In this topic, interest rates, I want to start off by talking about interest rate quoting conventions and I want to talk about how to compute the present value and future value of a string of cash flow when they arrived at the irregular time, non-annual and when the compounding is not annual as well. Now, I want to turn to interest rates in this lecture and this is not so much a new topic as much as it is really an extension of the time value of money to incorporate institutional details and make things a little bit more realistic. So we’re going to talk about interest rate quotes and we’re going to learn how to deal with cash flows that don’t arrive once a year, but may arrive monthly or semiannually. And we’re going to discuss how to deal with compounding of interest when it’s not just annually. So let’s get started. Here is a snapshot from December of 2014 of rates on five year jumbo CDs, where CD just stands for certificate of deposit. It is a savings vehicle most banks offer and here is one, two, three, four banks. Now the jumbo that just refers to, I think a minimum deposit of $100,000. So these are big deposits. And one of the things you notice when looking at these rates is when you’re looking at these savings vehicles is that each one has two different rates. It has a rate and an APY and these numbers 2.37, 2.4, they are different. So that begs the question is why are they different? How are they related? And most importantly, which one is going to tell me how much money I’m going to make when I invest in this product? Well, let’s go through this starting with the rate. The rate refers to the APR of the Annual Percentage Rate. That measures the amount of simple interest earned in a year. Simple interest is just the interest earned without compounding, ignoring compounding. And if you’re wondering what compounding is we’re going to talk about it, but just as a preview. Notice underneath rate, we have compounded daily. Simple interest ignores that compounding frequency.
Now many banks quote interest rates in terms of an APR. The problem is the APR is typically not, what we are going to earn or what we are going to pay. For that, we have to turn to the APY or the Annual Percentage Yield, which is really just another way of saying EAR or effective annual rate. See, the EAR that measures the actual amount of interest earned or paid in a year. That is what we care about the EAR, that is the number we care about. The rate or APR that is just a quoting convention. Now how are these different rates related? Well, before showing you the explicit mathematics, which frankly are almost trivial, let me emphasize the following. The EAR is a discount rate. The EAR is what matters for computing interest and discounting cash flows. The APR is not a discount rate, it’s a means to an end. It’s a quoting convention. So we’re going to use APR in conjunction with compounding frequency information to get at an EAR or at a periodic discount rate, which I’ll introduce in just a moment. So remember, EAR = discount rate. APR = quote.
Now how do we get from one to the other? How do we go from APR to EAR or vice versa? Well, here is that simple mathematical formula I referred to just a moment ago. The EAR is related to the APR by this equality. Well, what is going on here? K, that is just the number of compounding periods per year. So imagine we had monthly compounding, that would imply k = 12. How about semi-annual? That would imply k = 2 and I’m going to introduce a little bit of notation here. I is the periodic interest rate or the periodic discount rate and that = APR over k. Let’s do an example. So, imagine we are investing $100 in a CD offering 5% APR with semi-annual compounding. How much money will we have in one year? Well, there is actually two ways to approach this problem and we are going to do each in turn. The first thing we do, first thing we always do is we draw a timeline. And so today, period 0, we’re going to invest $100. And the question’s asking, how much money are we going to have in one year? Now, I’ve left this as a question mark to emphasize the fact that there are two ways to go about this. Let’s go about this the first way. Go about answering this problem. The first is to work in periods. With semi-annual compounding, that means there are two periods per year. Since I’m interested in how much money I have after one year, that’s after two periods. And these periods are every 6 months, period one, period two. So, if I’m going to work in periods, I better compute a periodic discount rate, that’s i. Which we know is APR over k and which in this case, reduces to 2.5%. In other words, I’m going to earn 2.5% over the first 6 months and I’m going to earn 2.5% over the next 6 months.
So I take my initial $100 investment, I multiply it by my periodic discount rate. And after 1 period, I’ve got $102.50, then I repeat. I take that $102.50, I multiply it by periodic discount rate to get $105 and a little over $0.6.
In other words, the future value 2 periods hence of $100 in this setting, it’s just 100 x(1+ i) 2. We are working in periods, so i is our discount rate and 2 is the number of periods removing the money forward in time. Now let us approach the problem from the prospective of years. Now we are looking for how much money we have after one year. But if we’re going to work in years, now we need to be consistent. So after six months, this is not one period, this is half a year. And because we are working in years, our discount rate isn’t going to be the periodic interest rate, i, it’s going to be the EAR, the equivalent with the effective annual rate, which we know from earlier is just one plus the periodic rate raised to the number of compounding periods per year. So 1 + i to the k, which in this settings comes out to be 5.0625%. In other words, over this entire period, I am going to earn 5.0625%, which turns out to equal $105.06. That is the future value one year, from now of $100 is $100 x (1 + the EAR) 1. We are working in years, which gives me the exact same answer we got before, the $105.0625. So the lesson, if you discount cash flows using the EAR, then you better measures time in years. If you discount cash flows using the periodic interest rate, then you better measure time in periods. And the equality between the two that we just showed with our simple example is much more general and given by this straightforward proof, which I’m not going to discuss in detail, but I’ll show you there if you’re interested in looking at it.
So just to summarize, we can work in periods.
One, two with a periodic interest rate or we can work in years with our Effective Annual Rate. And notice I measure everything consistently. Periods versus years.
So let’s go back to our original example. We had an APR of 2.37%. We have a compounding frequency of daily, which I am going to assume is 365 days. So you should be aware, it could be 360 days or it could be 252 business days. It depends on the convention used by that product in that institution. So these two pieces of information, the APR and k allow me to compute the periodic rate, i, which is 0.006714%. Now that’s a tiny number, but we’re computing interest every single day here, which means the effective annual rate, which equals (1+i) to the k. Or which in this case is (1+0.006714%) raised to the 365th power gets me to just under 2.4%. So when we round the 2.398, we get about 2.4%.
So let’s summarize. We learned that there are two discount rates depending on what unit of time you want to work with. If we want to think in term of periods, we want to use a periodic discount rate, that is i. If we want to work in year, we want to use the EAR, the effective annual rate, which relies on cash flows measured in years. Both of these are discount rates, they’ll both get us to the same goal or the same end result, but we have to be consistent in terms of how we measure time with which discount rate we use. APR, that is a quoting convention, that’s a means to an end. We use APR in conjunction with the compounding frequency to get our discount rate, whether it is the EAR or the periodic discount rate, i.
And we can move between the APR and I and the EAR by a couple of very simple mathematical relations that we discussed. Now, next are a bunch of great problems that I want you to dive into. And then after you are done with those, move on to the second part of interest rates in which we are going to investigate the term structure of interest rates and talk about the yield curve, but you know what these things mean.