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06BondValuation1.pptxTopic 4 bond valuation
Fundamentals of Finance
Fall 2017
Zhun Liu
What is a bond?
A bond is a debt instrument requiring the borrower to repay to the lender the amount borrowed plus interest over a specified period of time.
The bondholder generally receives a fixed interest payment, known as the coupon, each period until the bond matures.
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Today
We are starting the important topic of fixed income securities. In this topic we will analyze debt securities (all kinds of bonds). Debt securities are called fixed income securities because they pay a fixed stream of income that is determined at the beginning of the bond contract. Compared with the equity pricing we did last week, this is easier because there is almost no uncertainty about the cash-flows that a bond pays. The only exception is when the company that issues a bond goes in default. This credit risk is something we won’t talk about very much in this class. If you take fixed income class you will learn all the details about this.
This topic consists of 3 parts:
Today and Thursday: yield calculations (more complicated when coupons)
Maturity: yield curve (relationship between 1, 2, 5, 10 yr bonds)
Duration (relationship between change in interest rate and the change in the price of a bond)
We start by talking about some of the main features of bonds.
Then we go to yield calculations. We already know how to do these for zero coupon bonds. They are more difficult for coupon-bearing bonds. We will talk about YTM and holding period returns or realized returns.
In the next class we will talk about forward rates.
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Some terminology
Face value is the principal amount of a bond that is repaid at the end of the term, also called par value.
Coupon rate is the annual coupon payment, as a percentage of face value.
Maturity is the specified date on which the last payment on the bond is made.
Price (market value) of the bond is the amount paid to buy or sell the bond.
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Main Features of Bonds
Issuer:
US Treasury/Government
States, municipalities, and agencies
Corporations
Foreign governments (sovereign bonds)
Term (number of years to maturity):
Short (less than 1 yr)
T-bills, CD’s, Commercial papers
Long (more than 1yr)
T-bonds, corporate bonds
Consols
Market value (price) vs. face value (par value):
Par bond
Discount bond
Premium bond
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The federal government borrows money by issuing Treasury bonds.
Interest payments on state and local bonds (municipals) are tax-free from state-issuing and federal income taxes.
The two main other bond-issuing agencies are Freddie Mac and Fannie Mae, the two government-sponsored mortgage agencies. If you remember our discussion of asset-backed securities in class 1, we said that those two companies underwrite mortgages and pass these through to investors in the form of mortgage backed securities, a class of bonds.
In 2009.Q2, there was $14.4 trillion dollars of mortgage debt outstanding. About half of that ($7.5 trillion) was securitized in mortgage pools. Of that, about two-thirds ($5.1 trillion) is securitized by the government (Ginni Mae) and agencies (Fannie Mae, Freddie Mac). The market for mortgage-backed securities is still larger than the market for Treasuries.
http://www.federalreserve.gov/econresdata/releases/mortoutstand/current.htm
A par bond is a bond that is sold at time zero at par value; e.g. if the face value is $1000, the bond is issued at a price of $1000. Most T-bonds are par bonds.
Q: Why would you pay $1000 for a security that pays off $1000 in 10 years?
A: The coupon payments are set high enough to induce investors to pay par value for the bond.
If the bond is issued at a price below $1000, we say it’s a discount bond. Zero-coupon bonds are always discount bonds.
If a bond is issued at a price above par ($1000), it’s a premium bond.
5mins/15mins
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Main Features of Bonds
Coupon
Coupon rate: total annual interest payment per dollar face value
Period (usually semi-annual)
Fixed or variable (floaters and inverse floaters)
Nominal or inflation-indexed (TIIS / TIPS)
Possibly no coupons (zero-coupon bond)
Currency
Yankee bonds, Samurai bonds
Eurobonds
Credit risk
Risk free
Defaultable
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Usually the coupon rate is determined at the beginning of the contract (5% coupon rate = 5% of face value = $5). The coupon rate is an annual rate. Usually coupon payments are semi-annual: $2.5 after 6 months and $2.5 after 6 months
Floaters make interest payments that are tied to some measure of current interest rate (e.g. T-bill rate + 0.5% or LIBOR + 0.5%). If at the next coupon date the T-bill rate is 4%, the coupon rate over the next 6 months is 4.5% annually. Q: What happens to the price of the bond when the interest rate rises? A: the discount rate at which coupons are discounted rises (lower price), but this is offset by higher coupon payments (higher price). Floating rate bonds have no interest rate risk. They are very conservative investments.
