum.

Let's use the revised office project to see it in action.

So now, instead of selling after one year, we rent it out for two.

We get $30 ,000 in rent in year one.

And in year two, we get rent plus the sale price, so $870 ,000.

The discount rate is still 12%.

So we have to discount each of those cash flows separately.

First, the $30 ,000 from year one.

Divide that by 1 .12.

That's $26 ,786.

Now the big one from year two, 870 ,000 divided by 1 .12 squared.

That's $693 ,559.

Now you just add those two present values together.

And we get a total PV for the project of $720 ,344.

And the net present value is that total PV minus our $700 ,000 cost.

Which gives us $20 ,344.

A positive NPD.

We accept the project.

And you'll notice that this NPV is actually higher than the NPV from the one -year plan we calculated before.

So the two -year plan is not just acceptable, it's actually better.

It creates more value.

And this process, this DCF analysis, is the absolute bedrock of modern corporate finance.

Hashtag how to value perpetuities and annuities.

So we know how to discount cash flows one by one.

But if you have a project that lasts for, say, 50 years.

You are not going to do that 50 times.

It's tedious, and you'll probably make a mistake.

So we need shortcuts.

And we'll start with a cash flow stream that lasts forever.

Hashtag, tag, tag, value, and perpetuities.

Infinite cash flows.

This is a perpetuity.

It's an investment that pays the same amount of cash, C, every single year, forever, starting one year from now.

It sounds kind of theoretical.

Do these actually exist?

They do, or they did.

The British government used to issue bonds called consoles that never matured.

They just paid interest in perpetuity.

How can you possibly calculate the value of an infinite stream of payments?

It seems like it should be infinite.

It feels that way.

But because of discounting, the present value of the payments way, way out in the future gets incredibly small,

effectively approaching zero.

And that allows for a surprisingly simple formula.

A beautifully simple formula.

The present value of a perpetuity is just C divided by R.

The cash flow divided by the discount rate.

The intuition behind that is so cool.

It really is.

All you're asking is, how much money, PV, do I need to stick in the bank today so that the interest it earns each year, which is PV times R, is equal to the cash flow C that I want to pull out?

So PV times R equals C.

You just rearrange it to solve for PV?

And you get C over R.

It's the principle you need to generate that income forever without ever touching the principle itself.

Okay, let's use the malaria foundation example.

They want to fund an operation that costs $1 billion a year forever.

And they can earn 10 % on their investments.

So C is 1 billion, R is 0 .10.

The endowment they need is 1 billion divided by 0 .10.

10 billion dollars.

That's it.

That 10 billion dollar principle earning 10 % will throw off $1 billion in interest every year forever.

But the timing is really important here.

That formula, C over R, it calculates the value one year before the very first payment arrives.

A crucial detail.

If the foundation needed that first billion dollars immediately in year zero, we call that a perpetuity due.

And how do you value that?

You just take the regular perpetuity value, the 10 billion, and add that first immediate payment to it.

So they'd need 11 billion.

So a perpetuity due is always worth more.

It's worth one plus R times a regular one.

Exactly.

What if the payments are delayed?

Say the first billion is needed four years from now.

Great question.

It's a two -step process.

First, you recognize that a stream of payments starting in year four looks like a regular perpetuity from the perspective of year three.

So in year three, its value is still C over R, $10 billion.

Correct.

Then for step two, you just have to discount that 10 billion dollar lump sum value from year three back to today.

So you take 10 billion and divide by 1 .10 cubed.

Which comes out to about 7 .51 billion dollars.

The delay makes the required endowment much smaller today.

Hashtag, tag, tag, valuing annuities, fixed term cash flows.

Okay, from the infinite to the finite.

A lot more common in the real world is an annuity.

Which is just a stream of fixed payments, see, that lasts for a specific limited number of years, T.

Mortgages, car loans, bond interest payments, these are all annuities.

Yep.

And again, we could discount them all one by one, but we have a shortcut.

And the shortcuts logic is actually pretty clever.

It involves those perpetuities we just talked about.

It does.

You can think of a T year annuity as the difference between two perpetuities.

Okay, paint that picture for me.

Imagine perpetuity A starts in year one and pays forever.

Its value today is C over R.

Now imagine a second one, perpetuity B.

It also pays forever, but its first payment is delayed until year T plus one.

Okay, I'm with you.

If you own perpetuity A, but you sell perpetuity B, what are you left with?

You're left with the payments from year one through year T.

You're left with the annuity.

Exactly.

So the value of the annuity is just the value of the immediate perpetuity minus the present value of the delayed one.

