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3. Callable Bonds valuation The framework
Applied Investments (FTX4056S)
University of Cape Town
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- Callable bonds valuation_The framework
What I want us to do today is to look at the pricing of callable bonds. And before we... We are likely to look at the valuation of callable bonds in the next lecture. What I want us to do today is to thrash out the framework for the valuation of bonds with options. So that framework applies with any bond, with a kind of option.
So it can be a puttable bond, it can be a callable bond, or it can be a floating rate bond with a cap or with a flow. And before we get to the framework I want us to recap on the arbitrary free valuation technique. The reason why we’re recapping on that is because the arbitrary free valuation technique forms the foundation of the framework for pricing bonds with options. It’s based on that, and then secondly, so what you need to know is how to price a bond using the arbitrary free valuation technique. And secondly, you need to know the relationship between spot rates and forward rates. Once you understand those two it will be easy for you to price a bond embedded with options. So the arbitrary free valuation technique arose as a result of the shortcomings that exists with the traditional way we price bonds. Remember, we’ve got the traditional way of pricing bonds, which exists in South Africa. And then you’ve got the arbitrary free valuation technique, which is more popular in other countries. But you realise that there are some elements of the arbitrary free valuation technique that also apply here in South Africa. Especially when it comes to the valuation of callable bonds. And also let me say, this whole lecture, on the valuation of callable bonds, it’s mainly drawn from CFA material. There’s little on South Africa. If you look at the documents from the Johannesburg Stock Exchange, specifically from the bond exchange, they talk about pricing bonds using the binomial tree. And that’s what comes from the CFA material. But when you speak to the industry analysts and traders, they give you a different story. So the bulk of the material comes from the CFA material. And like I said, the arbitrary free valuation technique, it came about as a result of the shortcomings of the traditional method. And the shortcomings of the traditional method, they come from the assumptions that are implied by the way we price bonds using the traditional method. If you remember how we price bonds using the traditional method, you simply estimate your future cash flows, and then you find the present value of the future cash flows. Like this easy example. And the two key assumptions that are implied by this traditional method is they’re assuming a flat yield curve. Because you are using the same discount rate. You are using the same required date of return to discount cash flows that are coming at different maturities. So in simple terms you are assuming that investors require 10% for one- year instruments, and they still want the same 10% for five-year instruments. So you’re assuming a flat yield curve. So that’s the first assumption that is made by the traditional method. The second assumption that is made by the traditional method is it views a bond as one block of cash flows. That cannot be separated. That’s why it treats the same cash flow, the first cash flow, it gives it the same treatment as the last cash flow by discounting it using the same required date of return. So these are the assumptions that are implied by the traditional method. Obviously those assumptions have got limitations or their shortcomings to that kind of pricing. The first shortcoming is that it’s not necessary to assume a flat yield curve if the yield curve is actually upward sloping or downward sloping. So it’s an unnecessary assumption, that’s the first point. Why assume a flat yield curve when the yield curve is not flat? So that’s the first shortcoming of that first assumption. As far as that assumption is concerned as well, it’s unrealistic to find flat yield curve. So it’s quite rare. Even though they do exist, sometimes, but it’s rare to find a flat yield curve. So it’s kind of unrealistic. And then the second assumption is flawed in the sense that the cash flows of a bond can be stripped, can be reconstituted. So it’s not correct to view a bond as one block of cash flows, because it’s possible to split the cash flows or to reconstitute them. For example, you can take that first coupon that comes after one year, you can sell it as a one-year zero coupon bond with the first value of R80. The same happens with this coupon that comes after three years. You can strip it and sell it as a three-year zero coupon bond with a face value of R80. So in other markets a bond should not be viewed as a block of cash flows. So what are the implications of these flaws to a trader? The implications of maybe the first shortcoming, if you’re using a flat yield curve when the upward sloping is...