Inverse Floaters work the other way around. If the interest rate decreases the coupon rate increases and vice versa. Q: What happens to the price of the bond when the interest rate rises? A: The holder of an inverse floater is severely hurt because not only is the discount rate at which the CF are discounted higher (lower price), the CF themselves also go down (lower price). Interest rate risk is magnified.
Since 1997, the US government started issuing inflation indexed bonds (Treasury Inflation Indexed Securities). The value of both coupon payments and principal payment is tied to the Consumer Price Index. For example, take a 3-year maturity, annual coupon paying bond with face value of $1000 and a coupon rate of 4%. Assume that the inflation turns out to be 2%, 3% and 1% in years 1-3. The payments are:
Year 1: par = $1000*1.02 = $1020, coupon payment = 0.04*1020=40.80
Year 2: par = $1000*1.02*1.03 = $1050.6, coupon payment =0.04*1050.6= $42.02
Year 3: par = $1000*1.02*1.03*1.01 = $1061.11, coupon payment = $42.44 + principal = $1061.11=$1103.55. Total payments: $1186.4 (instead of $1120 for nominal bond)
Mexico has issued bonds that are tied to the oil price level.
Foreign governments or corporations can issue bonds and have them traded in the US. If a German firm issues a corporate bond in $ that trades in NY, such a bond is called a Yankee bond. Likewise, a bond issued in Tokyo in yen by the same German firm is called a Samurai bond. If that same German company issues a bond in $ that trades outside the US (e.g. in London) that is called a Eurobond (in the case of London a eurodollar bond). The eurodollar market refers to all $ denominated debt trading outside of the US. It falls outside of the jurisdiction of the Securities and Exchange Commission.
Corporations can go bankrupt, the US government cannot. But the Argentinian government can! Just like corporations, it is paying a premium above the risk-free rate. This credit risk premium is the default spread.
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Main Features of Bonds
Seniority and security
Senior, subordinated senior, junior…
Secured by properties and equipment, other assets of the issuer, income-stream, etc.
Sinking fund provisions (sinkers)
Covenants
Restrictions on additional issues, dividends, and other corporate actions.
Option provisions
Callability: After a certain period, issuer has the right to pay back the loan before it matures.
Putability: After a certain period, bondholder has the right to demand payment of the loan before maturity.
Convertibility: After a certain period, bondholder had the right to exchange the bond for stocks of the issuer.
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Seniority: who gets paid back the first when the company goes bankrupt
Security: Asset-backed bonds: e.g. mortgages, e.g. Walt Disney issued bonds whose coupon rates were tied to the income it got out of its movies.
Sinking Fund: Some municipal or corporate bonds are issued with a sinking fund provision. This means the issuer pays off the principal of the bond over time, for example 10% of the principal in each of the last 10 years of the life of the bond, rather than all at the end. The sinking fund provision can be either optional or mandatory.
Bond covenant: is a general term for all kinds of additional restrictions written in the bond contract. For example, the bondholders may want to limit the firm’s ability to pay out dividends to equity holders. They also may want to prevent the company from issuing more bonds or more stock. Q: Why is that? A: To protect their own cash flows. If the company goes bankrupt, they want to be paid back.
Option provisions: When a bond is issued, the issuer may have the option to call (redeem) the bond on specified dates and prices (call price) prior to maturity. The list of dates on which a specified bond can be called is shown in a call schedule. Since it’s an option, the firm can choose whether or not to call. Call protection refers to the amount of time from the current date until a bond can be called. For example, if the first call on a bond is 3 years from now, the bondholder will have 3 years of call protection, and they are assured that they can own the bond for at least 3 years. Call risk refers to the risk that a bond may be called when the investor does not want it to be called. Q: If you were a company, when would you call your debt?
A: Bond are often called when interest rates decline, so that firms can refinance their debt and pay lower interest rates afterwards (prepayment option for mortgages). Investors in the bond get their cash back and have to reinvest it at the lower rates. Call risk can be eliminated by buying non-callable bonds.
