Finance 527: Lecture 19, Bond Valuation V1 [John Nofsinger]: This is the first video for bond valuation. The previous bond topics were more the characteristics of bonds and different kinds of bonds. And this is more about the bond pricing and how it interacts over time. Changes in bond prices of course is how we would define risk and losses and also capital gains in returns. Bond Valuation V1 John Nofsinger Bond Valuation [Image of Investments: Analysis and Behavior textbook] [John Nofsinger]: So we ll talk about you know how we calculate the value or price of a bond. And we can reverse that if we know the price, we can calculate the yield. And talk about interest rate risk. And how we measure interest rate risk, and how it affects different kinds of bonds differently in the context of duration and convexity. Learn a little bit about convertible bonds and just sort of bond investment strategies at the end. Learning Objectives Calculate the value of a bond Compute the yield offered by various bonds Measure the interest rate risk of bonds using duration and convexity Learn the characteristics of convertible bonds Implement bond investment strategy [John Nofsinger]: So bond characteristics-most of this stuff was in the last topic. But bonds are of course the debt security. It s going to pay us interest payment that depends on the face value or par value of the bond. The coupon rate remember is the thing that tells us how much interest payment we re gonna get. But as interest rate changes in the economy-as interest rates go up or down-bond prices also therefore go up and down. And we ll see that dynamic. But in general, when interest rates go up, bond prices go down. Other things we ll keep in mind is that the interest payment that we get is typically twice a year. So even though it s reported as an annual coupon rate that is then divided by two and we get it every six months. And then at the very end, we re going to get the one thousand dollars back for the bond or whatever the face amount happens to be. As an aside, there is such a thing as a zero coupon bond. So if the coupon rate is the rate that tells us what interest payment we get, zero coupon means the rate is zero. And so indeed zero coupon bonds have no interest payments at all. They only have one payment. That s 1
when the bond matures, and we get our thousand dollars back and nothing in the middle. So how much do we pay for a bond like that? Obviously we would pay a lot less than a thousand dollars if we re gonna receive a thousand dollars in say ten years. So it s a very steep discount. Maybe we d only pay 150 dollars or 200 dollars for the bond to receive a thousand dollars ten years from now. Characteristics of Bonds Bonds: debt securities that pay a rate of interest based upon the face amount or par value of the bond Price changes as market interest changes Interest payments are commonly semiannual Bond investors receive full face amount when bonds mature Zero coupon bonds-no periodic payment (no interest reinvestment rate) o Originally sold at a discount [John Nofsinger]: That all depends on interest rates of course. Now the price of a bond-the price of a bond-is the present value of the annuity right. So we have all these interest payments that are exactly the same over time so that s an annuity. So we have the present value of those. And that s seeing right here the present value of annuity. If you want to use your equation, this is the present value of all those payments or we can actually put a specific annuity equation in there, but they would be the same. Then you get this thousand dollars back at the end right. So we want the present value of that principal so that s here or here and we ll add those up. So payment is the interest payment per six months usually we do this. That s usually a thousand for future value. This is usually the time to maturity, but it s time in six month intervals. Right so if it s a ten year bond, that s actually 20 six month intervals. And we re gonna also be careful with the interest rate as well. If payments are six month, and N is six month, then we want to make sure the interest rate is for a six month period as well. Bond Pricing Present of the bond = present value of interest payments + present value of principal PV of Annuity (pmt, I, N) + PV (FV, I, N) Where N=time to maturity NN PPPP = PPPPPP FFFF + (1 + ii) tt (1 + ii) NN tt=1 2
i=market interest rate PMT = semiannual interest payment FV = face value [John Nofsinger]: Alright let s just look at before we do you know examples of pricing and everything, let s just look at some examples. Let s just say that all five of these bonds have the same face amount of a thousand dollars. And here are the different coupon rates. So five and a half percent coupon rate is of course 55 dollars a year divide by two, and that s the semiannual interest payment that this bond gets. So some of these bonds have high coupon rates-9%. Others or much lower-5%. The market interest rate-now here I am showing the market interest rate to be a little bit different for each bond. And the idea here is not that these bond have different credit risk per say. But they do have different term to maturity. For example, this bond only has one year left before it matures. This one has five years, 13, 22, 30. So if we have an upward sloping yield curve, then a short term bond would have a lower interest rate that s being demanding than say a 30 year bond, which would have a higher interest rate. And that s just what I m trying to show here with the market interest rates being a bit dependent conditional on the length of maturity. Alright so if you were to do a bond price remember that bond prices are actually quoted in percent of par value. So this is 99.76. If you