Finance 527: Lecture 19, Bond Valuation V2

Transcription

1 Finance 527: Lecture 19, Bond Valuation V2 [John Nofsinger]: Hello this is the second bond valuation video. And what we re gonna do now is we re going to come up with measures of how sensitive a bond is to interest rate risk. And the first one is called duration. And we ve learned in the first video for bond valuation that the term to maturity that it is an important aspect to how a bond is volatile when interest rates change. In other words, long term bonds change in price more than short term bonds do when interest rates change. Therefore, long term bonds are riskier. Well as it happens, also there s another factor and that is the size of coupon rate. So for example, a zero coupon bond-remember it just has one payment at the very end. When interest rates changed, the present value of a zero coupon bond is very volatile-it changes a lot when interest rates change. Compare that to a bond with say a very high coupon rate so you re getting lots of interest payments along the way. When interest rates change, it also changes the price of this coupon bond, but not as much as a similar time to maturity zero coupon bond. So there is really two factors that cause sensitivity to interest rate changes for the bond price. And that is the time to maturity and that is the size of the coupons. So what duration is duration is an estimate that combines those two. And it s an estimate of the economic life of a bond. So a maturity-the time to maturity-is when the final payment is done right. But duration is also time. But it s sort of the average of when all the cash flows get paid. So some cash flows get paid really early, where some cash flows get paid very late. And so we take the average. When is the average payment made? Now all of the interest payments are the same size and then you have one really big one thousand dollar par value payment at the end. So we don t just take the average of when it gets paid. We take the weighted average where the weights are the size of the payments-the cash flows that we get. So this one counts a lot more than any one individual little interest payment. Okay so duration is an attempt to combine time to maturity and the size of the coupon into one measure of when the bond s average-i should say weighted average time-to maturity is. And so instead of having these two different things like maturity and coupon rate, we now just have duration. And the shorter the duration, the less sensitive is the bond s price to interest rate fluctuations. Bond Valuation V2 John Nofsinger Duration Term to maturity is an imperfect measure of bond risk because it ignores the valuation effects of differences in coupon rate and principal payment schedule Duration: an estimate of the economic life of a bond measured by the weighted average time to receipt of cash flows o The shorter the duration, the less sensitive is a bond s price to interest-rate fluctuations 1

2 [John Nofsinger]: Now how we do that is a bit of a complicated equation. And that s here. Now this part looks a little more complicated than it really is. This bottom part is simply the bond s price. Okay because the bond s price in and of itself is a weighted average of the value right. It s a present value of the bond, where it s weighted in size and time as well. So we re gonna standardize by the bond price. And then up here it s kind of the same calculation as down below right. This is just the each cash payment, what s the present value of it? Add them all up. That s the mon price. Up here in the numerator it s what s the cash payment and then when is the cash payment? Right half a year from now, two years from now. So it s weighted by when it occurs. And then we add them up. So if we have a bond with 7 and a half coupon. It s got 5 years left till maturity. And current interest rates are 6.75%. What is the duration? So you take the first payment is gonna be an interest payment of right that s half of the 75 dollars we get a year. And the first payment comes in six months so half a year. And then we find the present value of it. We do that then for the second payment, which happens one year from now. And the third payment which happens a year and a half from now. We keep finding the present values of those. All the way up until the very last payment, which is interest and the thousand dollar par value. It happens five years from now. So we find the present value essentially of all of that. Then we divide it by the bond s price. And we get 4.3 years. Duration is measured in years just like maturity. And a bond s duration will always be less than or possibly equal to the maturity-the time to maturity. The time the equal comes in is when the bond is a zero coupon bond. So you only have one payment way out at the end. Well the weighted time for when that comes is indeed the maturity date. So but all other cases the duration is gonna be less than the maturity date. Duration (or Maculay duration) calculation TT DDuuuuuuuuuuuuuu = tt=1 TT tt=1 tttttttttthpppppppppppptt tt (1 + YYYYYYYYYY) tt jj CCCCCChPPPPPPPPPPPPtt tt (1 + YYYYYYYYYY) tt jj = TT tt=1 tttttttttthpppppppppppptt tt (1 + YYYYYYYYYY) tt jj BBBBBBBBBBBBBBBBBB Example: Calculate duration for a 7.5% bond with 5 years to maturity and a yield of 6.75%. DDDDDDDDDDDDDDDD = 0.5xx$ xx$ xx$ xx$ xx($ $1,000) $1, = 4.3 yyyyyyyyyy [John Nofsinger]: So how do we use this duration that we ve calculated. Well the first thing we do is we modify it a little bit. We take that duration we calculated and we basically we adjust it or modify it by dividing it by one plus the yield-the interest rate per whatever the payments are. So if this is an annual yield to maturity, then we would divide by two if it s semiannual payment. So it d be a six month yield to maturity. So for example, once we have modified duration, we can estimate how much a bond price will change with whatever interest rate changes you want to 2

