equity & fixed notes boquet
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Equity & Fixed Notes BoquetTRANSCRIPT
1. LECTURE 1: FEATURE OF DEBT SECURITIES
1.1. What are we going to see?
-‐ What are the main components of a bond, and their characteristics -‐ Ways of which a coupon can be paid -‐ Look at the different ways in which a life of a coupon can end -‐ Understand the additional options included in the contract when it is issued -‐ The methods to finance the purchases
1.1.1. The bond
A bond = fixed income instrument ... See SLIDES ! Face value = 100. Always this !
1.1.2. Maturity Maturity = number of years, or he period that remains before the end of the contract. Maturity date = end of the contract. Maturity important for 3 reasons:
-‐ Interest rate payments: the bigger is the maturity, the more the interest we will have -‐ Yield curve = a function that gives us the level of interest for different types of maturity -‐ Price volatility = as the maturity changes, the volatility of the price changes.
Price of the bond: Pt = 100/(1+I)^T. So if the interest rate is higher, the price differs!
1.1.3. Par value, or principal value Trading at premium: is the price is above the par value Trading at discount: if below the par value At the maturity date, the bon price = par price.
1.1.4. Coupon rate Coupon rate = interest rate that the issuer agrees to pa each year to the bondholder. Interest rate = expressed in annual terms.
1.1.5. Coupon rate structures: different ways to pay the coupon -‐ The bond that do not pay coupon at all = zero coupon bonds. What’s the profit then ?
The price is not at premium. -‐ Step up note : the rate increase over time -‐ Differed coupon bonds -‐ Floating rate security: coupon rate = fixed value
o Coupon rate = reference rate (the coupon rate is coppeled to an other value) + quoted margin (fixed)
1.1.6. Accrued interest, full price, and clean price
1.1.7. Provision for early retirement of debt -‐ Bullet maturity: -‐ Iking fund provision: -‐ Call provision: option to rebuy the option before the maturity arrives.
o Why? When the interest rate -‐ Put provision: bondholder the right to sell the bond.
o When? If interest rate go up, he sell the bond and buy others -‐ Convertible bonds: Option that five the bondholder the possibility -‐
1.1.8. The embedded option granted to issuers The most common embedded options are:
-‐ The right to call -‐ The right of borrowers in a pool of loans to prepay principal early -‐ Cap on a floater
1.1.9. 2 measures financials institutions can use:
-‐ Margin buying: way of financing the buy of security. TUYAU If a financial institution want to by a 100$ bond, it can borrow 80 from a broker. Margin: 10$. He bought with 20$ something that has the value of 100$. If it goes up from 20%, the ROE = 100%. If it goes down from 20%, loss = 50%.
-‐ Repurchase agreement
A financial institution that owns an amount of bonds, sells it to an other institution, and promises to buy back a bond at a specific day, at a specific price. 2. LECTURE 2: RISKS ASSOCIATED WITH INVEXTING IN BONDS : 21/11/12 Nomination risk = related to the European community We will talk about how sovereign risk is traded by supervisor
2.1. Interest rate risk The price of the bond fluctuates as the interest rate changes. If the interest goes up, the price of the bond goes down. Because the cash flows are discounted at the market interest rates. We have an intuition why this is happening:
-‐ A bond, at par: the face value = the price. -‐ Coupon = c -‐ C = market rate that applies at the moment -‐ At one day, after the bond has been issued, the market interest ate goes up, è the
price? As the maturity of the bond didn’t change, and a new bond is issue at the par (which coupon is higher than the old one). They will be willing to buy the last one at a lower price. Because investors can buy the other bond and get a higher interest rate. The price of the bond of yesterday is such that the yield = the yield of the bond that is issued today! As we can’t either change the maturity or the coupon rate, the price change! So we have a relationship between the price of the bond and the current yield.
2.1.1. Featuring impact interest rate risk: -‐ Maturity: of the maturity increases: the bond that has been issued will be exposed to
the interest rate risk to a longer period of time. -‐ Coupon rate: as the coupon rate paid by the bond increase, than the interest rate
decrease: because of the price of the sensitivity. As the coupon increase there is the repayment will increase, so they will be less sensitivity to the price change.