Puttable bonds are bonds where the bondholder has the right to demand payment prior to maturity. Again, this is an option, not an obligation.
Convertible bonds allow the bondholder to convert the bond into stocks of the issuing company.
7mins/32mins
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Types of Bonds
Zero coupon bond
no coupon payments
pays face value at maturity.
sell at deep discount
Vanilla Bonds
coupon payments fixed for the life of the bond
repay principal and retire the bonds at maturity
contracts have the features and provisions found in most bond covenants.
annual or semiannual coupon payments
Convertible bonds
may be exchanged for shares of the firm’s stock
sells for a higher price than a comparable non-convertible bond
bondholders benefit if the market value of the company’s stock gets high enough
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Yield to Maturity (YTM)
YTM(y)
the rate that makes the present value of the bond’s cash flows equal the price of bond
the rate of return a bondholder earns if the bond is held to maturity, all coupon and principal payments are made as promised, and coupon payments are reinvested at the same yield
changes daily as general interest rates change
For a zero-coupon bond: (cash flow only at maturity)
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An Example
Suppose the following zero-coupon bonds are trading at the prices below per $100 face value.
Determine the YTM for each bond
Maturity 1 year 2 year 3 year 4 year
Price $96.62 $92.45 $87.63 $83.06
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Solution
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An Example
What is the price of a zero coupon bond with a $1,000 face value, 10-year maturity, and semiannual compounding? The yield to maturity on similar bonds is 12%.
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YTM on Coupon Bond
The yield to maturity (y) on a coupon bond paying a coupon of CPN each year and maturing in n years is:
Solve for YTM
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An Example – Coupon bonds
3-year, annual pay coupon bond, with coupon rate of 8%, face value of $1000
Draw cash-flow line
If the bond sells at P=$1000, what is YTM?
YTM solves the pricing equation:
Answer: YTM = coupon rate = 8% (par bond)
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Q: What is its YTM?
A: 8% (Compute the IRR with calculator.)
YTM = IRR = 8%,
Coupon rate = C/F = 80/1000=8%,
This is called a par bond.
We have that when a coupon bond is selling at par value: YTM = coupon rate
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Another Example – Coupon bonds
Consider a three year, $1,000 bond with a 8% coupon rate, annual coupons and a price of $950.26.
What is the YTM for the bond?
Solving for YTM = 10%
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Discount, par, and premium bonds
If the bond sells at $1000, what is YTM?
If the bond sells at $900, what is YTM?
If the bond sells at $1100, what is YTM?
If the YTM is higher than the coupon rate, then what must be true about the price?
| Price | Coupon rate | Current Yield | YTM | |
| Premium Bond | 1100 | 8% | 7.27% | 4.35% |
| Par bond | 1000 | 8% | 8.00% | 8% |
| Discount Bond | 900 | 8% | 8.88% | 12.17% |
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ANSWERS:
8.00% = 8.00% = 8.00%
8.00% > 8.88% > 12.17%
8.00% < 7.27% < 4.35%
YTM>coupon rate means a discount bond (the price appreciation makes the YTM higher than the coupon rate).
Q: What is the first bond called? A: Par bond. Q: What is its YTM? A: 8% (Compute the IRR with calculator.)
Case 1 – Par Bond
YTM = IRR = 8%, Coupon rate = C/F = 80/1000=8%, Current Yield = C/P = 80/1000=8%
We have :when a coupon bond is selling at par value: C/F=C/P=YTM
Case 2: Discount Bond
YTM = IRR = 12.17%, Coupon rate = C/F = 8%, Current Yield = C/P = 80/900=8.88%
We have : Coupon rate<Current yield<YTM. This relationship is always true for discount bonds.
Q: Why thus it make sense that the YTM is greater than the other two measures?
A: In case 2, for every $9 you spend you will be getting $10 back at maturity. You are realizing a capital gain. The YTM must be big to get this lower number (900) to grow up to $1000 in 3 years. YTM embeds price appreciation (capital gains). Other measures such as the coupon rate and the current yield do not. If the YTM is bigger than the coupon rate, you must be getting a capital gain, and so the bond must be a discount bond.
