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**FINC4101 Investment Analysis**

Instructor: Dr. Leng Ling Topic: Bond Pricing and Yields

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**Learning objectives Compute the price of a zero-coupon bond.**

Compute the price of a fixed coupon bond. Describe the price-yield relationship of bonds. Distinguish between a bond’s flat price and its invoice price. Compute different measures of bond returns. Calculate how bond prices will change over time for a given interest rate projection. Recognize default/ credit risk as a source of risk for bonds. Identify the determinants of bond safety and rating. Understand how default risk can affect yield to maturity.

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**Concept Map Foreign Exchange Derivatives Market Efficiency**

Fixed Income Equity Asset Pricing Portfolio Theory FI4000

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**Types of fixed-income securities**

Fixed-income security / bond: A security that obligates the issuer to make specified payments to the holder over a period of time. We focus on the pricing of two types of fixed-income securities: Fixed-coupon bond (FCB) Zero-coupon bond (ZCB) Assume that market is in equilibrium throughout. Thus intrinsic value, V0 = P0. Examples of FCBs: Treasury bonds and notes, Corporate bonds. Examples of ZCBs: Treasury bills (maturity = 1 year or less).

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**Fixed-coupon bond (FCB) 1**

Firm pays a fixed amount of interest (‘coupon payment’) to the investor every period until bond matures. At maturity, firm pays face value of the bond to investor. Face value also called par value. Unless otherwise stated, always assume face value to be $1000. Period: can be year, half-year (6 months), quarter (3 months). Coupon rate: annual coupon payment as a fraction of face value. Coupon payment = interest payment The coupon rate serves to determine the interest/coupon payment. The annual payment equals the coupon rate times the bond’s par value.

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**How to ‘read’ a fixed-coupon bond: Example**

A firm issues an 8% 30-year bond with annual coupon payments. Par value is $1,000. What does the above tell us? 8%: the coupon rate. Multiply coupon rate by par value to get annual coupon payment. Coupon= 8% x 1,000 = $80. Maturity = 30 years. Coupon of $80 is paid annually, i.e., period=annual. At maturity (end of 30 years), firm will pay $1000 to investor. What happens if coupon is paid semi-annually? Quarterly? This slide shows student how to get vital valuation information from the description of a fixed-coupon bond. The coupon rate of a bond is NOT the required rate of return; it simply establishes the amount of the periodic coupon payment. So, from this bond, an investor will receive the following cash flows in the future: an annuity of $80 (8% of face value) for 30 years and $1,000 at the end of 30 years. If coupon paid semi-annually, coupon = 8%/2 x 1,000 = $40 $40 coupon is paid every 6 months for 60 (30 yrs x 2) 6-month periods $1000 paid at the end of the 60 six-month periods. If coupon paid quarterly, Coupon = 8%/4 x 1,000 = $20 $20 coupon is paid every 3 months for 120 (30 yrs x 4) 3-month periods. $1000 paid at the end of the 120 three-month periods.

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**Fixed-coupon bond (FCB) 2**

FCB gives you a stream of fixed payments plus a single payment (face value) at maturity. This cash flow stream is just an annuity plus a single cash flow at maturity. Therefore, we calculate the price of a FCB by finding the PV of the annuity and the single payment, using an appropriate interest rate. We use the financial calculator to compute the price of the FCB. Note that the discount rate is also called the yield to maturity, cost of debt or interest rate. For the moment, we do not distinguish between price and value/intrinsic value, although from equity valuation, we know that the two concepts can be different.

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**Fixed-coupon bond (FCB) 3**

Reminder: The “interest rate” used to find the PV of a bond is also known as: Yield-to-maturity (“YTM” or “yield” for short) Discount rate Required rate of return Cost of debt

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**Fixed-coupon bond (FCB) 4**

Price of the FCB, PFCB Fixed periodic coupon Number of periods to maturity Face value Note that rd is the YTM, or interest rate or discount rate for each period. If the bond pays coupons semi-annually, then rd is the semi-annual rate. Yield to maturity (in decimals)