want that in dollar terms, that s 999 dollars. Now notice that this is a discount bond. It s trading at less than a thousand dollars. That is consistent with the fact that if you think about when this bond was issued, the discount rate or the interest rate in the economy must have been five and a half percent. That s what they usually do so they can sell each bond for exactly a thousand dollars. They set the coupon interest rate when it s first issued to whatever the market interest rate is for that bond. Well since that time, market interest rates have gone up a little bit-just a little. If that s the case, then bond prices must have gone down a little cause there s this inverse relationship that we will be seeing over and over again there. Here for example, when this bond was originally issued, interest rates must have been at 7.5%. But now, the interest rate for this bond is six and a half. So interest rates have gone down. So we would expect the price to go up. And sure enough, this bond is trading at over a thousand dollars. And so we can see-we can do similar things. And we can see the discount or premium bond depending on which one. Table 15.1 Bond Valuation Depends on Promised Cash Payments, the Maturity Date, and the Prevailing Interest Rate [Table of bond types and their face amount, semi-annual interest, coupon interest rate, yield, issue date, settlement date, maturity date, term to maturity, bond price, and bond valuation] 3
When the market interest rate is less than the bond s coupon rate, price is greater than the face value (Sold at premium, bonds 2,4). When the market interest rate greater than coupon rate, bond is sold at discount (bonds 1, 3, 5). [John Nofsinger]: Alright let s do a little pricing there. Let s say that we have a 5% coupon bond so that s gonna pay 50 dollars a year or 25 dollars every six months. Let s say it has three years and one month left to maturity. So the interest rate let s say the market interest rate is 6%. So again that s per year. So what we want to do is find the price of this bond, and we want to convert everything to semi-annual periods. So the number of periods even though it s three years and one month. We re gonna multiply that by two because it s 6.167 semi-annual periods. So I m going to use a financial calculator. And I m going to put in the N, I m going to put 6.167. The interest rate that I need to put in, the market rate that will discount cash flows is 6%. That s per year divide by two to get at six month intervals. So I ll put in 3 for I, 25 for payment, and 1,000 is what I m gonna get at the end. And I will compute the present value of the bond. Now actually when this actually computes, you ll see a negative sign here. Right if you re gonna receive payments and receive this payment, you have to pay money to buy that bond. And so the calculator would actually show a negative number there. The important part here is that this is answers that this bond sells for 972 dollars and 23 cents. That s the value of the bond. Does this number make sense because it is less than a thousand? If it s less than a thousand, prices have gone down from originally. And so that means interest rates must have gone up. And indeed this coupon rate of 5% and interest rates are now 6, interest rates did indeed go up. So that does check out. Bond price calculation The bond pays $25 semiannual coupon payment Maturity: three years and one month Market interest rate: 6% (APR) Solution: o Using Financial calculator N = 2 x 3 1/12 = 6.167 semi-annual periods I/Y = 6% / 2 =3% PMT = 25 FV=$1,000 PV = $972.23 4
[John Nofsinger]: Now some bonds have a callable feature that is when interest rates go down, I want to refinance my mortgage. Right I m willing to have the same debt, but end up with making lower payments after I refinance. Companies want to do the same thing. So some bonds have a provision that allows them to essentially issue new bonds at the lower coupon rate because interest rates have fallen. And use that money to pay off the old bonds. So they need to have that ability listed in the contract of the bond. But if that is done, then they are allowed to do it. Now they would do this at the exact time that we investors don t want them to do it right. If interest rates have fallen, the price of the bond has gone up. I ve had a nice capital gain. But they want to call the bond. So as compensation for just giving me the bond back in this now low interest rate environment, the call price is typically the face value-a thousand bucks-plus one year of interest payments mkay. So in the previous example where the coupon rate was 5%, that s fifty dollars a year. The call price would be 1,050 dollars. So instead of giving me back a thousand, they re gonna give me back this call price to help compensate me a little bit for the fact that they re calling the bond when interest rates are low. So when I get this money, I m gonna have poor alternatives to invest in. Callable Bonds Call provision allows the issuer to repay the investors principal early Issuers call the bond when they want to refinance their debt at the lower interest rate Call price is commonly the face value plus one year of interest payments Call protection: amount of the time before the bond becomes callable [John Nofsinger]: So we might want to know well we can take the price of the bond is given right. Call our broker. What do I want to buy? Here s my bond. The price is given. So what return am I getting if I buy that bond at that price? And that is called the yield to maturity. We calculate that by solving that equation where before we were saying what s the present value of annuity and the present value of the face value of the stock. And we came up with a bond price. But now we have the bond price. But we re trying to do is solve for I with the equations. This is a bit messy. But it s not a big deal with the financial calculator. It is tells us if we buy at this price and we hold it all the way to maturity and get a thousand dollars back, what rate annualized rate of return are we getting? Expected Yield Calculation Yield to maturity (or yield to call): expected total rate of return if investor were to buy and hold the bond until maturity or until call date 5