-‐ Embedded option: hard to say -‐ Floating rates:
2.1.1.1. Maturity: See the example on the slide. As the maturity increase: the sensitivity decrease
2.1.1.2. Impact of coupon rate See example in the slides. As the coupon rate increase we have the other phenomena.
2.1.1.3. Embedded options impact interest rate risk
Example of a callable bond -‐ A price of the bond with imbedded option is composed of two parts.
o Option free bond (PF) o Value of the embedded option (PO)
-‐ To option: o Price of the option free bond o Price of the option
What is going the price of callable bond: see slide. Why will the value of the call option go up? As the interest rate will be closer to the call value of the yield to called. In the callable bond: we have the called yield: the yield under the issuer can call back the bond. As the IR goes close to this value it will the increase in the value in the option will be higher than the increase in the value of the option free bond. Why? Because it will be sure the issuer will call back this bond
2.1.1.4. IR risk for floating rate securities When the coupon is link to an other bond in the market. Every 6 month, the coupon will be adjusted by an other interest rate. What is going to happen if IR goes down? It will go down, the only thin that constitutes a risk:
-‐ Length between the setting of the coupon: every 6-‐month, 1-‐month. The smaller it is, the smaller is the risk
-‐ Whether the margin will change.. Interest rate of the bond = interest rate normal (=floating rate) + margin. If the margin goes down, it means that the investor will receive an additional part with respect to the floating rate that would be lower than the one prevailing in the market.
-‐ If this floating securities have a cap. After a certain value, the interest rate will not change. Example: mortgage for the interest rates. If the IR increases by more than 2%, it will not go up of more than 2. Ex : if IR= 1,9, cap = 1. So it cannot change more than between 2,9 and 0,9. Otherwise it will become a normal bond, because the coupon would remain constant.
Below the cap the coupon adjust to the market rate. So there is no change in the price of the bond, because there is no interest free rate, thus no need to adjust the price of the bond because the IR paid by the bond will fold. The price of the bond in the case the IR is fixed changes because it has to adjust to the yield on the market. If there is a floating component, this will happen immediatly, but the price of the bond will remain constant. If there is a model component it will change immediately. Once the cap has been reached, and the interest rate will be above the cap, the price of the bond will be affected by interest rate. Between the margins in which the IR changes, the price of the bond will not be affected by the IR. As the cap is exceed, the price of the bond will be related to the yield, and we will get back to the standard case of a bond that does not have a floating component. What will happen to the graph if we have lower value for the cap? If the price is between the two sides of the cap, the price will not change, just the coupon will change.
2.1.1.5. Measuring interest rate risk Duration! It gives us an idea of how much price will change as the IR will change by 1%. We have to compute the price of the bond when the IR declines by 1%, we have a value, and then we compute the price of the bond when the IR goes up by 1%. We devise this change in the past by 2.initials price.
2.1.1.6. The Yield curve risk An other risk that has to be considered = the yield curve risk; refers to the fact that investors have bond from different maturities and the yield would be different. How change the volatility of the bond when coupon rate change, or ... When we have a big portfolio, it is important to consider the IR at different maturities. Ex : we have different bonds, what is the percentage change in the price of the bond if the yield of the price change. As the maturity is high and the coupon paid is low, the change in the price is the highest. Important to see how the change in IR is spread in different maturities. Graph of how the yield will change: relationship between the maturity of the bond and the yield. Parallel shift of the yield: for the yield of each maturity the market interst rate is going to increase. Parallel shit = the exception. In other case, change in IR will be higher for high maturity yield. The change in the price of the bond will be higher when we consider higher maturity. If we have a big portfolio, we will have huger change in the market value of the portfolio.
2.1.1.7. Call and reinvestment risk Risk related to bond that have call options. Why is there a risk for this type of bond:
-‐ it may happened in the future that the issuer will call back the bond. When it happened when the IR in the market would be low, when the investment perspectives available in the market are the worse. The change in the price would be reduce to a comparable option feel (??) bond).
2.1.1.8. Credit risk:
1) default risk : the issuer is not able to accomplished what he as promised. : Payback IR or the full ...