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Find FCB price Consider an 8%, 30-year coupon bond that pays coupons semi-annually. Compute the bond’s price if the yield to maturity is a) 6%, b) 8%, c) 10%. If YTM = 6%, verify that bond price = $1,276.76 If YTM = 8%, verify that bond price = $1000. If YTM = 10%, verify that bond price = $810.71 All three cases share the following common inputs: FV=1000, N=30 x 2 = 60, PMT = 0.5 x 0.08 x 1000 = 40, YTM=6%, then I/Y = 6/2 = 3, CPT PV = or (to 2 decimal places) YTM=8%, then I/Y = 8/2 = 4, CPT PV = -1000 YTM=10%, then I/Y = 10/2 = 5, CPT PV = or

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**Inverse relationship between price & interest rate**

Notice that as YTM (interest rate) increases, bond price decreases. Conversely, as YTM decreases, bond price increases. This is the inverse relationship between bond price and yield to maturity (interest rate). This is a crucial general rule in bond pricing based on time value of money. When yields approach zero, the value of the bond approaches the sum of the cash flows Recall that bond price is just the pv of future cash flows (coupons and face value). When interest rates rise, the discount rate or YTM increases and the pv of future cash flows decreases. Thus the bond price decreases. Conversely, if YTM decreases, the pv of future cash flows increases and the bond price increases.

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**Figure 9.3 The Inverse Relationship Between Bond Prices and Yields**

Graphically, the price-yield relationship can be depicted by this graph, which shows the price of the 30-year, 8% coupon bond for a range of interest rates including 8% and 10%. The negative slope illustrates the inverse relationship between prices and yields. The shape of the curve implies that an increase in the interest rate results in a price decline that is smaller than the price gain resulting from a decrease of equal magnitude in the interest rate. This property of bond prices is called convexity because of the convex shape of the bond price curve.

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**US Treasury Note and Bond Quotes, Figure10.1**

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**Invoice price, Flat price, Accrued interest**

when you buy or sell a bond between coupon payment dates, the invoice price must incorporate accrued interest. Invoice price = Flat price + Accrued interest (actual price paid by buyer) (price quoted in the financial press) For a semi-annual payment bond, accrued interest between two coupon payment dates = The prices you saw in the previous slides are quoted or flat prices. These prices do not reflect the invoice price (i.e., actual price) paid by the buyer because the flat prices do not include the accrued interest. The invoice price formula is given here. Accrued interest formula assumes a semi-annual coupon payment bond. In Finance Project 2, you will get a chance to compute the invoice price of a bond. To keep things simple, we assume that we compute prices on the coupon payment dates. Thus flat price is equal to invoice price. In reality, you should know that when you buy or sell a bond between coupon payment dates, the invoice price must incorporate accrued interest.

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example Suppose the coupon rate is 8%, face value is $1000 and coupon is paid semiannually. 30 days have passed since the last coupon payment. The quoted price is 990. What is the invoice price? 990+(1000*8%/2)*(30/182) = =996.59

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Exercise Suppose the coupon rate is 10%, face value is $1000 and coupon is paid semiannually. 125 days have passed since the last coupon payment. The asked quote is 100:11. What is the invoice price that the investor has to pay? solution

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**Corporate Bonds Figure 10.2**

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Find YTM, Coupon rate 1)A $1,000 par value bond sells for $ It matures in 20 years, has a 10 percent coupon rate, and pays interest semi-annually. What is the bond’s yield to maturity on a per annum basis (to 2 decimal places)? Verify that YTM = 11.80% 2) ABC Inc. just issued a twenty-year semi-annual coupon bond at a price of $ The face value of the bond is $1,000, and the YTM is 9%. What is the annual coupon rate (in percent, to 2 decimal places)? Verify that annual coupon rate = 6.69% What happens if bond pays coupon annually? Quarterly? FV=1000, PV= , N=40, PMT=0.1x0.5x1000=50. Press CPT then I/Y. I/Y = this is the YTM on a semi-annual basis. To get the YTM on an annual basis, multiply by 2. Therefore YTM = x 2 = 11.80%. Unless otherwise stated, when question asks for YTM, it usually wants answer on a p.a. basis. 2) FV=1000, PV= , I/Y = 4.5 (because semi-annual pay), N=40. CPT then PMT. PMT = This is the amount paid every 6 months. To get annual coupon rate, multiple PMT by 2 and express the product as a proportion of par value. Therefore, annual coupon rate = ( x 2)/1000 = or 6.69% For both types of questions, if bond is annual pay, use the I/Y and PMT as is. Don’t have to do any multiplication. If bond pays quarterly, then multiply I/Y and PMT by 4 to state answer on an annual basis.

