# Interest rates: compound frequencies (FRM T3-8)

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“We re told. The interest rate is 8 percent per annum. We actually don tn t have enough information to discount or compound forward. I d like to compare a compound frequency to a continuous compound frequency and show you the formulas for translating between the discrete and continuous compound frequencies.

So i start with a simple example that only requires three assumptions first what is the future value i m going to assume 100 dollars. That we will receive in the future second how far in the future let s assume i m gonna receive the 100 dollars three years in the future and third what is the interest rate. We will use to discount. So what is the discount rate.

We will use to discount the 100 dollars to be received in three years. If we want to discount. It to today s present value that s the pv stands for present value so let s assume we re told that the discount rate is 8 per annum. The per annum is typical.

In general unless. We have a special need interest rates should be specified in annual terms. That is to say in this case 8 per annum. So here s an interesting thing that even some finance students forget so far we haven t been told enough information.

That s sort of interesting. Which is to say that the 8 percent per annum as a discount rate is an incomplete specification until. We are told what the compound frequency is so that s why i have the present value solved in three different ways for three different compound frequencies. Although these first two are just examples of a discreet compound frequency annual monthly.

There are any number of discreet compound frequencies. We could go in between with quarterly and we could go all the way up to a daily compound frequency. So we have several different discreet compound frequencies. We can choose and then we only have one continuous compound free see so my annual compound frequency for example you can see the excel is very straightforward.

But i ll also draw it out it s going to be just equal to my. 100 divided by one. Plus my interest. Rate so that s 108.

Raised to the third power. So that s pretty familiar method for discounting at 8 over three years. But if we look at that formula. We know that implicitly it s with annual compounding and then this is also called the effective annual rate.

So the effective annual rate is a special case of discrete compound frequency. Where the compound drink we ll see is annual but if i want to go to monthly compound frequency. How do i get that well i m just using 100 divided by the quantity. 1 plus.

My 8 rate. That s 8 not very good. But my point oh eight and i m dividing it by in this case. 12.

Because there are 12 months per year. So my denominator is the number of periods per year. And then that quantity instead of being raised to the 3rd power to be consistent. I need to raise it to the third years times the number of periods per year in this case.

12. So be 36..

So that my general form here for the discrete now i m just going to put it in general terms is a future value divided by 1 plus. The rate divided by m is the number of periods per year. And i m raising that to the n is a number of years times the number of periods per year. So that s my general form where you can see we can substitute any number of periods per year and as i went from annual the monthly my present value decreased okay what if we kept increasing them if we went to 250 that would be 250 trading days per year.

We could go to 365 days per year. If we kept increasing me number of periods per year. We would approach the limit which would be continuous with continuous compounding in this case were continuously discounting and it has a very elegant expression. It s just 100 times e raised to in this case my negative.

8. Multiplied by 3 years so that my general form here is future value multiplied by e raised to the negative rate times. Number of years so very elegant expression for the continuous compounding in this case. Continuous discounting and it s going to have the lowest present value of any of the alternative discount frequencies.

So that remember that point here was that we re told 8 percent per annum. But it has a different significance as with respect to 8 percent per annum with annual compounding with monthly compounding and with continuous compounding. So my second sheet. Here just replicates.

John holes table and shows this just from another angle. Now we go compounding forward in time as opposed to discounting. We say the initial value is a hundred eight percent is the eight percent per annum. Is the stated also called the nominal rate and again now this time.

We re compounding forward three years. So. What is the future value of 100 in three years. If we compound at eight percent per annum.

Okay so far incomplete. But with various compound frequencies. So now with m equals. One to four all the way up to 365 daily and you can see that our general form here for discrete compounding is the present value x.

So this time. We re multiplying instead of dividing one plus. The interest rate divided by m for the number of periods per year. So you can see annual as one sum annual is two quarterly is four and we re raising this to the power of n number of years times.

The number of periods per year to be consistent that s the general form and you can see in the excel spreadsheet. I ll provide the link these labels are dynamic. So we have different future values and as we increase. The compound frequency the future value as we might expect is increasing such that we get to the limit.