2) Credit spread risk: the interest rate paid by bond = I = risk free interest rate + spread. The spread is due to the fact even if the RF remains constant, the other part can change. Why?
a. Investor will think the issuer has more probability to default b. Downgrade risk: the risk that is related to the fact that the rating who is
attached to the bond by the issue, it will go down. Each bond get a rating, when it goes down, the yield of this bond will go up because the probability of default will go up, and the price will go down.
3) Downgrade risk: every bond that is issued receive a rating, it reflect the quality of the bond, as the rating is going down, the risk premium will go up, the yield will go up, and the price will go down.
Two parts: investment grade bond and not investment grade bond. For not investment grade bond, the probability of default would be much higher. Transition matrix = the probability that has a certain grade will stay triple A, for example, or will change his rating to AA, or to A. This is computed by looking at historical downgrade.
2.1.1.9. Liquidity risk; When the inventors would like to sell it, they will sell it to a price that is below to the average price. They are two prices:
-‐ Bid price: price at which they are willing to buy it -‐ Ask price: price they want to sell the price
-‐ Bid-‐ ask spread = difference between the two.
2.1.1.10. Inflation risk Has to do with the fact that investors when they receive the coupon, want to have nominal value, but in term of real price, they have to deduct the inflation.
2.1.1.11. Volatility risk Important for bond that has embedded options. As the volatitly increase, the value of the option component of the bond is going to increase. Example: suppose we have a call bond. Graph with yield at which the bond can be sold: Yc, and Yx=yield right now. It can be represent with a normal curve, with Yx as the centre of the normal curve, and Yc at the left of this. So P(Y<Yc)= Yc. But how is going to become the function if the volatily
2.1.1.12. The sovereign risk Risks that a sovereign institution will not pay back the coupon o the face value or will reduce this value. Reasons:
-‐ The government don’t want to pay back -‐ The government don’t have the money to do it
We will talk about How does regulator dealt with those risks? Stress test: the look at value component and how they change when there are extreme measures. One thing at which they look at is: ... If they are extreme macroeconomic condition, if the banks will be able to handle those types of risks. How? They look at the financial balance sheet of the bank, and how much asset they have in each institution. For each of those they will ask a total % that will be covered by the equity. That percentage that have to put aside, as to be used in any case of default of risk. They will assign a percentage of the bonds, with the baseline scenario (no extra events), they should put aside 2.55 % aside. They declare in the case of no stress they should put aside 2.55 % of the bonds. In the case of extreme scenario, they should put aside more (99%): 4,78%. Chapitre 2 Maité : Chapter 2 – risk associated with investing in bonds:
1) Interest rate risk The price of the bond fluctuates as the interest rate changes: if r goes up, the price of the bond goes down. Why? If a bond is issued at par (price = face value), and the cupon c=r. And suppose that one day after that bond has been issued, the interest rate goes up. What will happen to its price? It will go down, nobody will want to buy such bond at the same price because if you buy the same bond today, you get a higher rate of return for the bond issued today compared to the bond issued before (when the interest rate was lower). Negative relationship between the price of the bond and its yield. As the yield goes up, the price of the bond goes down. Impact of maturity:
1) If the maturity of the bond increases, then the bond issued will be exposed to the
interest rate risk for a longer period.
2)
The higher the coupon rate, the lower its price sensitivity to interest rate changes.
1) Embedded option impact interest rate risk:
Option free bond PF= bond that doesn’t have a call option (less risk for the investor, because the issuer cannot call back the bond).
The lower the interest rate, the more interesting it becomes to call back a bond (a
callable bond!). The lower the interest rate, the more probability there is for a callable bond to be re-‐called by its issuer, so the lower will be its price. (je pense).
2) Interest rate risk for a floating rate security: Every six months the coupon rate changes, reason why there is a interest risk in the bond price. The smaller the time period between two coupon resetting dates, the higher the price sensitivity to the interest rate change. R(floating) = R + margin. The margin could be +/-‐ 1%, which means that the interest rate can vary by this amount. Once the interest rate goes above R+margin, the bond becomes a normal bond. Below the cap, the coupon rate adjusts to the market rate, there is no need to adjust the price of the bond because the interest rate paid by the bond adjusts to the yield prevailing in the market. Once the cap is reached and the interest rate fo the market is above the cap, then the price of the bond will be affected by the interest rate (we go back to the situation where there is a negative relationship between the bond price and the yield of a bond).