When the yield curve is upward sloping. And also if it is possible to split the cash flows as zero coupon bonds. Then that means the price that you get here is going to be incorrect. So it can be much higher or lower than the true price. So that creates opportunities to make profit. And then the flaw of the second assumption is that if using a flat yield curve can give you incorrect price, then that means it’s possible to buy this bond, strip it, and then sell it at a higher price. So the main implication of these shortcomings is that they create arbitrary opportunities by buying a bond at this particular price, stripping it, and then selling it. Or reconstituting a bond and then selling it at this particular price. Depending on which one is higher. But that’s not the main focus of this lecture. The main focus is simply to show you where the arbitrary free valuation comes from. And what it does. Yes? Yes. So those opportunities that I’m talking about... Yes, the transaction’s cost have an impact on the amount of profit that you’re going to make, and also if you’re going to go ahead with the trade. Those have to be taken into account. And when I’m talking about this, the traditional technique having arbitrary implications, it basically means that after taking into account the transaction cost you can still make money. If you can’t make money, if you include transaction costs and not make money, then you can’t do the trade, and therefore we can’t really talk of the traditional technique as having shortcomings, if that is not possible. Okay? If you look at reality, for example, if you look at the huge differences between a bond priced using the South African way, the traditional method, and then just take a swap curve, and then price the bond using the swap curve. Look at the huge differences that you see. There are huge differences that actually exist even if you include transaction costs. So what the arbitrary free valuation technique does is that it addresses those shortcomings. How does it address the shortcomings? One, it views a bond not as a block but it views a bond as a package of cash flows. So it looks at that one year, that coupon that comes after one year. It looks at it as a one-year zero coupon bond. And it will be discounted not at 10%, but it will be discounted at the yield maturity for similar one-year zero point bonds. It does the same thing with the five-year. The coupon that comes after five year. Plus the face value. It doesn’t look at it as a block, it looks at it as a separate one-year zero coupon bond with a face value of 1 080. Therefore the required rate of return that it uses is going to be the yield maturity of similar five-year zero coupon bonds. So that’s what it does. So these are the details of the arbitrary free valuation technique. It views a bond as a package of zero coupon bonds. Which means each coupon is viewed as a zero coupon bond. And given that, it then requires us to discount each cash flow using the appropriate yield to maturity that corresponds to the timing of the cash flow. Which means the coupon that comes after one year is discounted using the one-year spot rate. The one that comes after two years is discounted using the two-year spot rate, or the yield to maturity of zero coupon bonds. So spot rate is simply the yield to maturity of a zero coupon bond. And it says the fair value of the bond should be equal to the package of the zero coupon bonds that make it up. If that is not true then there will be opportunities to strip or reconstitute and sell. So those are the details of the arbitrary free valuation technique. I do have a very short example. Not an example, but the formula. So there is the formula which says the present value of a bond is equal to the present value of the future cash flows. But however, the future cash flows are discounted at the appropriate yields of zero coupon bonds, which are the spot rates. So the first one is discounted using the one-year spot rate. The two-year spot rate. And the N-year spot rate. So now you are using different required rate of returns for coupons that are coming at different time periods. This is just a quick example where you’ve got your time to maturity and you’ve got your spot rate. So in simple terms, this is the yield maturity of a one-year zero coupon bond. Yield to maturity of a two- year. Yield to maturity of a three-year zero coupon bond. And then for a four-year. And what we have there are basically the forward rates that have been derived from this curve. We covered that last week. So these are basically the forward rates. So if you’re pricing this particular bond, for example, which has an annual coupon of 6%, and a face value of 100, so you’ve got your coupons for four periods and then you discount them using the spot rates. So the first coupon, which comes after one year, is discounted using 3. The coupon that comes after two years is discounted using the 4. And 4 for the coupon that comes after three years,