o Internal rate of return of the bond that equates the present value of the cash flow with the price of the bond Solve for I in Bond price = PV of Annuity (pmt, I, N) + PV (FV, I, N) [John Nofsinger]: So let s go back to that bond that s a 5% coupon bond. And it pays 25 dollars every six months with three years and one month left to maturity. If the price of the bond is 972 dollars and 23 cents, what is its yield to maturity? If I buy it at this price, what annualized return am I getting? So I put in just like before, the number of semiannual periods. Here I m putting in the price. I m putting in a negative sign because I have to buy it. And I m going to receive 25 dollars every six months and a thousand dollars at the end. I compute i and it tells me 3%. But I know that s for a six month interval. So I multiply by two to get the 6%. The yield to maturity is 6%. So notice I just reversed the problem. Before I was calling that 6% the market interest rate for that bond. When we re solving it this way, we re calling it the yield to maturity. Example: The bond pays $25 every six months. The bond matures in 3 years and one month. Price of the bond is $972.23. What is the bond s yield to maturity? Solution: Using financial calculator N=6.167 PV= -972.23 PMT= 25 FV=1,000 i 3% (or 6% annually) [John Nofsinger]: Now if you can see that interest rates are very important for the calculating bond price. And when interest rates change, bond prices change. Therefore, one of the biggest risks in bonds is changing interest rates. So let s just kind of see how that works a little bit. Interest rate risk Bond prices are sensitive to the market interest rate If interest rates rise, the market value of bonds fall in order to compete with newly issued bonds with higher coupon rates Sensitivity to the interest rate chance becomes more severe for longer term bonds Percentage rise in price is not symmetric with percentage decline [John Nofsinger]: Let s say that we have a 6% coupon bond. And let s say it is a short termpretty short term-bond of only two years. If interest rates from 6 percent, if they increase-if they 6
go up one percent, this bond s price will fall about 1.8%. If interest rates go up 2%, this bond s price will fall about 3.6. Not too bad of a loss. K but what if we own a long term bond instead a 30 year bond? If interest rates go up just one percent, you re gonna lose over 12 percent-almost 12 and a half percent-of the bond s value. Notice what happens when interest rates go up 3 percent. You lose almost a third of the value of the bond. Now the opposite is also true. When interest rates go down, prices of bonds go up. The last thing I want to-well one of the last things I want to illustrate right-is the longer the term, the more the price changes whether it s dropping or whether it s going up. That the longer term bonds are riskier. Their bonds go up and down more than the other bonds in the short term bonds. So that s the first thing. Also there s kind of an asymmetry here. If the price in this particular example goes down one percent, I d make 15 and a half percent. But if it goes up, I lose 12 and a half right. These numbers are not the same. So the change in prices a little bit asymmetrical. This is one of the reasons why if fixed income investors think interest rates are gonna go up, they try to keep their money in short term bonds. If you think that interest rates are gonna go down, you try to put money in long term bonds to get all this capital gain if you can. Table 15.3 An Illustration of Interest-rate Risk for Treasury Securities With a 6% Coupon Selling at Par of $1,000 [Table of list of bond types and their term to maturity, decline in bond value following an increase in rates, and their rise in bond value following a decrease in rates] [John Nofsinger]: And interest rates-this is just an illustration of how interest have changed. One of course interest rates have incredibly declined over the last three decades. But also, it s pretty volatile within the time periods. So as these interest rates go up and down up and down, those bond prices go up and down. And that is an indication of the volatility of those bond prices for the ten year treasury. 10-year Treasury Interest Rate [Graph of years vs. interest rates] [John Nofsinger]: Currently-well currently as the end of March 2013-the curve, which was illustrated earlier. This is an upward sloping yield curve we have. We get almost no interest at all from short term right. Less than point one percent for bonds that mature in 3-6 months. And this annualized so that s per year. If you go out ten years, you re closer to two percent is your yield to maturity. And if you go out 30%, this is-so you can actually see how longer term bonds usually offer us higher returns because remember they have higher interest rate risk. Now the yield curve 7
is not always upward sloping like this. Sometimes it has a different yield curve. Alright so this concludes the first video for the bond pricing topic. Term structure of interest rate Yield curve: line describing the relationship between yield to maturity and term to maturity [Graph of years of U.S Treasury versus interest rates] [Table of US Treasury Bonds and the maturity, yield, yesterday, last week, and last month] 8