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# Term structure of interest rates

Last time we talked about interest rates. In particular, we talked about how interest rates are quoted versus how interest rates are used to discount cash flows. We also talked about how to deal with cash flow streams when the cash flows arrived more than once a year or less frequently than once a year, and when the compounding frequency was not annual.

This time I want to talk about how interest rates or discount rates, can vary over time and how that relationship is captured via term structure of interest rates and the yield curve. Let’s get started.

Hi, everybody. Welcome back to Corporate Finance and our 2nd lecture on interest rates. So last time we introduced the topic by talking about interest rate quotes versus discount rates, and we talked about APR, annual percentage rate, which was, means of quoting interest rates, the financial institutions often use. That is often distinct from what we care about for discounting cash flows, which is, an EAR, effective annual rate, or a periodic discount rate. And we talked about how to move between these concepts or these constructs, and then we showed how to apply them to non-annual cash flows and in situations where we have non-annual compounding. Say monthly, semi-annual, whatever you might have.

Today I want to talk about the term structure of interest rates and the yield curve. And this lecture’s a little bit different in that it’s not geared toward solving problems per se. At least not directly, because that is going to take us into a fixed income valuation which is beyond the scope of this course. Rather what it is going to do is it is going to be important to understand what these concepts are, because they are going to be used for corporate decision-making later on. So let’s get started.

Thus far, we have assumed discount rates are constant through time. They just do not change. And what do I mean by that? Well, if I look at my present value formula right, I take my cash flows and I discount them by one plus the discount rate. But notice that’s the same discount rate. It is the same number for every cash flow regardless of when the cash flow arrives. Now, in reality, it seems as if Interest rates vary with the term of the investment. Let me give you some examples. Here is a screenshot of home mortgage refinancing rates that I took not too long ago, and you can see that as the term of the mortgage refinancing varies, so too does the rate or APR, and consequently, so does the EAR, the discount rate.

Likewise, when I looked at fixed term CD rates where, remember, CD’s are just certificates of deposits. Their rates tend to vary with the term of the investment as well, and there’s a lot of numbers here, so let me focus your attention here we have the term of the investment.

And here we have the APR.

And the EAR.

And you can see that as the term increases or changes, so to do the interest rate.

Now, what is the point here? Well, as the term of the investment changes, quite often, but not always, quite often, the interest rate will change. And the term structure is nothing more than the relation between the investment term and the interest rate. The yield curve is just a graph of that relation, so let me show you a treasury yield curve from July 24, 2014. On the horizontal axis we have the maturity of the treasury security, so here’s a one-month T-bill. Right here is a 30-year T-bond. Here is a five-year T-Note.

So these are just different treasury securities that vary by maturity across the horizontal access. And on the Y access is the yield, which for the time being and in this context, you can think of loosely as the discount rate, R.

And the point is, is that as the maturity of the security varies, so too does the discount rate, right? In other words, when the government borrows for 30 years, it is getting. Or paying an interest rate just above three percent. Whereas when it is borrowing over a short horizon, say thirty days. It is paying basically zero.

Now, let me come back to this notion of a yield, and what a yield is. A yield, y, is the one discount rate that when applied to the promised cash flows of the security recover the price of the security, and that is a mouthful. So, let me actually show you a little formula that you are familiar with to hopefully clarify this. See, I take the price of the security. On the last, slide the price of the T-bill, the price of the T-bond. And I lay out all of the promised cash flows to the security. I discount them back and I ask what is the discount rate such that when I discount these cash flows, I get the price?

That is the yield or yield to maturity.

So to build the yield curve, that’s just a matter of simply computing the yield for securities of different maturities.

So without getting into the institutional details and the semiannual compounding and quoting conventions associated with treasuries, let me just talk conceptually, so if I want the one year, the one-year yield, I would just take this say cash flow at year one, divide by one plus y1, set it equal to the current price and solve for y.

If I want the second yield, I would take the second security with maybe annual cash flows of one and CF1 and CF2, and I would solve for the one discount rate, y2 let’s call it, such that when I discount these cash flows, CF1 and CF2, I get the price, P2. And we do that for all different maturities. The Y’s that come out, Y1, Y2, Y3… Dot, dot, dot, dot all the way down to YT. That represents the yield curve. Those are the points on the yield curve. But that’s the same as computing the discount rate for securities with different maturities. Hence, the length between discount rates and yields. Now they are doing a difference between discount rates and yields have to do with promised versus expected cash flows. But that’s a little bit outside the scope of the course, so let’s leave that aside for now. And in the context of treasuries yields and expected returns are relatively close.