concerned. But when you’re dealing with a bond with options the future cash flows are not known in advance. We don’t know whether you’re going to receive a coupon at the end of the 50 year or at the end of the 40 year, because the bond might be called. So for bonds with options the cash flows are uncertain. Why? Because the cash flows depend on whether the option has been exercised or not. That’s why the cash flows are uncertain. And whether the bond is exercised or not depends on the levels of future interest rates. So when you are dealing with callable bonds, the first thing that you need to do is to predict future interest rates. So that’s the first thing you need to do, because that determines the cash flows. Which are uncertain because you don’t know whether they’re going to be called or not. And that’s a difficult job, to determine future interest rates accurately. So what analysts would do or traders do is to kind of estimate various interest rates. Because you don’t know which one is going to exist. Or they try to determine forward rates, and then from the forward rates they’ve got various possible interest rates that may exist. So that requires us to construct an interest rate tree, which is basically a framework that shows us the possible different rates that may exist in future. So because we have got different possibilities of interest rates, and because we have got to focus interest rates, which would then determine the cash flows of the bond, we have to value the bond at each node using the backward induction method. And that’s important because that will show us the cash flows of the bond. So the key point is that we have to value the bond at each node, to take into account the various possible interest rates that may exist. And the reason why we have to take into account those various possible interest rates is because those determine the cash flows of the bond. Which are uncertain, which depend on the levels of interest rates. So the approach that we’re going to use, which is the backward induction method, it draws heavily from the arbitrary free valuation technique. Why do I say it draws heavily? Because you’re going to be discounting each cash flow using a particular interest rate that exists for that period. If you get the average of all those one period interest rates that you’re going to be using, you get a spot rate. So in simple terms, you’re pricing the bond using the arbitrary free valuation technique. So it’s like just valuing a bond using the forward rates that you are getting, one by one. The same way I showed you in the previous example. So this is just an illustration of the backward induction method. You don’t have to draw this diagram. I think it’s probably similar to one of the diagrams that I showed you. So what I simply want to show you here is the value of the bond at different nodes, and just to show you that it’s exactly similar to the arbitrary valuation technique. So you can either stand here (I am pointing at Node 0) and discount the first coupon using the 5%, which is one-year spot rate, then if you’re still standing here, the R5 here (I am pointing at the R5 at Node 2) can be discounted using 5. That R5 that comes in year three, using 6%. Or alternatively you can start from there (Node 4) , where the value of the bond at this node is 105. And then the discount, to get the value of the bond at node three, which would be the discounted amount and the coupon that comes in that period. So in other words, the value of the bond at node three is simply the present value of that number (The number I am referring to is 105). And the value of the bond at node two would be the present value of that (here I am referring to the answer you get at Node 3 after discounting R105) plus that coupon (This is the R5 coupon at Node 2) , and then that’s the number that you’d get, plus that coupon, that gives you the value of the bond at node two. This method, the backward induction method, it doesn’t make much sense if interest rates are stable. It only makes sense if interest rates are volatile. Why does it make sense if interest rates are volatile? Because if interest rates are volatile we are unlikely to have that 10%. So the interest rate in that period can be higher or lower. If interest rates are stable we can simply use spot rates because if interest rates are stable that means our interest rate in that period will be 10. In that period it will be 7, or 5. This one will remain the same. If interest rates are volatile we can actually have a rate higher than 10, or a rate lower than 10. So that would require you to create a binomial interest rate tree where at that rate (I am pointing at 10%) you’re going to have a higher interest rate or a lower interest rate, and here (I am pointing at 7%) you can have a higher or lower interest rate. But whenever, if you’re creating the model that we use to estimate various possibilities of interest rates, that we
use to price callable bonds, it assumes that the forward rate that you get is the midpoint between the two different rates. Between the high and the low. So the forward rates, so the yield curve is going to be your benchmark, where you get your forward rates, and then your forward rates are going to be the midpoint. You’re going to be high and below. In fact, when you’re pricing callable bonds you need to calibrate your model to the yield curve so that it reflects current market conditions. So are there any questions so far? Okay. So let’s move on to the binomial interest rate tree. The idea behind the binomial interest rates tree is it’s more useful when interest rates are volatile. That’s when you’re going to have different possibilities of interest rates. If they’re not volatile, work with your forward rates, because it’s stable, there’s no risk, so those forward rates will exist. If interest rates are volatile, then you need