Here we just have a higher value for the cap, and we see that the price of the bond is affected by the yield only once the interest rate is >= cap (upper value of the cap is exceeded). If there is a lower value for the cap, when the interest rate prevailing in the market goes below the lower value of the cap, the bond price will increase. Measuring interest rate risk:
Formula for estimating the approximate % price change for a +/-‐ 100 basis point change in yield. Yield curve risk:
That is what hurts most the investor. The change in the bond price will be higher when we consider higher maturity and lower yields. Call and reinvestment risk: Related to bonds that have a call option. We don’t know what the future cash flowns of this kind of bonds will be, because it may be called back. When the interest rates of the market are low, the probability for the bond to be recalled increases. Credit risk: 1) Default risk = issuer is not able to accomplish what he has promised, he cannot pay
back either the interest rate or the principal. The default rate is the probability that the issuer will default : ratio between the number of issuer that won’t pay back divided by the total number of issuer.
2) Credit spread risk = mkt value of the bond falls as the return demanded by the market increases. The spread of a bond has an impact on the interest, and thus on its price. The risk premium can change for several reasons, and this can change the value of the bond for several reasons.
3) Downgrade risk = if a bond is downgraded, its price will probably fall. Divisions between ‘investment grade bond’ and ‘not investment grade bonds’.
Liquidity risk: Bid-‐price = price at which people are willing to buy the bond Asked-‐price = price at which people want to sell the bond. The difference between these two prices is the bid-‐ask spread.
The liquidity risk is the risk that the investor will have to sell the bond below its indicated value, where the indication is revealed by a recent transaction. It is important for investors that are willing to re-‐sell their bonds before their maturity. Inflation risk: As the inflation rate increases, then the risk is higher. Volatility risk: Really important for embedded option bonds. As the volatility increases, then the value of the option component of the bond is going to increase. If the volatility of the yields increases, the probability that (y*<yc) will be bigger. And if this probability that the yield of the market will be below the call price increases, it is because the standard deviation (volatility of the yields) is higher. Event risk: Refers to the events that can not be prevented. Sovereign risk = risk that a sovereign institution will not pay back the coupons or the face value, or will reduce these values. See: stress test – sovereign debt haircuts. There are different types of risk: macroeconomic (decrease in GDP, government cannot pay back etc) microeconomic risk etc. The higher the risk, the higher the amount of capital that banks have to put aside. Market prices all type of risk, even the convertibility risk. See: example of Belgium, historical decomposition of bond yield spreads. Three components of risk = economic, idiosyncratic and redenom. Risk. Germany is below the risk free rate of the market, thanks to the economic and political situation, but also thanks to the redenomination risk part. Germany’s risk components are below the actual risk components. Chapter 3 – overview of bond sectors and instruments:
1. Sovereign bonds are the biggest components of that market, they are issued by the
government. The government can issue bonds both on the internal and/or external markets.
2. Regular auction cycle method: -‐ multiprice method: institutional investors can choose between different prices -‐ single price method: only one yield -‐ ad hoc auction method -‐ tap method
3. Type of US treasury securities traded in the market:
-‐ Treasury bills -‐ Treasury notes: maturity <10years -‐ Treasury bonds
-‐ Tips (treasury inflation adjusted securities)= value of the coupon paid to
investors is adjusted to the inflation every six months. Inflation adjusted principal.
In the case of a bond that is not linked to inflation the coupon rate that should be paid is 3%.
-‐ Strips: it is another type of security. Every six months you receive a certain amount.
The tax treatment for this type of treasury security is paid at once (in the US).
-‐ Semi government or agency bonds: agencies of the federal government that can issue bonds as if they were government bonds (without really being gov bonds). Ex: US agency mortgage backed securities = institution that creates liquidity in the market, they take the mortgage that they have in their portfolio and create securities to be issued. Mortgage loan = contract that goes on until the end of the mortgage. Risks related to that kind of security is the incertainty of the borrower’s payment, the reinvestment risk for the lender/investor. Prof parti aie aie aie voir ca par nous meme.