Case 3: Premium Bond
YTM = IRR = 4.35%, Coupon rate = C/F = 8%, Current Yield = C/P = 80/1100=7.27%
We have : Coupon rate>Current yield>YTM
Now comes a little complication: semi-annual pay bond (standard in US).
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Negative relation YTM and price
Recover YTM from price and vice versa we find a negative and convex relation between YTM and price:
Price
YTM
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P = 949.2 => IRR = 0.05 = 5% per half year = 10% YTM
P = 1000 => IRR = 0.04 = 4% per half year = 8% YTM (par bond!)
P = 1054.2 => IRR = 0.03 = 3% per half year = 6% YTM
Negative relation between prices and returns . Convex (we will return to this convexity next week).
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Effect of time on bond prices
If YTM stays constant at 5%
Premium Bond
Discount Bond
This is the “pull to par” effect
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Resources: Pearson Education
YTM and price fluctuations
For a 30-year zero coupon bond
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Resources: Pearson Education
Semiannual Coupons
Semiannual Compounding
Most bonds issued in Europe pay annual coupons, most issued in the U.S. pay semiannual coupons
The price of bonds that pay semi-annual coupons is:
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Example
(Following previous example in annually compounded case)
What is the YTM if the bond made semi-annual coupon payments?
Solving for YTM = 9.96%
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Compare
What is the price of a three year, $1,000 bond with a 5% coupon rate, annual coupons and a yield to maturity of 8%?
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Compare
What is the price of a three-year, 5% coupon bond with a market yield of 8% and semi-annual coupon payments?
Semi-annual coupon payment =CPN = $50/2 = $25
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Effective Annual Yield
In bond trading, the EAR is called the effective annual yield (EAY). The way to annualize a bond yield
Simple annual yield is yield per period multiplied by the number of compounding periods; for bonds with annual compounding, simple annual yield = semiannual yield * 2
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Yield to Maturity on Semi-Annual-Pay Coupon Bonds
For a semi-annual-pay coupon bond, the YTM is computed in 2 steps:
Find the semi-annual IRR, that is, the rate R that solves:
The YTM is the corresponding annual percentage rate (APR): YTM = 2*R
Read as “YTM equal to a APR with semiannual compounding”
The corresponding effective annual yield (EAY) is (1+YTM /2 )2 –1
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First explain general principle, next slide is example.
Step 1: First find the R that sets the price equal to the coupon payments using the standard formula. You can think of this as computing the IRR on an annual pay bond with twice the maturity and half the coupon rate. This rate R is a rate per half year.
Step 2: To find the YTM, which is an annual rate, multiply the R by 2 :
YTM = 12.07% (bond equivalent yield, APR).
The YTM takes into account the interest you earn on the coupons over the years. However, it does not take into account interest-on-interest earned within the year.
Step 3: If you want to account for compounding within the year, and calculate the effective annual yield. We have already seen how to convert a 6 month HPR into an annual return: EAY = (1+R)^2 -1.
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Example: Bond Yields
Yield to Maturity and Effective Annual Yield
An investor buys a 30-year bond with a $1,000 face value for $800. The bond’s coupon rate is 8% and interest payments are made semi-annually. What are the bond’s yield to maturity and effective annual yield?
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The Solution
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Yield to maturity/ effective annual yield need calculator or spreadsheet with solver
Yield to maturity = 10.14%
The Solution
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Yield to maturity/ effective annual yield need calculator or spreadsheet with solver
Yield to maturity = 10.14%
Effective annual yield = 10.40%
Example: YTM on a Semi-Annual-Pay Coupon Bond
Suppose that a 3-year bond has a face value of $1,000 and pays semi-annual coupons of $40. If the price is 900 then what is the YTM?
Step 1: use calculator to find R=6.036% that solves:
Step 2: YTM = 2*R = 12.072%
EAY = (1+R)2 – 1 = (1+YTM/2)2 –1 = 12.44%
-900
40
40
40
40
40
1040
t=3
t=0
t=1
t=2
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Let’s take the same 3 year discount bond (case 2) with face value 1000, time zero price 900 and coupon rate 8%. But now the cash-flows are semi-annual, as is customary for US government and corporate bonds.