to come up with an interest rate tree. So it simply gives us potential interest rates that are likely to exist in future. And it plays two roles. Firstly, it helps us to determine the cash flows. Because by looking at the levels of interest rates that it predicts you can tell what kind of cash flow you’re going to get. Are you going to get the core price, are you going to get your normal coupon? It also gives us the discount rates. In other words, it gives us the yield to maturities that we use to discount the cash flows. Those are the two purposes of the interest rate tree. I explained this part. What I just want to emphasise here, don’t draw this, you do have it already, what I just want to emphasise is that the key input in coming up with our binomial interest rate tree, the key input is the forward rate. Or let me say the key input is the yield curve. Which then gives us the forward rates. And whatever various possibilities that we’re going to have are going to be a function of the forward rate or the midpoint is going to be the forward rate. Okay? When you are using the forward rate as the midpoint, the price that you’re going to get is going to be slightly different. I will then show you the more accurate way of coming up with these various possibilities of interest rates. So the key input is your forward rate, which you can use if interest rates are stable. So the implication of stable interest rates is that you use the forward rates to price the callable bond. So take note of that. So one, forward rates are the key input, or the yield curve is the key input in building our binomial interest rate tree. And if interest rates are stable, then those forward rates are going to be the interest rates that you’re going to use to price your bond with options. If interest rates are volatile then you’re going to have either a higher or a lower interest rate. And we just assume that there’ll be an equal chance of having a high interest rate and having a lower interest rate than 10%. So let’s look at an example of an interest rate tree. So here is just an example of a binomial interest rate tree, we call it a binomial because it takes two possible values. So what you see here are the various interest rates that you’d have estimated, or that you expect to happen. That represents a node (here I am referring to the squares, all the squares represent Nodes). So the node will be one period or one year from the node going to the right. So here (i 0 ) you’ve got node zero, which is one year from today, to next year. Here (i1H and i1L) you’ve got node one, which is after one year, one period, after one year. Here ( i2HH i2HL and i2LL) you’ve got node two, which is after two years, and it’s a one period or a one year after two years. So that’s as far as the nodes are concerned. You can look at the nodes this way (look at the timeline at the bottom of the page). So node zero is basically from here to there (from or between time 0 to time 1), and then node one (from or between time 1 to time 2), is basically covers, it’s one period from time one to the end. And then your node two is basically from there to there (from or between time 2 to time 3),. So what we have here (i 0 ) is basically the current interest rate. So I0, node zero, is basically the current interest rate. And then what we have here (i1H and i1L) is basically the one-year forward rate after one year, or the one-year forward rate in period two. So this is the highest possible interest rate that could exist at that time, and this is the lowest. The midpoint would be the forward rate. Which you are not worried about because interest rates are volatile. So there’s a less likely that the forward rate will exist. And then here, in period three, you have got three possible interest rates. So in period two you’ve got a high and a low, and then in period three, which is from node two, you have got a high interest rate that comes from the high. And then you’ve got low interest rate that comes from the high. So H means it’s coming from a high rate, and then another H means it’s the highest interest rate for that particular period. And then here you’ve got HL, which is it’s coming from a high rate in the previous period, and then it’s a low, the
I also showed you the relationship between the highest rate and the forward rate, and the lowest rate and the forward rate again. So these two are very important. And keep in mind, this is an approximation. It leaves a small difference. Because at the end of the day, whichever method that you’re going to use, you need to calibrate it. You need to calibrate all these numbers with the yield curve to make it suitable for... Or to make it reflect the current economic conditions, and also to ensure that there are no arbitrary opportunities that as a result of the interest rate that you’re estimating. Or the interest rates that you’re going to trade your bond on. Because at the end of the day you’re going to trade your bond based on these. So if you trade on these without calibrating them to the yield curve you are still creating the same problem that arises with the traditional method, arbitrary opportunities. Okay? So these are the inputs that you use to estimate your binomial interest rate tree. The first input is basically your yield curve. So you need your yield curve because you’re going to calibrate it to the yield curve, given the way we’re going to do it. You also need the