3. CHAPTER 3: OVERVIEW OF BOND SSECTORS AND INSTURMENTS Faire tout le CHAPITRE! Two types of bond market:
-‐ National bond market: o Domestic bond market : legislation that is applied depend on the state of the
bond issue o Foreign bond market; if the issuer is a foreign company
-‐ External bond market: legislation will be in the US, the legislation of the US
How does central bank distribute new bonds?
3.1. Regular auction cycle method? -‐ Multiple price method
We have: Y1>Y2>Y3 : Investor who wanted Y2 get it, Y3 get it, but Y1 doesn’t get it as the yield is to low.
-‐ Single price method -‐ Us treasury securities are issued using a regular auction cycle/single price method
Cours du 5/12/12 Example of the exam. Consider the following two bond issued :
-‐ Bond A : 15% year bond : 5% -‐ Bond B : 30 year bond : 5%
Neither bond has an embedded option, both bonds are trading in the market in at the same yield. Waich bond xil lfuclutae more in price when interest rate changes : Beacause of madurity duration
3.2. Understanding the yield spread Slide 7/44 YTM: interest rate that make the future payment of the bond equal to the current price of the bond.
What are the questions we want to ask?
-‐ Which market instrument can we choose as risk free -‐ What are the drivers of the movement of the risk free rate? The policies that the
central banks uses to model the movement of the risk free rate -‐ Theorise about the yield curves
-‐ What are the drivers of spreads: corporate and swap spreads Slide 8/44 Bonds are considered do be risk free rate so they will Yield curve refers to the yields as defined as the YTM Terms structure of interest rate refers to the yield of 0 coupon bonds. Difference between the two,
So the first part of this (with the C) disappear in he term structure of interest rates. Slide 10/44 If they want to prifce an instrument oof 10 years = you check the 10 years. This is the benchamkrk that every instrument needs to pay n order to be traded in the market. As time evolves the yield shape curve changes The yield curve can stay flat if the yield is the same at all the maturities. See explications at Tanguy. Slide 11/44 What is the value of the risk free rate is important as it is the benchmark, and how can it be controlled by the central banks. Slide 12 : Total market size of the whole bond market, and securities, and stocks, etc We see that in 08, there is a drop in stock market. ¾ of the market = fixed income. The risk free rate is the driver of those. This is why central banks try to control risk free rates. Slide 13/44 If the central bank can control the risk free rate, it can influence the cost of finance corporation, cost of mortgages (in order to do so, they should control the longer part of the yield curve : mortgage have a long term maturity, so in order to control it, they have to control the long part. Slide 14/44 USA Federal Reserve, Europe ECB tryies to control the risk fee rate. Operations commonly used are the open market operations, which are the most common. Exceptional operation used by the central bank to stimulate the economy is a Maturity extension program: buy short-‐term bond. : They replace short-‐term bond by long-‐term bond. Why? If they sell short bonds, the price will decrease. The yield will increase. With these they will try to buy long-‐term bonds. The price will increase. Therefore the yield will decrease. So the curve of the yield will change Usually they work on the open market operations. Of the whole deposit that the bank have, a part of that needs to be deposited at the central banks. Example: Bank A = 100 of deposit Bank B = 100 of deposit
Central bank tell them they have to put 10% of this in the central bank, order to be sure that if people claim that money that can do it. So 20 go to the central bank. This is done everyday, what is the total amount of the deposit, and put 10% of that in the central bank. But this is very costly. In T+1 : A = 110. Put 11 in CB B = 90., put 9 in CB. Two options :
-‐ Either the bank A transfer 1 to the central bank, and bank B transfer 1 from the central bank.
-‐ They exchange the deposit. What is the Irate for this exchange = overnight interest rate.