Step 1: First find the IRR that sets the price equal to the coupon payments using the standard formula. You can think of this as computing the YTM on a 6 year bond with coupon rate 4%. If you do this, you will find IRR = .06036. The correct interpretation is that this bond has a 6.036% rate per half-year.
Step 2: Now find the YTM, which is an annual rate. If you use simple interest, you multiply the r by 2 to find the YTM = 12.07% (bond equivalent yield, APR).
Step 3: If you want to be more precise, you account for compounding within the year, and calculate the effective annual yield. We have already seen how to convert a 6 month HPR into an annual return: (1+0.06036)^2 -1 = 12.44%.
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Bond Valuation
Calculate bond price
Determine the required rate-of-return
Determine expected future cash flows – the coupon payments and face value
Compute the current market value, or price (P) by calculating the present value of the expected cash flows
P = PV(Annuity of N payments of CPN) +
PV(Single cash flow from repayment of face
value)
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Bond Valuation
General equation for the price of an annual bond
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Realized Bond Yields
The return earned on a bond given the cash flows actually received by investor
The interest rate at which the present value of actual cash flows generated by the investment equals bond’s price
The realized yield is important because it allows investors to see what they actually earned on their investments.
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Realized Return vs. YTM
Suppose that you buy a bond.
Will the return on your investment be equal to the YTM ?
Realized return = YTM if and only if :
you can re-invest the coupons at the same rate, and
you hold the bond until maturity
In most cases it will be different because :
you must re-invest the coupons at a different rate, or
you sell the bond before maturity at a price that corresponds to a different yield-to-maturity. (Market yields can change.)
Lesson: Bonds are risky!
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In class 4, we talked about the annual return on a zero coupon bond sold before maturity.
We said that generally, the rate of return of a zero sold before maturity will not be equal to the YTM. The reason that the HPR and the YTM are not the same when you sell the bond before maturity is that the interest rate can and typically will move in between, even though the face value in year 10 is known with certainty. We will do another example of zero coupon bonds in 2 slides.
Just as with zero coupon bonds, the HPR on a coupon-bearing bond is typically not equal to the YTM if you don’t hold the bond until maturity.
However, there is a second reason why the HPR is typically not equal to the YTM, which is specific to coupon bearing bonds. That second reason is that you may not be able to reinvest the coupons at the YTM.
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Realized Holding Period Return
Suppose that
at time 0, you buy a bond for P0
you re-invest all coupons until date t (potentially < T)
at time t, you sell the bond and the re-invested coupons for a total price of Pt
The annual holding period return (HPR) is the solution to:
where represents the interest or dividends paid during the period
Hence, the annual HPR is:
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Let us first review our general formula for the holding period return from class 4 (2).
The holding period return measures by how many percent per year your initial investment grows to the final value of the investment.
We can apply it to our current application of fixed income securities. We’ll do this in 2 steps: first we revisit the case of a ZCB and then we’ll talk about the realized HPR on a coupon-bearing bond.
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Practice 1: Realized Return on Zero-Coupon Bond
Suppose that a 3-year zero-coupon bond has a YTM of 5%
What is the bond’s current price?
Next year, the YTM changes to 7%
What is the price in that year?
What is the realized (holding period) return over the one year period?
What if the YTM in year 1 had remained 5%
What would be the price that year?
What would be the realized return over the one year period?
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Let us first review the case of a zero coupon bond.
Draw CF line for 3-year zero
V(t) = final value of the bond at maturity; V(0) = initial value of the bond at time zero = price at time zero.
V(0) = 1000/(1.05)^3 = 863.8376
V(1) = 1000/(1.07)^2 = 873.4387
Annual HPR = (873.4387 / 863.8376 )-1=0.0111 = 1.11%. HPR = 1.1% < YTM = 5%.
If you sell a zero before maturity, the HPR is different from the YTM. The HPR is uncertain. It depends on what the interest rate tomorrow turns out to be, or equivalently, the price of the bond next year. Increasing interest rates make the bond price fall, we saw that before.
V(1) = V(t)/(1+r)^2 = 1000/(1.05)^2 = 907.0295.
Annual HPR = (907.0295 / 863.8376 )-1 = 0.05 = 5% = YTM.
Point is : HPR = YTM for zeros if the interest rate stays constant.
Please review previous slides if you are not 100% sure on this.