3 come up with in your scenario. The equation is the bond s price will change, we multiply it by negative one because an increase in interest rates will cause a decrease in price change. And then it s the how much we-the what if. Times modified duration. So here we go. Our bond has a 4.3 years duration. What if interest rates were to go down by half a percent? So we need to convert our duration into modified duration. So 4.3 divided by 1 remember the yield was what 6 point. Can t remember exactly what the yield was So that divided by two gives us what we divide by. And our modified duration is Put that into the equation. You can say what if interest rates were fall by half a percent? Here's our modified duration. Then our bond-this bond should go up by just over two percent. So duration is used as-can be used as an estimate for how will your bond prices change when interest rates change. Again the larger the duration, the more change you ll get. MMMMMMMMMMMMMMMM dddddddddddddddd = DDDDDDDDDDDDDDDD YYiiiiiiii 1+ CCCCCCCCCCCC PPPPPPPPPPPPPPPP pppppp YYYYYYYY Direct estimate of the percentage change in bond price for each percentage point change in the market interest rate % change in bond price= -1 x %Yield Change x modified duration Example: given that duration is 4.3 yrs, if interest rates fall by 0.5% what is the change in the bond price? Modified duration= 4.3/(1.0337)= 4.16 % change in price= -1 x (-0.5%)x4.16=2.08% [John Nofsinger]: Now what the duration is actually doing is let s have this graph right here. It s different yields to maturity or interest rates in the economy and the different bond prices. And let s say we have a long-term bond with a 6 percent coupon. So when interest rates are 6%, it should trade at exactly a thousand dollars. And we see that it does. And if interest rates go up, bond prices go down. We see that. If interest rates go down, bond prices go down. We see that. Essentially what duration is is that at any point in time, this is why you use modified duration, cause you also have to adjust for where you are in the graph. But at any place on the graph, if you calculate duration under those interest rate conditions, it is calculating the slope of the line. So then I say what if prices changed a little bit. Interest rates change a little bit, then we can calculate how much the price changes when interest rates change at that much using this estimate of the slope of that line. Now you can see that the slope is not a straight line. It is a curved line. And it is more curved for long term bonds than it is for short term bonds. So what this means is that is for large changes like if I say well what happens if interest rates move two percent or 3 percent. For large changes, that slope is not gonna fully get the price change right. So we can increase our accuracy by adding a piece called convexity. So what is the slope of the line? And then how convexity it is-how is that slope changing as the curve moves? 3