How are they going to influence this? If they want to get the interest rate down: They will buy bonds from banks: so the central bank will put the money n the account of the banks at the central banks. So it goes for the bank A from 10 to 20. This increases immediately the supply of reserve (FYI: min 35). So in the graph, the curve supply will go lower. This makes the quantity higher, and the interest rates lower. See graphique at Tanguy Other way to do it : change the required amount of reserves imposed to banks. But not really used. What they usually d is what we have seen: open market operations. Slide 16/44 Forward rate formula: shows that investing in the yield wit the maturity M is the same as investing in the yield of maturity N as the risk free rate F(T+N,M-‐N) If the future forward short-‐term interest rate is expected to increase, the yield curve will be increasing sloping. Voir le reste chez Tanguy
12/12/12 Overview of the last lecture
3.3. Talk about the yield spread 3.3.1. Several types or yields:
Yield to maturity = interest rates which solve the equation P=98 F+100 C=8% Semi yearly coupon. Maturity 2 years. 98 = 4/(1+X) + 4/(1+X)^2 + 4/(1+X)^3 + 104/(1+X)^4. (Ps at the exam he will ask if X is gonna be bigger or smaller than 4%? Bigger). First part = risk free component:
-‐ Which market do we have to choose for the risk free rate? It depends on the market. In the US: treasury. In Europe: German bonds. In any case, government bonds.
-‐ Refers to the type of market, but also to the maturity. In the investment horizon. If we want to evaluate a cash flow that will be paid on 3 years, we need a risk free rate of that horizon
-‐ Important to understand how this RF changes with the maturity of the bond: called the yield curve. Movement of this:
o It can go upwards. o Steepening of the yield curve o Inversion. It can go from a y=X to a y=-‐x. The slope is reversed
Risk free rate is the basis of fixed income instruments. On top of that we have to pay a spread: additional component related to riskiness of the enterprise. Risk free rate is the basis of fixed the interest for corporation corporation of mortgages. It can slow down or accelerate the economy. What is the intuition that is responsible for that? The central banks.
3.3.2. 3 types of monetary policies : 1) Maturity extension program
Thrives to change the shape of the yield curve. They are selling short terms securities and buying long terms securities. (Pour info, min 19).
-‐ So the Offer goes up. The price will then get down as the demand is the same. So the yield of short term goes up
-‐ The demand of long term goes up, and the price goes up, thus the yield goes down. This changes the shape.
2) Open market operations There are based on the act that banks have to put at the level of the Federal Reserve a fraction of their deposits. If we consider banks X and Y, at time T, the level of deposit = 100 for both banks. And the chink of the deposited required to be in the fed = 10%. At XT+1 there is 11 for X and 9 for Y. So X will buy 1 and Y will by 9, in order to respect the percentage of deposits required. The bank Y lends for 1 day at the overnight rate. To lower interest rates, ECB and the FED will buy loans from banks X and ... Graphically, supply or reserve is upward sloping while demand is downward sloping.
3) Reserve requirements Etc.
Second part = spread component. We talked about Different kinds of yields and the most important one is the YTM. Spread refers to the sector. We may have different spreads regarding the market where you are: Intramarket spread. Corporate spreads can be divided into two components = default probability, and rate of liquidity. Swap spreads: nothing more than the difference between the swap rate minus the risk free rate. Determinants of swap spreads: credit risk + convenience yield + swap specific components. Main driver of swap spread is convenience.
4) Valuation of debts instruments 3 steps to follow:
-‐ Compute the cash flow -‐ At which interest ate to discount -‐ Compute present value of those cash flows.
Graph (CF/t). If coupons are paid every six months, the person who sold the bond after only three months will have the right to receive half of the coupon that will be given after six months (because he has been owner of that bond for three months. Need to take the ratio between the buyer and the seller to calculate the present value of the bond. Pt+3m = C/(1+y)^0,5+C/(1+y)^1,5+C/(1+y)2.5 + (100+C)/(1+y)^3.5 This is the price of the coupon when you want to sell it or buy it at a certain point of its life.
3.4. Valuing a bond between coupon payments P = 4/(1+0,04)^0,43 + ‘/(1+0,04)^1+0,43 + … This is the price that the buyer will have to pay.