This same point holds for coupon bonds; selling a coupon bond prior to maturity exposes you to interest rate risk!
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Practice 2: Realized Return on an Annual-Pay Coupon Bond
A 4-year coupon bond has face value 1000, coupons of 80, and a YTM of 8%
What is the bond’s current price?
What is the bond’s future value if coupons can be reinvested each year at YTM = 8%?
What is the bond’s annual return?
What if next year, the reinvestment interest rate falls to 4%?
What is the bond’s future value?
What is the bond’s annual return?
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Let’s start with an annual pay bond. We know that coupon rate C/F = 0.08 = YTM, then the price of the bond must be P=1000. It’s a par bond. How do we calculate the annual HPR? We want to use our annual return formula, so we need to get the future value of the bond V(t). This future value now includes all the reinvested coupons. Here the bond pays intermediate cash-flows.
Scenario 1
A: Year 1: coupon = $80, worth 80*1.08^3 = 100.78 at the end of year 4, Year 2: coupon = $80, worth 80*1.08^2 = 93.31 at the end of year 4, Year 3: coupon = $80, worth 80*1.08^1 = 86.40 at the end of year 4, Year 4: coupon = $80 + $1000 principal = 1080.00 at the end of year 4. Future value of all CF = $1360.49. Annual return = (1360.49/1000)^(1/4) = 0.08 = 8% =YTM. If you reinvest the coupons at the YTM and you hold the bond until maturity, then the HPR = YTM!
Scenario 2
Year 1: coupon = $80, worth 80*1.04^3 = 89.99 at the end of year 4, Year 2: coupon = $80, worth 80*1.04^2 = 86.53 at the end of year 4, Year 3: coupon = $80, worth 80*1.04^1 = 83.20 at the end of year 4, Year 4: coupon = $80 + $1000 principal = 1080.00 at the end of year 4. Future value of CF = 1343.2. Annual return = (1339.7/1000)^(1/4) = 0.0759 = 7.59% < YTM=8%. If you reinvest the coupons at a lower rate than YTM, then the annual HPR < YTM even if you hold the bond until maturity!
Lesson: The annual return on a bond crucially depends on the reinvestment assumptions. If and only if you can reinvest at the YTM is the annual HPR = YTM. If you can’t, the two are different. This makes the YTM a somewhat flawed yardstick.
Do this slide in the 2nd class on FI - 12min
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Why Bond Prices Change
Interest Rate Risk and Bond Prices
Effect of time on bond prices is predictable, but unpredictable changes in rates also affect prices
Bond prices are inversely related to interest rate movements
As interest rates decline, prices of bonds rise; as interest rates rise, prices of bonds decline
Bonds with different characteristics will respond differently to changes in interest rates
Resources: Pearson Education
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Interest Rate Risk – Maturity
Investors view long-term bonds to be riskier than short-term bonds
Interest rate risk increases as maturity increases, but at a decreasing rate.
Result: For a given change in interest rates, prices of longer-term bonds change more than prices of shorter-term bonds.
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Example The Interest Rate Sensitivity of Bonds
Problem:
Consider a 10-year coupon bond and a 30-year coupon bond, both with 10% annual coupons.
By what percentage will the price of each bond change if its yield to maturity increases from 5% to 6%?
Solution:
We need to compute the price of each bond for each yield to maturity and then calculate the percentage change in the prices.
The only difference is the maturity: 10 years and 30 years.
Resources: Pearson Education
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Example The Interest Rate Sensitivity of Bonds
The price of the 10-year bond changes by (129.44 - 138.61) / 138.61 = -6.6% if its yield to maturity increases from 5% to 6%.
For the 30-year bond, the price change is (155.06 - 176.86) / 176.86 = -12.3%.
Resources: Pearson Education
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Example The Interest Rate Sensitivity of Bonds
Evaluate:
The 30-year bond is twice as sensitive to a change in the yield than is the 10-year bond.
In fact, if we graph the price and yields of the two bonds, we can see that the line for the 30-year bond, shown in blue, is steeper throughout than the green line for the 10-year bond, reflecting its heightened sensitivity to interest rate changes.
Resources: Pearson Education
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Relation Between Bond Price Volatility and Maturity
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Interest Rate Risk – Coupons
For a given change in interest rates, prices of lower-coupon bonds change more than prices of higher-coupon bonds.