Now one thing I want to emphasize is that yield curves, in another words, interest rates they move around a lot. Or at least they can. So to illustrate that fact I’ve plotted three yield curves here, the purple one is from 2012. The blue one is from 2000, and the red one is from 1981. And you can see that the rate at which the government was borrowing has very dramatically, over time. Right. Today For short-term loans, they are not paying any interest. The interest rate is basically zero. But back in 1981, the interest rate was around 15%. The other thing I want to point out is that the relationship between the short end of the yield curve, short-term interest rate, and the long end of the yield curve, long-term interest rates that can vary over time as well. Here, at least in 2012, we see that the curve is upward sloping, so that interest rates, short-term loans to the government, are less costly than long-term loans.

Whereas back in 1981, we see that long term interest rates were actually below shorter term interest rates, at least over a stretch here. So it was cheaper for the government to borrow with longer rates, or over longer terms at least, at a lower interest rate. Now, this raises the question, well what does this mean?

What does the upward sloping curve in 2012 mean as opposed to the downward sloping curve in 1981? Well, there is a lot of academic debate about that, but one popular opinion is that the slope reflects expectations of future interest rates. So when the yields curve is upward sloping, as it is in 2012, this suggests that future interest rates are likely to be higher. Whereas back in 1981, this downward sloping portion, also actually in 2000 it looks slightly downward sloping as well. The downward sloping curve suggests that future interest rates are going to be lower.

Treasury yield curves, what they are doing is they are graphing the relationship between interest rates on, on risk-free loans and loan maturity. So they, they have a very special meaning. When we talk about the risk-free rate, we are really talking about.

The interest rate, or the yield on treasuries. But we can plot yield curves for a host of different securities, and that’s what I’ve done on this next slide, is I’ve got three different yield curves. The blue curve is the yield curve for high-quality corporate debt, all right. So you can see that for, what do I mean by high quality? That is investment grade, think triple-B or higher, okay? So when high quality or highly credit rated firms were borrowing for say, 29 years, it’s costing them about 5% per annum at least as of July 2014. Whereas when they are borrowing short terms, say two years, it’s a little over a percent. The green curve is the yield curve for municipal bonds; AAA rated municipal bonds as of July 2014. And then the red curve is the treasury curve. And what’s sort of interesting to note Is that the yields or the cost of borrowing appear to be higher for the federal government over at least certain periods, than it was for municipalities. Which is strange if you think that our federal government is a much safer bet than a municipality, even a triple A rated municipality, but what’s going on there is mostly a tax differential as well as some liquidity issues.

All right, so what is the lesson here? Well, look. Yields vary by maturity and risk. So to illustrate that point, I’m going to look at the yields on corporate bonds. And you can see as the credit rating improves from triple B to triple A, which is the best rating, the yield goes down and I know drawing the up arrow isn’t helping, but the numbers are getting smaller. And they’re getting smaller within each maturity bucket.

One to two, two to five, five to ten, ten plus.

And that’s the result of decreasing credit risk. These are safer bonds, these are safe. These are less safe, but still relatively safe.

The other thing to notice is that within a credit rating, say AAA, the yields, or the interest rates, are different. Now, in this case, they’re increasing, but as we showed a few slides ago right, they don’t always have to be increasing with maturity. They could be decreasing.

The point is that they are different across the maturity spectrum.

Now all of the interest rates that we have discussed thus far are referred to as spot rates. And what are spot rates? They are the interest rate for a loan that’s made today. Now typically there is a different spot rate for loans of different maturities and different risks. That is what we showed right here, these are spot rates. They vary by credit risk, and they vary by maturity.

Now, what is one of the big punch lines or takeaways from all of this, aside from just general knowledge and information and understanding interest rates? Well, this present value formula that we’ve been working with is in many ways just an approximation, because we know interest rates vary over time. So it’s really an approximation for this. Now it is not such a bad approximation when the yield curve is flat. So when we have a yield curve, here is maturity.

Here’s yield.

When the yield curve is flat R1, R2, R3 are the same. So these two things would be equal. The problem, let’s see if I can erase some stuff here. The problem is when the yield curve says, upwards sloping severely or even downwards sloping.

Now there’s going to be a pretty big difference from using some sort of average discount rate R as a proxy for all of the different spot rates.

Okay.

So let’s wrap up here.

What did we talk about today?

• We talked about the term structure of interest rates, which captures the relation between interest rates and the investment term.
• Right loans or savings of different maturities different terms will often have different interest rates.
• And that relation is captured by the term structure.
• The yield curve simply graphs the term structure, right. It plots on the horizontal axis the maturity, on the vertical axis the yield or the interest rate. And that shows us the relationship between investment term and interest rate.

We also learned that interest rates would vary by the risk of the investment. We discussed that earlier on, back in I think our first lecture. And we talked about spot rates, which are the interest rates for a loan that’s made today.