3.5. Yield measures, sport rates and forward rates Example : M = 2 FV = 100 C = 8% Coupon payement every six months, and thus every time equal to 4. Capital gain, assuming that the price of the bond P = 90, the the capital gain = 100 – 90 = 10. From the cash flow receiver, the investor can reinvest on the market : 4*(r+i6m)^3 after six months. After 12 months the reinvestment equals R = 4*(r+i12m)^2. Where i12m is the interest rate that will be available in 12 months. Current yield and yields on ZCB (zero coupon bond) Current yield Y = C/Pt,m Yield of a ZCB d = (1-‐p)*360/Nsm where Nsm is the number of days to maturity. Yield to maturity
See formula above. It’s going to be the yield that will equal the PV of future cash flows to the current price of the bond. Coupons are paid every six months, thus for M=1 we are talking about the first period of six months. The yield to maturity is realised only if two conditions are completed. See slides. The YTM of the bond in the case of the slides is equal to 8% which is equal to the interest rate of the investment number one. The investor invests at time t an amount of 94,17, and after six months his investment will be worth 94,17*(1+0,04). After one year the bank will calculate the interest rate on this amount, thus after one year the holder of this contract will have 94,17*(1+0,04)^2 in his bank account. See the example of the YTM calculation. As we don’t know the interest rate that ill apply in the future, the investment income has a risk, called the “reinvestment risk”. When the coupons are paid every six months, we know that we can reinvest this money, but e are not sur about the rate that will apply at that point in time! Having information on the YTM on the coupon paying bond (=8%) doesn’t give you enough info to compare it to a bond without coupon, because of the reinvestment possible for each coupon received. Goal of this example is to understand that we cannot compare a certificate of deposits (no coupon paid) with a coupon paying bond, because of the reinvestment. Bootstrapping step 1: Obtain the YTM of all maturities: Graph YTM/M. Data you can get from traded bonds. Zero coupon bonds give information for the two first periods (thus 6months and 12 months). Then all the other info you can get comes from coupon yielding bonds. Remember the graph with the red crosses, the white points and the blue carrés. The blue are zero coupon bonds (yields on bond that do not pas coupons). 1.5 3% 2.5 5% D =( 2.5 – 1.5)/(4-‐2) = 0.5 calculated in periods, knowing that the unit of each period is six months. (voir graphique Camille). Y of period 3 = 1.5 + 0,5 = 2 ===> 4% Step 2: Obtain the zero yield curve Find the X which is the only inconnue of the formula on the slides.
4. COURS 19/12/12 4.1. Bootstrapping.
Goal = obtain yields on zero coupon bonds for all types of materials. Yield we compute by looking at the formula of the YTM will be a yield that will have incorporated the effect of a coupon To have the yield on a zero coupon bond, we have to do this procedure. The ultimate goal = yield for all types of maturities, also for bonds that are not traded. Second issue: for types who are traded, there are two types of bonds:
-‐ Zero coupon bond -‐ Coupon Paying bonds.
First: compute the yield : on maturity in which bonds are not traded. Once we have that, we need to compute the zero coupon bond for the whole yield curve.
4.1.1. The procedure Year 1: got something, as the year 3,4, 7, and 8. The yields that we will observe in DataStream. We will do 2 things:
-‐ Fill the gaps -‐ Transform the yield years in zero coupon yields.
How do we do that?
1) Compute the yield for the bonds that are not traded 2) Compute the yield of the zero coupon bonds
4.1.1.1. Bootstrapping step 1
Consider that we have two bonds that are not traded -‐ Maturity of 4 years -‐ Maturity of 1 year.
What we need to do = have the yield on the bonds that are within 4 years and 1 year. The only information that we have = yield on the bond of 4 years = 8%, and on 2 years = 6%. What we do = linear approximation: we draw a line between these two points. The yield that we do not observe in the market is on the line that we draw. A yield of a 6 period bonds (=3 years) = yield of period of 2 years + something. The something = 6%/2 + 1/6.*4 (because we go 4 period forward). = 3,67%. The slope of the linear regression = see slides.