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Example Coupons and Interest Rate Sensitivity
Problem:
Consider two bonds, each pays semi-annual coupons and 5 years left until maturity.
One has a coupon rate of 5% and the other has a coupon rate of 10%, but both currently have a yield to maturity of 8%.
How much will the price of each bond change if its yield to maturity decreases from 8% to 7%?
Resources: Pearson Education
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Example Coupons and Interest Rate Sensitivity
The 5% coupon bond’s price changed from $87.83 to $91.68, or 4.4%, but the 10% coupon bond’s price changed from $108.11 to $112.47, or 4.0%.
You can calculate the price change very quickly with a financial calculator. Taking the 5% coupon bond for example:
Solution:
| Given: | 10 | 4 | 2.50 | 100 | |
| Solve for: | -87.83 | ||||
| Excel Formula: =PV(RATE,NPER,PMT,FV)=PV(.04,10,2.5,100) |
Resources: Pearson Education
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Example Coupons and Interest Rate Sensitivity
What we can learn:
The bond with the smaller coupon payments is more sensitive to changes in interest rates.
Because its coupons are smaller relative to its par value, a larger fraction of its cash flows are received later.
As we learned in the example, later cash flows are affected more greatly by changes in interest rates, so compared to the 10% coupon bond, the effect of the interest change is greater for the cash flows of the 5% bond.
Resources: Pearson Education
45
45
Relation Between Bond Price Volatility and the Coupon Rate
46
46
Bond Prices and Interest Rates
Resources: Pearson Education
47
47
Default Risk (Credit Risk )
Yield on T-bills is a proxy for the risk-free rate
However, there’s risk that a borrower may not make payments as promised
Credit risk is the risk of default, so that the bond’s cash flows are not known with certainty
Lenders are paid a default risk premium for purchasing securities with default risk
The default risk premium (DRP) is the difference between the yield on a security with default risk, idr, and the risk-free rate, irf
The credit spread is the difference between the yields of corporate bonds and Treasuries
Thus corporations with higher default risk will need to pay higher coupons to attract buyers to their bonds
48
48
Corporate Bonds
Corporate Bond Yields
Yield to maturity of a defaultable bond is not equal to the expected return of investing in the bond
A higher yield to maturity does not necessarily imply that a bond’s expected return is higher
You might, or might not, receive the scheduled payments
49
49
Corporate Bond Rating
Bond Ratings
Several companies rate the creditworthiness of bonds
Two best-known are Standard & Poor’s and Moody’s
These ratings help investors assess creditworthiness
Category
Investment-grade bonds
Speculative bonds
junk bonds
high-yield bonds
The rating depends on
the risk of bankruptcy
bondholders’ claim to assets in the event of bankruptcy
50
50
Corporate Bond Rating Systems
51
Default Risk Premiums for Selected Bond Ratings
52
Figure 6.6 Corporate Yield Curves for Various Ratings, July 2013
53
53
Figure 6.7 Yield Spreads and the Financial Crisis
54
Review - Yield Measures for Coupon Bonds
Coupon Rate = C/F
Current Yield = C/P
Yield to maturity = IRR
Effective Annual Rate
Holding Period Yield
1 coupon payment at end of holding period
Coupon payments throughout holding period
55
Coupon rate: when a bond sells at par, this equals yield to maturity, otherwise it ignores the effect of price paid differing from F.
Current Yield: although it improves on C/F by replacing F by P, it ignores the capital appreciation or depreciation associated with P moving to F at maturity (pull to par).
The yield to maturity implicitly includes all effects of P, C and F on yields. If the coupons are annual, compute the IRR with number of periods = number of years of the bond. When semi-annual pay bonds, compute IRR with number of years 2x number of years of bond and double the IRR.
EAY = YTM for annual pay bonds. EAR = (1+YTM/2)^2 -1 for semi-annual pay bonds
Holding period yield = return per year
When coupon is at end of holding period: (P’+C-P)/P, P’ = selling price at end of year
When coupon payments occur throughout holding period: HPY = YTM only if annual coupon payments are reinvested at YTM. If coupons are reinvested at some other rate and/or if the bond is semi-annual pay bond, then you must calculate the final value of all cash-flows and HPY = (Final value of all CF/initial price)^(1/t’) -1, where t’ is the years the bond is held.