And that logic gives us the annuity formula, which can look a bit scary.

It looks complex, but it's just doing what we described.

It's PV equals C times the thing called the annuity factor.

And that factor is just one over R minus one over R times one plus R to the T.

Let's make it real.

An airline is leasing a plane for five years.

The payment is five million a year at the end of each year, and the discount rate is 7%.

So we need the five -year annuity factor at 7%.

You can look it up in a table or calculate it, and it comes out to 4 .1002.

So you just multiply the $5 million payment by that factor.

And you get a present value of $20 .501 million.

That's the all -in upfront cost of that lease in today's money.

And again, timing matters.

That's for payments at the end of the year.

If the payment started immediately, it's an annuity due.

And the adjustment is the same as always.

You just multiply the result by one plus R.

So 20 .5 million times 1 .07.

Which would be 21 .94 million.

The sooner you have to pay, the higher the present cost.

Hashtag amortizing loans and calculating payments.

Now we can flip the problem around.

This is super practical.

Most loans, like your mortgage, are amortizing loans.

Meaning each payment you make has two parts.

Some of it is interest, and some of it goes to pay down the principal.

Right.

And in this case, we know the present value, that's the loan amount, we need to solve for the payment C.

So we're just rearranging the annuity formula.

Exactly.

The payment C is just the loan amount divided by the annuity factor.

The example is $1 ,000 loan for four years at 10%.

First, we find the four -year annuity factor at 10%, which is 3 .170.

Then we take the loan amount, $1 ,000, and divide by that factor.

And you get an annual payment of $315 .47.

And that level payment, every year for four years, will pay off the loan completely.

The way it gets paid off is really interesting though.

That's what the amortization schedule shows.

Let's walk through year one.

Your starting balance is $1 ,000.

The interest for the year is 10 % of that.

$100.

But your payment is $315 .47.

So after you pay the $100 of interest, the rest.

$215 .47.

Goes to reduce the principal.

So your new balance is now only $784 .53.

So in year two, the interest charge is smaller.

It is.

It's only 10 % of the new lower balance.

It's 78 .45.

Since your payment is still the same,

a much larger chunk now goes to paying down the principal.

And this is the key thing for anyone with a mortgage to understand.

In the early years, almost your entire payment is just gonna interest because the balance is so high.

It feels like you're not making any progress.

But as time goes on, that flips, and in the final years, almost your entire payment is going to principal and you pay it off really fast at the end.

We can see that with the big mortgage example, $250 ,000.

For 30 years, at 12%.

The annual payment is over $31 ,000.

And if you look at the graph, after 15 years, halfway through the loan, the homeowner has paid off less than 20 % of the principal.

It's brutal.

It just shows how much value is transferred to the lender in the early stages of a long -term loan.

Hashtag tag future value of an annuity.

Okay, let's look at it from the other side.

Not borrowing, but saving.

Right.

We wanna know the future value of an annuity.

If I save a certain amount every year, what will my pile of cash be worth at the end?

This one's actually easy.

We don't need a whole new formula.

We don't.

We already know how to calculate the present value of the annuity.

So we just take that PV.

And compound it forward to the future using our standard future value formula.

The FV of an annuity is just its PV times one plus R to the T.

Let's say you're saving for a sale, though.

You put away $20 ,000 a year for five years and you earn 8%.

Step one.

Find the present value of that savings stream.

We use the five -year annuity factor at 8%.

The PV is $79 ,854.

So that's what those future savings are worth in today's money.

Step two.

Now we just need to find out what that lump sum would be worth in five years.

So we take the 79 ,000 and multiply it by 1 .08 to the power of five.

And the final number is $117 ,332.

And there's your sailboat fund.

It's the combination of your consistent savings and the power of that compounding interest.

Hashtag how to value growing perpetuities and annuities.

Up to now, we've assumed all the payments are level.

But in the real world, things grow.

Costs go up with inflation.

Companies try to grow their dividends.

So we need a way to handle cash flows that increase at a constant rate, which we'll call G.

Hashtag, tag, tag, growing perpetuities.

So a growing perpetuity is a stream of payments that lasts forever.

But each payment is bigger than the last, growing at a constant rate G.

Right, let's go back to our malaria foundation.

If their costs, the $1 billion,

are expected to rise by, say, 4 % a year due to inflation.

Then that $10 billion endowment we calculated is not gonna be enough for long.

It's not.

The payments will quickly start to outpace the interest earnings, so we need a new formula.

And this formula is one of the most important in all of finance, especially for valuing stocks.

It is.

The present value of a growing perpetuity is C1.