4.1.1.2. Bootstrapping Step 2: Second step in order to have the zero coupon bond Suppose that we have:
-‐ Zero coupon yield for period -‐ ZCY 1 and 2. -‐ YTM for period 3 = 4%
What we want to do = transform the date point number 3to zero coupon yield. Therefore we have to do:
A) Compute the cash flow in every period a. Period 1 = 2 b. Period 2 =2,2 c. Period 3 = 102
B) Compute the present value a. Discount the cash flow by the corresponding zero coupon bond b. Z1 and Z2 is divided by 2 because it is computed in year return.
C) Equate the present value to the current price The only thing that we miss = Z3. This is not a zero coupon yield, that is why we don’t take it. To discount the ash flow properly, we don’t take the YTM. We cannot compare the zero coupon bond which give nothing during 3 years, and the YTM. The YTM =: interest rate that will apply for different maturity. Second cash flow we divided by the interest rate that apply for a period of one year. In Z3, the return will be higher than the YTM. Because the 4% refers to a bond that pay coupon during 3 years, while Z3 refers to a bond that doesn’t pay coupon. In the exam = the most difficult part = compute something last this !! He will go maximum to 3 period, but he may ask us to compute the zero coupon yield.
4.2. The forward rates The forward rate is nothing more that the interest rate that make. The future value is equalt to the 1+zero coupon yield to the power of two. There might be two questions. Here it is a zero coupon bonds. Or to compute the zero coupon bonds. And then given the answer, compute the forward rate! Compute the forward rate with the YTM = WRONG. We need the zero yields to compute the forward rate. These parts is the questions we will not find in the book.
4.3. Interest rate risk management Interest rate risk = most important risk for bonds.
4.3.1. Motivations Refers to the fact that interest rate can change over time. They give to their member mortgage or other type of personal loans. Characteristic o those institution (Saving and Loan: S&L)
-‐ Borrow short -‐ Lend long
Because their deposit = short investment. Why long? When they give a mortgage to one member it is like a investment for them. For the deposit, the interest rate = s =short term interest rate + the risk related to this institution for a short period Investment (mortgage) = m = risk free rate for a period of 10 years + a spread related to the riskiness. Profit = difference between those two interest rates = difference between the 10 year interest rates and the 3 month interest rates. What is the key issue? When they sell a mortgage, they will do it like many years in advance, and the financing of those types of finance will be down every quarter. They will receive the M interest rate at every period, but it is established years in advance. What happened in the 80, the IR that were earned, were established in 1970, but what they had to pay in 80, jumped up to 16%. So all of them were in trouble.
4.3.2. Price volatility 1) The changes in the prices as the yield change, is gonna be different from bonds to
bonds. Not all the bonds will change their price in the same way as the IR change. If the IR increase by 1M, the change in a price will be different for a bond with maturity of 1 year, than for a one of 5 years.
How does bonds change when IR goes up of 1%? Slide: IMPACT OF COUPON RATES How can we approximate the change in price?
-‐ With the duration -‐ With the formula of the present value, we just add 1% to the actualization factor
Price that pay a higher coupon will be less subject to the change in interest rates Slide: IMPACT OF MATURITY As the maturity increase, the price sensitivity of the bond will increase as well.
2) For small change in yield the percentage change in price is almost symmetric. We can see it by the shape of the curve
3) If the curve is gonna be flat, then the change is price is gonna be more asymmetric.
How much asymmetric this is gonna be will depend on the shape. If the level of convexity is low.
4) The bondholder is going to ask a discount with respect to an option free bond just
because there is an option. When IR goes down, the bond is going to be Change in price is asymmetric, but the flatter the steeper (plus pentue) the curve, the less asymmetric it will be).
Embedded option: callable bond will be called at the worst moment for the bondholder and at the best moment for the issuer of the bond. The bondholder will ask a discount with respect to a non-‐callable bond. The bond is going to be called when the IR is low. He will have to Bond holder will ask a discount, I will pay you less with option free bond, because there is a option you will use f exactly when I don’t want. So there are two types of prices :
-‐ Types of normal bond. We have seen the call option, but NOT the PUT option! Duration : 10,66 = if the IR go down by 100 basis point, the price of the bond, will go up by 10,66%. There is an error with this approximation. The more convex is this curve, the higher is gonna be the mistake. Don’t do duration and immunisation.