4 Convexity Convexity measures the sensitivity of modified duration to changes in interest rate (the rate of acceleration in bond price changes) o The degree of bend in the price-yield curve Figure 15.3 The Price-yield Curve for a 30-year 6% Bond is More Convex to the Origin than the Price-yield Curve for a 5-year 6% Bond [Graph of Yield to maturity vs. bond price in dollars for 30-year 6% bond and 5-year 6% bond] [John Nofsinger]: Convexity is even a little more complicated. We still have price down here. But we re multiplying it by 1 plus the yield squared. Instead of just multiplying each payment by when it occurs, we also have when it occurs squared. So it s a little more complicated. But you know put it in a spreadsheet or whatever we can do this. So then what we get is again we still want to ask the question what happens when the bond prices interest rates change, we want to know what the bond price change is. So it equals again this is what we saw from the duration part right. Now we re adding a little piece to it here. And it s one half of this convexity with an adjustment okay. Bond pricing using convexity and duration TT (tt 2 + tt) CCCCCCh PPPPPPPPPPPPtt tt tt=1 (1 + YYYYYYYYYY) tt jj CCCCCCCCCCCCCCCCCC = BBBBBBBB PPPPPPPPPP (1 + YYYYYYYYYY) 2 % bond price change = -1 % Yield change modified duration + ½ convexity (yield change) 2 Using both duration and convexity allows for a more accurate estimation [John Nofsinger]: So I m gonna go ahead and calculate convexity for this same bond. Not gonna go through all the numbers. You saw the equations. Just putting them all in there and cracking them out. Convexity comes out to about 21 and a half. So again what happens when bonds change they go interest rates change by going down half a percent. Now here you notice that it s actually more convenient to use the decimal format of the interest rate. So when I say interest rates go down half a percent is also saying down point zero zero five in decimal format. That s because this part of the equation is squaring the percentage. And so we want to be a little careful there. So let s go with the decimal format. But anyway here again is the duration portion of the equation and then we add this little convexity change. And we get that the bond will actually go 4

5 up in price 2.11%. When we use duration only to estimate this, we got 2.8%. So I guess the question is for estimation purposes, is doing all this other work for convexity, is it worth it just to get this much more accuracy in your estimate? Well I guess everyone has their own yes or no on that. CCCCCCCCCCCCCCCCCC = Example: Compute the convexity for the 7.5% bond with 5 years to maturity and a yield of 6.75% ( ) $ (12 + 1) $ ( ) $ (52 + 5) ($ $1,000) $1, (1.0675) 2 = o % change in bond price: -(-0.005) / (-0.005) 2 =2.11% (% change approximation using only duration was 2.08%) [John Nofsinger]: Alright so we can use-if you re interested in buying if you think interest rates are gonna go down for example, then you want capital gains and the most you can get. So you buy bonds with the largest duration. If you think interest rates are gonna go up, then they ll be losses in bonds. So you want to get into the lowest duration bonds. So that s an investment strategy for that sort of a thing. Alright let s talk just a little bit about another kind of bond. It s called a convertible bond. These are issued by corporations because it is a regular bond-corporate bond-that you have the option to convert it to another security. It s usually stock-common stock. So it s like saying I own a Harley Davidson bond that at any time I can convert into 20 shares of stock. That s the idea. You would only want to convert it of course if Harley Davidson stock goes way up. If Harley Davidson stock is down, then you just like to keep the bond. The conversion value is the number of shares you get to convert to in that case 20 as I said. What you actually find is that the convertible bond itself will be trading at a price that is higher than if the value you would get out of converting. Why is that? Let s say that if I convert it to 20 shares of stock that would be 1,400 dollars of stock value. Let s just say that s the case k. The stock s trading at 70 dollars per share. This bond-convertible bond-is probably trading at one thousand let s say 500 dollars a share. So if you wanted to get out all your profit, you can just sell the bond. You get more money. If you want the stock for the long term, you can convert it. Why is it that the bond-convertible bond-is selling for more right now? The answer is that you will make more money by holding the bond and getting the interest payments, which are usually higher than the dividend payments on a stock. So it s better to hold the bond up until right before it matures and then you convert it. And you ll make more money that way. And that s recognized as in the price of the bond. 5