That's the cash flow at the end of the first year, divided by the difference between the discount rate and the growth rate.

So PV equals C1 over R minus G.

Let's plug in the numbers for the foundation.

C1 is 1 billion.

The discount rate, R, is still 10%.

And the growth rate, G, is now 4%.

So the denominator is 10 % minus 4%, it's 6%.

The calculation is 1 billion divided by .06.

And the required endowment jumps to $16 .667 billion.

It's almost $7 billion more than what was needed for the level of perpetuity.

The economic logic makes sense.

The investment doesn't just have to cover the payment, it has to earn enough extra to fund the 4 % growth in the payments forever.

And that leads to a critical constraint, a rule that you can't break.

Which is that R has to be bigger than G.

The discount rate must be greater than the growth rate.

If they're equal, the denominator is zero and the math just blows up.

You get an infinite present value.

And if G is bigger than R, you get a negative value, which makes no sense.

So you can only sustain a perpetually growing stream of payments if your investments are earning a higher rate of return than the rate at which your costs are growing.

Hashtag, tag, tag, growing annuities.

And now the finite version, the growing annuity.

A stream of payments for T years where each payment grows by G%.

And the best way to understand this is with the lottery prize example.

Oh yeah, the Mega Millions prize.

The headline number was $1 .54 billion.

Which is the nominal sum of all 30 payments.

But the fine print is what matters.

The first payment was $23 .18 million.

And then each payment after that grew by 5 % a year for 30 years.

And here's the interesting part.

The market interest rate at the time, the discount rate R, was only 3 .4%.

So in this case, for this finite period, the growth rate G is actually higher than the discount rate R.

Which is okay for an annuity because it doesn't go on forever.

Right.

And to value this, we need the big complicated growing annuity formula.

We don't need to recite it.

But it essentially takes that C1 over R minus G term and then adjusts it to account for the fact that the stream stops after 30 years.

When you plug all those lottery numbers in, what's the PV?

Assuming the payments come at the end of each year, the present value comes out to $847 .7 million.

But hold on, lottery winners get their first check right away?

They do.

So it's technically a growing annuity do.

We have to make that familiar adjustment.

You multiply by one plus R.

We take that 847 million and multiply it by 1 .034.

And the final true value of the prize is $876 .5 million.

Now stop and think about that.

The advertised prize was 1 .54 billion.

The actual take it today cash value was less than 900 million.

The time value of money just vaporized almost half the advertised value.

It's a brutal financial lesson.

Nominal numbers spread out over decades mean almost nothing.

It's the present value that tells you the true economic worth.

Hashtag tag how interest is paid and quoted.

Okay, through all of this, we've kind of just assumed we know what R is.

But in the real world, the way rates are quoted can be incredibly misleading.

This is a huge source of error for consumers and even for some financial managers.

It's all about compounding frequency.

Hashtag tag annual percentage rate, APR, versus effective annual rate, EAR.

So when I see an ad for a car loan that says, I don't know, 10 % a year,

what am I actually looking at?

You're usually looking at the annual percentage rate, or APR, and the APR is the simple interest rate over a year.

It ignores the effect of compounding within that year.

So if a bond says it pays 10 % APR, but it pays semi -annually.

That means it pays 5 % every six months.

Let's trace the money.

I invest $100, after six months, I have 105.

Right, and then for the second six months, you're earning 5 % on that new higher total.

So 5 % of 105,

which means at the end of the year, I have $110 .25.

So even though the quoted APR was 10%, you actually earned 10 .25%.

And that 10 .25 % is the true rate?

That's the effective annual rate, or ER.

It's the rate you would have needed with annual compounding to get the same result.

That extra quarter of a percent is the interest you earned on your interest.

Why do they even quote APR then?

It seems deliberately confusing.

Sometimes it is.

It makes loans look cheaper than they really are.

If you're comparing two loans, you have to convert both of their APRs into years to do a fair apples to apples comparison.

What's the general formula for that conversion?

The ER equals, in brackets, one plus the APR divided by M, where M is the number of compounding periods per year.

And then you raise that whole bracket to the power of M, and finally subtract one.

Let's try it with a car loan that's compounded monthly, an APR of 12%.

So M is 12.

The monthly rate is 12 % divided by 12, which is 1 % a month.

So the ERR is one plus .01 raised to the 12th power minus one.

Which comes out to 12 .68%.

So the true cost of that loan is almost 13%, not the 12 % in the ad.

The rule is simple then.

The more often interest is compounded, the higher the effective annual rate.

Hashtag tag continuous compounding.

So what do we take that compounding frequency M and just push it to the absolute limit?

Not daily, not hourly.

