It is important to understand continuously compounded rates. These rates are rarely encountered in day-to-day life, but are relevant to a finance professional. You will never see, for example, a bank advertise ‘continuously compounded rates’ for its deposits. (In fact, it may even be against the law to do so as they may be required to disclose easier to understand APRs).
To understand continuously compounded rates, think about natural processes. Think about how, for example, population grows. It does not grow in discrete steps. It just grows all the time.
It is important to understand continuously compounded rates. These rates are rarely encountered in day-to-day life, but are relevant to a finance professional. You will never see, for example, a bank advertise ‘continuously compounded rates’ for its deposits. (In fact, it may even be against the law to do so as they may be required to disclose easier to understand APRs).
To understand continuously compounded rates, think about natural processes. Think about how, for example, population grows. It does not grow in discrete steps. It just grows all the time.
Or how, if you remember primary school physics, a sugar crystal grows around a suspended thread in a supersaturated sugar solution. Say our crystal grows at a rate of 100% annually (very bad example, but you get the picture). But this does not mean that till the end of the year the crystal stays where it was, and then boom, suddenly it doubles as the clock strikes 12. In reality, it just grows. It grows at a natural rate, which if left to itself, would make it double in size in a year. It is not growing at monthly rests, or quarterly intervals, but is growing all the time at what may be called an ‘instantaneous’ rate.
Examples from the physical world
Let us take the example of a snow ball that is rolling downhill and growing bigger every minute. Say it has a diameter of 10 cms to begin with, and a nanosecond later it is just a little big larger, and has a slightly larger surface area that allows it to catch & pack even more snow, and then the next nano second later it has grown just a wee bit bigger, and so it rolls on. For this snow ball, we could take the difference between the size of the ball at two points a minute apart in time, and come up with a rate of growth per minute. Say that rate turns out to be 50%. But that rate of growth would be misleading because it would be based upon the initial 10 cm size of the ball. After the first nano second, the ball had already grown to more than 10 cm, and was growing faster. Towards the end of our hypothetical minute, the rate at which snow was added was different from the rate in the beginning. Our calculation of 50% masks that difference. If we have to get the ‘instantaneous’ rate, we have to take the log (with e as base) of the ratio of the size of the ball at the end and at the beginning of our minute.
This ‘instantaneous’ rate is what a continuously compounded rate is.
Discrete rates, on the other hand, as expressed in the form of (1 + r)^n. Everything grows at a constant rate till n arrives, and then the ‘growth’ gets added to the principal, and the cycle starts again. In reality, that is not what is happening. The growth is actually happening all the time, only that to measure it between two time intervals we ignore the ‘interest-on-interest’ for periods shorter than our interval.
Continuous compounding
Let us take another example. If we start at time 0 with principal P0, and after a while, we get to time 1 (say the end of one year from time 0) and find that our principal has grown to P1, we can say either of the following:
- We can say that we earned [(P1 – P0)/P0] rate of interest on a simple interest basis.
- If we like quarterly compounding, we can say that the rate of interest earned during each time period would be ((P1 – P0)/P0)^(1/4). We will then multiply it by four to get the total return during our time period.
- If we like continuously compounded rates, we will say that the rate of interest earned was ln(P1/P0). Now this is a big leap. How did this happen? How did ‘ln’ come into the picture? What happened to time periods? What does ‘ln’ mean in the first place? Let me try to answer these to the best of my abilities – but there is no guarantee you will get it. Try to follow me, but if you get lost, that is okay, so long as you remember that the continuously compounded rate is ln(P1/P0).
e and ln
What is ln? ln is the logarithm to the base e. What is e? Suppose you have $1. It ‘grows’, spinning out fractions of a cent every second. In turn, each fraction starts generating more income, more micro-fractions of cents every second. A bit like a little pyramid scheme. (only that this one would be real and work). Say that the rate of growth is 100%. What does that mean? When I say 100% rate of growth, it means my “original” dollar will grow by 100%. I am not counting the interest on interest, just the interest on the original. That is what I mean by 100% (This would be like a situation when I placed money in a bond, and instantly bond yields fell to 0% so my earned interest makes no money. Another way to think about it: assume we have a rabbit colony where the original parents reproduce at the rate of 100% during the period, not counting the second or subsequent generation rabbits). Of course, in reality my interest would start earning more interest the moment it is generated, and more would follow.