2mins/80mins (class 1)
55
10/10/2017
Review – Price vs. Rate
What will happen when a bond is traded at par? Below par? Above par?
Coupon Rate > Yield to Maturity
Coupon Rate = Yield to Maturity
Coupon Rate < Yield to Maturity
56
56
10/10/2017
Coupon RateFace Value
Number of Coupon Payments per Year
CPN
´
=
1/
1
n
n
Face Value
YTM
Price
æö
+=
ç÷
èø
P = FV
1+ yn m
⎛ ⎝⎜
⎞ ⎠⎟
mn =
$1000
(1+ 0.12
2 )2×10
= $1000
(1+ 0.06)20 = $311.80
P=
FV
1+
y
n
m
æ
è
ç
ö
ø
÷
mn
=
$1000
(1+
0.12
2
)
2´10
=
$1000
(1+0.06)
20
=$311.80
P = CPN ×
1 y
1− 1
(1+ y )N ⎛ ⎝⎜
⎞ ⎠⎟ +
FV (1+ y )n
P=CPN´
1
y
1-
1
(1+y)
N
æ
è
ç
ö
ø
÷
+
FV
(1+y)
n
Annuity Factor using the YTM ()
Present Value of the
Present Value of all of the periodic cou
pon payments
Face Value repayment
using the YTM (
11
1
(1) (1)
y
NN
y
FV
PCPN
y
yy
æö
=´-+
ç÷
ç÷
++
èø
644474448
144444424444443
)
14243
3
2
1
)
1
(
1080
)
1
(
80
)
1
(
80
1000
YTM
YTM
YTM
+
+
+
+
+
=
P = CPN × 1 y
1− 1
(1+ y )3 ⎛ ⎝⎜
⎞ ⎠⎟ +
FV (1+ y )3
950.26 = 80 × 1 y
1− 1
(1+ y )3 ⎛ ⎝⎜
⎞ ⎠⎟ +
1,000 (1+ y )3
P=CPN´
1
y
1-
1
(1+y)
3
æ
è
ç
ö
ø
÷
+
FV
(1+y)
3
950.26=80´
1
y
1-
1
(1+y)
3
æ
è
ç
ö
ø
÷
+
1,000
(1+y)
3
EAY = (1 + Quoted rate/m) m - 1
EAY = (1 + Quoted rate/m)
m
- 1
T
T
t
t
R
R
)
1
(
F
)
1
(
C
P
T
1
t
0
+
+
+
=
å
=
P = CPN × 1
( y m
) 1−
1
(1+ y m
)mn
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ +
FV
(1+ y m
)mn
800 = 40 × 1
( y 2
) 1−
1
(1+ y 2
)2 X 30
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ +
1,000
(1+ y 2
)2 X 30
P=CPN´
1
(
y
m
)
1-
1
(1+
y
m
)
mn
æ
è
ç
ç
ç
ö
ø
÷
÷
÷
+
FV
(1+
y
m
)
mn
800=40´
1
(
y
2
)
1-
1
(1+
y
2
)
2X30
æ
è
ç
ç
ç
ö
ø
÷
÷
÷
+
1,000
(1+
y
2
)
2X30
EAY = (1 + Quoted rate/m)m - 1
= (1 + 0.1014
2 )2 - 1=0.1040
EAY = (1 + Quoted rate/m)
m
- 1
=(1 +
0.1014
2
)
2
- 1=0.1040
(
)
(
)
(
)
6
5
1
1
1040
1
40
...
1
40
900
R
R
R
+
+
+
+
+
+
=
P = CPN ×
1 y
1− 1
(1+ y )n ⎛ ⎝⎜
⎞ ⎠⎟ +
FV (1+ y )n
P=CPN´
1
y
1-
1
(1+y)
n
æ
è
ç
ö
ø
÷
+
FV
(1+y)
n
1
.
/
1
0
-
÷
÷
ø
ö
ç
ç
è
æ
+
=
t
t
t
P
I
P
HPR
ann
t
t
t
I
P
HPR
ann
P
+
=
+
)
.
1
(
0