But infinitely, every instant.

That's the idea behind continuous compounding.

It's a theoretical concept, but it's incredibly useful in finance, especially for pricing complex things like stock options.

This sounds like it involves some heavy math.

It involves Euler's number, E, which is about 2 .718.

As M approaches infinity, that ER formula we just used converges to a much simpler expression involving E.

So the formulas change.

They do.

The future value becomes PV times E raised to the power of R times T.

And the present value formula becomes the cash flow, C divided by E to the R.

If I have $1 and it's earning 11 % continuously compounded for one year.

Its future value is raised to the power of .11.

Which is about 1 .1163.

So 11 % continuously compounded is equivalent to an ER of 11 .63%.

It's the highest possible effective rate you can get from an 11 % APR.

When would a manager actually use this?

You'd use it when your cash flows aren't just coming in at the end of the year.

Think about a supermarket.

Sales are happening all day, every day.

It's a continuous stream.

Or someone in retirement spending money every month for living expenses.

Exactly.

The retirement planning example shows why this matters.

A person wants to spend 200 ,000 a year for 20 years.

If we assume that's a single payment at the end of each year and the rate is 10%.

It's a standard annuity.

The PV, the amount they need to have saved is about $1 .7 million.

But it's much more realistic to assume they spend that money evenly throughout the year.

To model that, we have to use continuous discounting.

And when we do that, the amount they need to have saved jumps.

How much does it jump by?

It rises to almost 1 .79 million.

Wow, that's a difference of over $80 ,000.

$83 ,000.

Just from modeling the timing of the cash flows more accurately, it shows you that these details, the compounding frequency, they have a massive impact on the final numbers.

Okay, we've gone through a ton of math.

And doing this by hand, especially an amortization table or growing annuity.

It's a nightmare.

You'd never do it by hand in the real world.

You'd use a spreadsheet,

Excel, Google Sheets.

They have all these functions built in.

Powerful functions, PV, FV, rate, PMT for payment.

They can solve for any of the variables in these equations almost instantly.

But there are some big gotchas, some warnings people need to hear before they start plugging numbers in.

Absolutely.

First, you have to be careful with positive and negative signs.

A cash outflow, like a loan you receive or an investment you make, should usually be entered as a negative number.

And rates always go in as decimals.

5 % is .05.

And then there's the big one, the most dangerous trap, especially for students learning this.

The NPV function in Excel.

The NPV function in Excel is, frankly, misnamed.

It does not calculate the net present value.

Wait, what?

It calculates the present value of a stream of cash flows.

Assuming the first cash flow in the series you give, it happens one period from now.

So if I have an investment where I pay money today, C zero, and then I get money back starting in year one, and I highlight that whole range of cells and use the NPV function.

You will get the wrong answer.

Because the function will discount your C zero payment as if it were happening a year from now, which it isn't.

So what's the right way to do it?

The right way is to use the NPV function only on the future cash flows from C one onward.

That gives you the present value.

Then you go to a separate cell and you manually add your C zero, which is usually a negative number.

So PV minus the initial cost.

That will give you the correct NPV.

Not understanding that one little quirk about the software could lead a company to make multi -million dollar mistakes.

Hashtag tag outro.

Okay, so let's just pull it all together.

We have built this incredible toolkit from the ground up.

First, we established future value and saw the absolute power of compound interest, all wrapped up in that formula.

FV equals PV times one plus R to the T.

Then we flipped it around to get to present value and the most important decision rule in finance,

the NPV rule.

Except projects only when the PV of what you get back is more than what you put in.

We learned that the right discount rate is always the opportunity cost of capital.

It's all about the return your shareholders could get elsewhere on an investment of similar risk.

It's about time and it's about risk.

Then we learned the shortcuts, the super simple perpetuity formula, C over R and the annuity formula.

And then we made them even more powerful by adding constant growth with that famous C one over R minus G formula.

And finally, we got really practical.

We learned not to trust quoted interest rates, to always distinguish between the misleading APR and the true ER because compounding frequency can dramatically change your returns.

All these calculations, this whole way of thinking, it's the foundation for everything that comes next, for valuing stocks, for valuing bonds.

Just think back to that lottery prize,

$4 billion on the billboard, but only $876 million in actual present day value.

Now turn that idea on yourself.

Think about your education, the cost of tuition, the time you spent.

That's a huge investment.

The payoff is a higher salary for the rest of your career.

What's the present value of that future stream of higher earnings?

Is it NPV positive?

That's something to think about on your own.

Thank you for sharing your sources and for taking this deep dive into the fundamentals of valuation with us.