So the question is, what does my $1 grow to at the end of one year given a such a ‘instantaneous’ growth rate of 100%? I can think about this as being a compound interest rate where the period of compounding is very short. Say daily. Thus, my $1 would have grown to (1 + 100%/365)^365 = $2.714. Now if I increase the compounding frequency to say, 1 second, I would find that my ultimate number goes up by a little. In fact, as I keep increasing the compounding frequency to mimic ‘continuous compounding’ every nano second, I find that my number ends up being 2.7181 which is ‘e’. (Try it out in Excel. Put a formula in for (1 + 100%/n)^n, and keep increasing the value of n. As n approaches infinity, the value approaches 2.718, which is represented as e).
In other words, a growth rate of 100% ends up producing a value of 2.718 and that is really what e is. A different way to say it is that a growth rate of 100% (whether achieved over 1 year, or 1 second) produces e^100% in the end. If I had a growth rate of 200%, I would have ended up with e^200%. Now think that you started with $1 and ended up with $x. What was the rate of growth? Well, it was whatever power you need to raise ‘e’ to to get $x. Say that power is y. I started with $1. Ended with $x. If my ‘instantaneous’ rate of growth was y, then I can find y if I solve e^y = $x. (knowing that e = 2.718). Or in other words, y =ln($x)
Now think you started with P0, and ended with P1. What was the continuously compounded rate of growth? If I started with P0 and ended with P1, that means for every $1 I ended up with P1/P0. So what was my rate? ln(P1/P0).
That is how it works. Try reading it again to see if it makes sense.
Now on to thinking about what’s the deal with ‘geometric rates’?
Geometric rates are different from continuously compounded rates. They tell you what the discrete rates were for each period, with a certain compounding frequency. If you earned 10% in year 1 (discrete), and 15% discrete rate in year 2, then your real rate earned each year is √[(1+10%)*(1+15%)]. This is still not the continuously compounded rate though, it just gives you the equivalent annually compounded rate.
At the end, it does not matter what rate you use so long as you understand what’s really going on and you have a strong conceptual understanding of the matter.
What happened to time periods?
So far, talking about the continuously compounded rate of return I did not talk about time periods. Why is that? Well, when we say something has ‘grown’ by 100%, this is regardless of the time period – because time is ‘absorbed’ away in this one number. Think of the rabbit colony, say that grows at the rate of 100% every year and this 100% is only for the first generation, and not counting the offsprings of the offsprings. This first generation 100% comes about in an even fashion – 50% in the first 6 months, and 50% in the second six months. So I could have said, the rabbit population grows at the continuously compounded rate of 50% for 2 periods, or 100% for 1 period, and it would have meant the same thing. In the first case, I would have ended up with E^50% at the end of 6 months, and that would have grown to whatever it started off with (ie e^50%) multiplied by e^50%, or (e^50%)^2, or e^100%, which is the same as for the other case.
So future value of an amount growing at a rate r per time period is nothing but e^rt. If we know the future value, we can calculate the continuously compounded rate by doing ln(future value/present value). The rate we get will be the ‘growth’ during that period, and we can convert it to a continuously compounded rate for another period by simply multiplying by the new time period.
Decay rates
Now e^rt is really cool because it applies not just to growth processes, but also loss causing processes ie where r is negative. These are technically called ‘decay’ rates. For example, consider a large portfolio of car loans. A bank knows that 10% of these will go bad. Now if it starts the year with $100 of loans, at the end of the year it will be left with only $90 worth of loans. You can calculate the ‘instantaneous’ rate of default in this case to be ln (90/100) = -10.54%. Why is this higher than 10%? Because the loans are reducing in value every day, and every day there is a smaller principal to decay. So the continuous rate has to be higher than the discrete rate of 10% (which applies to the initial $100, even though $100 was the principal only for a nano second after which it just kept falling).
Got it? Well, if not, it will come eventually. Exam III will grind it in.