Which is continuously compounding and it s similarly it had an elegant expression in discounting. It has an elegant expression in compounding forward. It s just the earase to the rate multiplied by the number of years. Very elegant and it will be the highest future value it will be higher than any of the discrete alternatives.

Okay. So my final sheet shows. The tool. The tool that up finance candidate in the frm especially wants to have this tool that comes in handy.

And that is the ability to translate a discrete rate to its equivalent continuous rate or vice versa a continuous rate to its equivalent discrete rate. So in the that s what i have an epic panel going from discrete to continuous with the formula here that s solving for continuous and here going from a continuous discrete..

Which is to say solving for the discrete rate as a function of the continuous rate. So in the top row. Here for example and i ll just bold this what i have here is eight percent per annum with a compound frequency of once per year. So that means with annual pounding and so we have an 8 annual rate and the question is what is the equivalent continuous rate.

And that s solved for by using this expression c. Where we plug in the discreet rate of 8 in this case for when it s an annual discreet rate the m is one we take the natural log. We multiply that quantity by m in this case. 1.

And we get the continuous rate. That is equivalent to 8 percent per annum with annual compounding. What that means is that if we were compounding forward to a future value or discounting back to a present value we would get the same result. If we use this continuous rate as we would if we use this dis.

8. Percent with annual compounding similarly just for example. If we if we if we are given 8 percent per annum with monthly compounding the equivalent continuous rate. You can see here would be seven point nine seven three percent such that if we use that we would get the same present value or future value.

And so we can also go from continuous to discrete so in this case. A continuous rate of 8. That s fully specified. It s already got that it s telling us what the compound frequency is if we want to convert it to a discreet rate.

It has several different translations right there s only one continuous 8 percent. But there s a whole variety of different discrete rates. So if we want to get the annual equivalent. That would be a higher number.

It s eight point three to nine and so that s the interest rate. We would use to get an equivalent future value of four compounding forward or an equivalent present value if we re discounting back. So this is a key skill to be able to apply these formulas that translate or that in this case solve for the continuous rate that s the rate in c. For continuous as a function of the discrete rate that s mm and vice versa.

The we want to know how to solve for solve for a discrete rate as a function of the continuous rate and i just noticed that i was missing the m. So i just caught that before i finished the video and sort of the m back in and and made this formula match you can see the correct calculations below. So i hope that s and right now that is the g. That is gbtc.

It is the grayscale bitcoin investment trust. That s actually run by dcg. Which is barry silbert company. Which is essentially a holding company for a number of different blockchain.

Crypto related organizations. Now the problem with gbtc is that it because and we ve talked about this a little bit before in the channel. But because gbtc is basically the only place where you can go in a brokerage account. And even though.

It s it s an otc. So it s not traded on the new york stock exchange. It s not traded on the nasdaq gets traded on otc markets. But even though it s traded on otc.

It s one of the few places that you can go with a brokerage account to get exposure or direct exposure to the price of bitcoin and because of that because it really is the only place to go it trades at a significant premium to the underlying assets. So with that being said you are paying in some cases..

You paid a 30 40 50 percent premium for the gbtc bitcoin. Essentially that you would be paying. If you had bought bitcoin directly. And that is because you can acquire it directly through a brokerage account.

Now the interesting part about this is there is now what is called an exchange traded note. That american investors are able to buy this came out about two days ago. And this is something that was traded on a swedish exchange. But it is the cxb tf and something to take a look at here as well.

But the interesting thing about this too is that it has not previously traded at a premium to bitcoin. So. The gbtc investment instrument has typically traded at a significant premium to bitcoin. So you re basically paying more because it s the only place you can get it now with cxb tf.

You do not essentially have to pay a premium for it and cxb tf. Does say one of the one of the individuals. There does say they do see this as a directly competitive product. What gbtc has and they say historically their assets have not traded at a premium and are liquid.

So this could be another opportunity. No it s not an etf. I m not gonna sit here and say it is i m not gonna sit here and say it has any bit of the you know of the the level of institutional respect per se that a that that s something like an etf would have but this etn this exchange traded note. Which is again also on the otc markets similar to gbtc.