yield curve so that you can estimate the forward rate. And then you need an assumed level of volatility. So those two rates, the high and the low, they are separated by two standard deviations. So you need to have an estimate of the standard deviation of the volatility. And then you need your model. These are just further details of the model, but I’ve just explained it already. So the model simply shows you that the difference between the high and the low are basically the multiple of this ULS number and two standard deviations. And keep in mind that whichever model that you’re going to use, the forward rates are basically the midpoint. So if the forward rates are basically the midpoint, then the relationship between the highest rate is just one standard deviation. In the positive. And then one standard deviation in the negative for the lowest interest rate. So that’s your model. There’s a lot of theory that is related to that model, which we are not going to discuss. Maybe next year I’ll bring it in the first chapter, just to discuss various models that we have and why that model is the best choice. But keep in mind that there’s a lot of theory that comes with this model. As far as this course is concerned, in this year, let’s take the model as is. So one of the inputs that goes into the model is basically the volatility and there are many ways of determining volatility. I would focus on the first one. You can estimate the historical volatility. In other words, you just look at the benchmark yield curve, look at the historical volatility for the past 15 years. Whatever number that you get, you take that as your volatility number. So if you get a volatility of 8%, for the past 20 years, you just assume that history would repeat itself. So you work with a volatility of 8%. There are also various other ways of estimating the volatility from derivatives, but we will not focus on that. I think I will give you up to node three probably. Node two and node three. I won’t go beyond that. So what I was basically saying was if you look at this high and the low, your forward rate is the midpoint (the midpoint between i1,H and i1,L). If you look at the... In node two, the forward rate would be that number, the HL (the midpoint between i2,HH and i2,HL). If you look at the last one, your forward rate would be the number over here (the halfway mark between i3,HHH and i3,LLL which is also the midpoint between i3,HHL and i3,HLL). So from the forward rate, one standard deviation above, one standard deviation below. And then from this point (from i3,HHL) to the next point ( to i3,HHH) we then applied that model 𝒊𝟏,𝑯= 𝒊𝟏, 𝑳𝒆𝟐𝝈 )****_._. To say for you to estimate this HLL, you work out a number such as that, the
difference between the two (i3,HHL and i3,HLL) , would be two standard deviations. And then the same happens here, okay?
So how do you determine the value of a bond at each node? If you get to a situation like this where you have got a bond value based on a high interest rate, and a bond value based on a lower interest rate. The value of the bond at this particular node is going to be the average of these two (the average of the value based on the H and the L). You just take the average, because the assumption we make is there’s an equal chance of the rate being higher or the rate being lower. So you simply take the average. And this is how you calculate the average. You are saying the value of the bond at the node plus the coupon that comes at that particular node, you discount it. So this will be based on... The value based on the high rate. And then this will be the value based on the low rate. So remember, you have got your bond value which is, say, 100. Based on the high rate. And then the value based on the low rate which is, say, 120. So the value of
the bond at this particular node will be the value of the bond based on the high rate (VH is the value of the bond based on the High interest rate) discounted at say 2%. Plus the value of the bond at the low rate (VL is the value of the bond based on the High interest rate) , plus the coupon that comes at that period, discounted at 2%. So the average of those two numbers. So VH is simply the value of the bond based on the high interest rate, and then VL the value of the bond based on the low interest rate. And then your discount rate is simply the one period forward rate for that particular period. And then C is your coupon payment. So you work with averages in a situation where you’ve got two values for you to get to the value of the bond at that particular node. Okay. You did this in derivatives, right? Did you value a callable bond in derivatives? Okay. But you did some binomial model somewhere, right? Okay. So this is not new stuff. Is there anything new here? Are there any questions before we take a break? So let’s take a five minute break. So we’ll start... We’ll resume the lecture at four o’clock. Okay. Thank you. I think the nice thing about taking a break is I get some questions, which I get to clarify before the lecture ends. So I just want to clarify a few points. So... And okay. So I just want to clarify a few points. And I think some of you kind of... Some of you are already rushing. So, and for me to clarify that, let me go to this tree, right? So what I’m basically saying is that some of you are saying practically how do you come up with those interest rates? So the first point is that what you are