6 Convertible bonds A type of bond that can be exchanged into some more junior grade of securities (usually common stock). Conversion value is the number of equivalent common shares multiplied by the current share price Premium to conversion is the percentage over conversion value at which the convertible trades Break even time is the numbers of years needed to recover the conversion premium with the convertible s higher income [John Nofsinger]: Example. Let s say that we have a stock that pays 35 cent dividend. It has a current price of 55 dollars a share. It s a 6% convertible bond, which means 6% is the coupon rate. Selling for 118% of par value. We can convert to 50 dollars a share, which means you get 20 shares of stock if you convert the bond. No matter when you convert it, you get 20 shares. K if you converted it right now, that is one thousand one hundred dollars worth of stock value you could get. However the stock is actually-i mean the bond-is actually trading at 1,180. So if you converted it, you would actually lose 80 dollars in value. Again why? Well here it is. A 6% bond, you get 60 dollars a year in interest. If you convert it to 20 shares of stock that s getting 35 cents a share, you re only getting seven dollars of dividend. Therefore, you make more money holding the convertible bond as a bond and collecting interest until later when just before it matures instead of getting the thousand dollars back. That s when you say I want my 20 shares of stock. Example: A common stock pays a 35 dividend and has a price of $55/share. The company also has 6% convertible bond selling at 118% of the par value, convertible into common at $50/share. What is the conversion ratio, conversion value, and premium to convert? Solution: The conversion ratio is $1,000/$50=20:1 The conversion value is 20 $55=$1,100 The premium to convert is $1,180-$1,100=$80 Interest from convertible bond: $60 Dividend from stocks when converted: $7 Therefore, holding convertible yields higher income [John Nofsinger]: Bond investment strategies. Why invest in bonds? We saw in the risk area-risk return stuff-that bonds often have a little lower risk and a little lower return. And that s fine. A lot of people like that. A lot of people like a nice, stable income. But you also get diversification 6

7 right. Just adding some bonds to a stock portfolio gets a more stable type of return and risk in the new portfolio. Bond Investment Strategies Why invest in bonds? o Stable income and diversification Asset allocation: the process of diversifying an investment portfolio across various asset categories, like stocks and bonds and cash to balance the risk/reward tradeoff. o Prime benefit is daily risk reduction o Even a modest amount of diversification can sharply dampen portfolio risk [John Nofsinger]: Here s just some examples where if you had all stocks, you would have earned 13% and had about a 17% level of risk. If you-there were 21 years that had a loss. This was the best way to go about 64% of the time. Now let s go down and add about 40% bonds. Of course the rate of return that you re gonna get goes down. But the risk goes down quite a bit as well. And even your worse day was a lot better than if you had 100% stock portfolio. So you get more stabilization out of this. Figure 15.4 Asset Allocation Can Help Achieve a Balance Between Risk and Return, [Table of different asset allocation and their annual return, risk, risk-reward, how often the best mix, years with loss, and worst loss] [John Nofsinger]: Lastly what are some other trading vocabulary-bond investing vocabulary? People have a bond portfolio. They usually just don t have one bond right. So one way to do it is a laddering approach where I buy a two year bond and I buy a four year bond and I buy a five year bond and I buy a six year bond etcetera, etcetera. Right and so in two years-in two years-this bond will come too and I ll get a thousand dollars back. Well this two year bond-this four year bond-will be a two year bond left to maturity right. This I didn t mean I meant to go two, four, six, eight-every two years. So this six year bond becomes a four year bond after two more years right. So over time, the maturities get smaller. And when I get my thousand dollars back from when this one matures, I take that money and I go out and I buy the one on the end right. So every two years I m gonna get a thousand dollars from that bond from the bond that matures and I m gonna go out and I m gonna buy a long term bond with it. And then wait two more years and then this one will mature. And I ll take that money. And so it s like a ladder you have at every 7

8 step in time or in maturity-time in maturity-you have a bond. So that s one way to do it. And then it has some management because every year, every two years you get the principal and then you buy a new bond. Another way to do it is a Barbell strategy. Visualize a barbell. Well you have a lot of short term bonds and then you have a lot of long term bonds. And year after year, these all mature. So you gotta buy new ones that are just one or two years to maturity. And you just do that cycle for a while until you decide that your long term bonds are not long enough anymore. That they ve now become midterm bonds or something. So that s another way to have a portfolio of bonds. Bonds also are traded periodically right. You just don t have to hold. You can change into low duration bonds during times when you think interest rates are going up. You can get into long term bonds when you think they re going down etcetera, etcetera. And so you can have a more active strategy as well. Okay well this concludes the second video and last video for the bond valuation topic. Maturity-Based Strategies Laddering o Appropriate for investors seeking greater interest income with minimum price volatility o Strategy: construct a portfolio using bonds with a series of targeted maturities, resembling a bond maturity ladder Barbell o Another strategy used by investors seeking greater interest income with minimum price volatility o Strategy: concentrate portfolio on both very short term and very long term bonds (six month T-bill and 30 year T-bonds) Bond swap o Complex risk/return strategies o Strategy: the simultaneous sale and purchase of fixed income securities 8