But the fact that it does not trade at a premium will be interesting to watch this pan out now. Spencer noone. Who has it was an investor in the crypto investor in the space will pull this over here. But he s been keeping an eye on it and on the volume as well as the premium.

And what is happening to the premium of gbtc as cxb tf comes out and as you re looking at this so he tweeted this about a day ago. Buddy joked around but seriously free bitcoin alert. If you own gbtc you re paying about a 50 premium to the underlying spot price of bitcoin. As of yesterday.

There is a new way for retail investors to get exposure with no premium. So gpgc holders can sell for cxb tf and get 50 more bitcoin instantly makes a decent amount of sense to me and in the past about 24 hour volume range. He tweeted on the 16th it was about 3 million dollars in volume right now it s about 27. Million dollars in volume.

I believe but the gbtc interestingly enough the gbtc premium over the past couple days has actually come down a little bit because of the competition from cxb tf. So something to keep an eye on just another potential instrument for people to get exposure to bitcoin that is not necessarily spot bitcoin. That is not self custody bitcoin. But you can actually buy it through a brokerage account.

So that s something that you know i ve kept my eye on there as well and lastly. So i ll pull this up. And we ll pull up. Et.

Cie and there s a lot of you folks know. Et. Cie was added to et. Cie was added to coinbase so here i pulled up on pitch rex so etc.

Was added to columbus and had when that announcement happened it was a actually a pretty nice little bit of a run up recently and then sold off hard and then we bottomed out here a little bit during that crash. But bottom doubt..

It wasn t a specific date. That was provided as soon as that date was provided if we zoom in a little bit. Here. The price of etc just popped up and a second basically the second.

It hit coinbase or columbus pro. You saw another dump off there as well so one of the things that not necessarily diving. Too too far into depth on the exact time frames of aetherium classic. And what happened here.

But the interesting thing to me is that in in some cases in the past few months. There have been certain times where certain assets even if they ve had something big happen they ve had a big piece of news. Whether it s in this cases of it s a rarity. But getting out at the coin base is obviously a rarity maybe they wouldn t mean that maybe they had a product launch whatever it might have been but the market hasn t reacted in a positive or negative way in this case.

You can directly see the market reacting. I will pull this up here. You can directly see the market. Reacting about 25.

Over 25 percent about 30 percent in the course of two days. When the news that quota theorem classic was going to get added to coinbase and then it since sold off. Obviously another case that by the you know buy the rumor sell the news or by the announcement. So when the announced thing actually happens that s usually the better scenario.

But definitely something to you know something to think about here. And the same can be said for polymath. A little bit as well so polymath polymath jumped up and it hasn t even really sold off yet. But polymath had a nice bounce actually when they announced so they tweeted out something jumped about 30.

They tweeted. I ll see if i could pull it up here. But they basically tweeted hey. We re announcing something at the i forget.

The conference s conference in toronto. They said hey we re announcing something in the conference in toronto. They actually launched on may net and i actually went through the process of creating a buff token b. Uff.

I created my own security talk and i paid 500 polymath tokens. You create a buff token and then i went through the entire process of actually creating the security. Token offering and then the areand you actually have to pay 20000. Pali tokens.

Which is about three thousand seven hundred dollars give or take yesterday was and i didn t feel like paying three thousand seven hundred dollars to launch a pali to launch a buffalo chicken token. So i did not do that but again going to the point of a theorem classic and pali you re seeing some of these assets start to maybe stray from the pack. A little bit. Which i think is you know is is a good thing you re seeing a maybe a little bit of a disconnect in the correlation that we ve seen in the past.

So that s happy to happy for me to see some of these assets when when news is announced when something seemingly positive is announced for them to actually react in a negative way and not just get dragged down by the rest of the market now outside of that i m just keeping an eye on things in general this weekend. So i hope you have a fantastic one if you enjoyed this video hit the thumbs up on hit the like button would really really appreciate that and if you re new to the channel by any chance. We d love for you to subscribe thank you so much for your time krypto bobbi. ” .

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