going to have is this is your timeline. So this is the current. So this is not a forward rate. We know it already. So it can be 5%. And then here, these are forward rates, forward rates, forward rates, forward rates. Okay? So the forward rate is the midpoint, that’s point number one. Point number two, there are two standard deviations between the nodes. And there’s one standard deviation between the forward rate and a particular node. Okay? So if your forward rate is 6%, if interest rates are volatile, they can either be higher than that or lower than that. So you take your model and your model to get the high number, okay? I’m not going to use the real notation. So to get the high rate you simply take your 6% multiplied by that ULS number which is 2-something. What’s the correct number? Can somebody give me? Can somebody shout it? It’s two point...? So 2. So you are getting the high number. So you’re getting the high number, so it will be to the power, the standard deviation, let’s say, is 8% to the power 8% or to the power 0 positive. To get the lowest number for that particular period you do the same thing. 6% multiplied by 2 to the power minus 0. Okay? One standard deviation between. Now, when you get to that node (Node 2), that HL is the midpoint. So that will be the forward rate for that particular period. So the forward rate, if the forward rate is 7%. So that HL is going to be 7%. So once you get the HL there are two standard deviations between the nodes. Then you apply that other formula where you’re going to have two standard deviations here. It’s going to be that multiplied by two. Because there are two standard deviations between the nodes. And then when you come to here (Node 3) you are going to have your forward rate. If that forward rate is 16%. So you have got your 16% as the midrate (the forward rate will lie halfway between i3,HHL and i3,HLL) , so you can get one standard deviation to get to that number (to get to i3,HHL) , one standard deviation below (to get to i3,HHL). Once you get these numbers you know that there are two standard deviations from each node. So you basically apply that same formula to get all the other numbers. Keep in mind, it’s an approximation. What makes this a good approximation is no matter which way you are going to get the possible interest rates, you have to fit that to the yield curve. So that’s what makes this approximation very close to that. And how you do it is basically something that is almost closer to bootstrapping. With bootstrapping you take the market price of a bond and then you assume that that price will... For us to get that price, we’re discounting the bond using the spot rate. And then that allows you to solve for the missing last spot rate. Almost same concept. Why? When you use the iteration process you estimate the lower rate. And then you use the model to estimate the high rate. Right? So to check whether those numbers are correct you take a two-year coupon bond issues by the government. Take the price, use those interest rates to price the bond. If the price you get is equal to the market price, then your estimations are in line with the yield curve.
These are the working that gives you those answers that I gave you. So you can copy down the workings. I didn’t give you the answers to the workings. So copy down the workings. The first part, this is the discounting the value of the bond to node two. We were using those T3 different interest rates. And then after that we’ll go onto the workings for discounting the value of the bond to node one, and then to node zero. Sorry. So here we haven’t started looking at callable bonds as yet. We are still looking at the framework for valuing bonds with options, okay? So let’s go to probably the last part. Okay, are you done copying? If you haven’t finished copying then you are delaying yourself. Okay, so this is just additional detail on how you construct the binomial interest rate tree in industry. You use Solve in Excel, analytical tools, basically to estimate. So these are basically some of the steps that I summarised. So you select the lowest rate, arbitrarily, in the first trial, and then you find the corresponding higher value. So you estimate the low value and then estimate the high value, and then after that to check whether the numbers are correct, compute the value of a bond. That is trading in the market. Take the price, compare the price using the numbers that you’ve estimated. And if the two are the same, if the two are giving you the correct price, that is trading on the market, then you know that it fits the yield curve and the numbers are correct. So I don’t know whether I should end here or I should continue. You are tired? So why is it that you get tired more than me, yet you are just sitting there? I will be talking, I should get more tired than you. And also, I get to benefit if I leave early because I’ll be paid for nothing. So why do you complain when, one, you are being short changed, you paid money, you’re not getting value for your money? So why would you want to rush home? Yes? Okay, so this is what I budgeted to cover today, so I’m releasing you because I finished what I wanted to cover. I’m not releasing you because you want to leave early. Okay? Thank you.
3. Callable Bonds valuation The framework
Course: Applied Investments (FTX4056S)
University: University of Cape Town
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