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**Zero-coupon bond (ZCB) 1**

Zero coupon rate, no coupon paid during bond’s life. Bond holder receives one payment at maturity, the face value (usually $1000). Price of a ZCB, PZCB F = face value of the bond The pricing equation is just the formula for finding the PV of a future cash flow. It’s that simple. N = number of periods to maturity Yield to maturity (in decimals)

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**Zero-coupon bond (ZCB) 2**

As long as interest rates are positive, the price of a ZCB must be less than its face value. Why? With positive interest rates, the present value of the face value (i.e., the price) has to be less than the face value.

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**These problems are just basic TVM problems where you receive a single cash flow in the future.**

ZCB Problems 1) Find the price of a ZCB with 20 years to maturity, par value of $1000 and a yield to maturity of 15% p.a. Assume annual compounding. N=20, I/Y=15, FV=1000, PMT=0. Price = $61.10 2) XYZ Corp.’s ZCB has a market price of $ 354. The bond has 16 years to maturity and its face value is $1000. What is the yield to maturity for the ZCB. Assume annual compounding. PV=-354, FV=1000, N=16, PMT=0. YTM = 6.71% p.a. What if semi-annual compounding? These problems are just basic TVM problems where you receive one lump sum in the future. For (1), PV is the price. For (2), I/Y gives the YTM. Some problems require semi-annual compounding so students should be careful.

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**U.S. Treasury Bills (T-bills) page 24**

Short-term, issued at a discount from par value, return the par value at maturity. The cash flow pattern looks like a ZCB. The discount from the par value is annualized based on a 360-day year.

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Figure 2.2

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Measures of return We measure the rate of return from investing in a bond in several ways: Yield to maturity (YTM) Current yield (CY) Yield to call (YTC) Realized compound yield (RCY) Holding period return (HPR)

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Yield to maturity The discount rate that makes the present value of a bond’s payments (coupons & par value) equal to its price. Interpretation: it is the compound rate of return that will be earned over a bond’s life if It is bought now and held until maturity All coupons are reinvested at the same YTM. From the earlier slides, we know that YTM just the discount rate that allows us to compute the bond price. Here, I give the formal definition of YTM. The YTM is the IRR on an investment in the bond. It can be interpreted as the compound rate of return over the life of the bond under the assumptions that (a) all bond coupons can be reinvested at that yield and (b) the bond is held to maturity. In this sense, the YTM is used as a measure of ‘average’ return over the bond’s life. Note that, if the interest rate at which the coupons can be reinvested differs from the YTM, the actual realized compound rate of return will DIFFER from the YTM.

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Annualizing YTM (1) If coupons are paid semi-annually, then the YTM we get from the financial calculator is a six-month YTM. We can convert this six-month YTM to an annual YTM using Simple interest => bond equivalent yield to maturity (or bond equivalent yield for short) OR Compound interest => effective annual yield to maturity (or effective annual yield for short) Compound interest accounts for the effects of compounding (or interest on interest) while simple interest does not. Bond equivalent yield is just the ‘stated’ or ‘nominal’ interest rate that we came across in FI3300. Effective annual yield is just the effective annual interest rate in FI3300.

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Annualizing YTM (2) In general, if a coupon bond pays coupons m times a year, then: Bond equivalent yield = periodic YTM x m Effective annual yield = (1 + periodic YTM)m – 1 Note: periodic YTM is the value of I/Y you get from the financial calculator. Periodic ytm is the value of I/Y you get from the financial calculator. Stated in decimals

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Annualizing YTM (3) A 20-year maturity bond with par value $1000 makes semi-annual coupon payments at a coupon rate of 8%. Find the bond equivalent and effective annual yield to maturity of the bond if the bond price is $950. Verify that Bond equivalent yield = 8.53% Effective annual yield = 8.71% BKM EOC Q10 part (a). First, solve for the semi-annual ytm. FV=1000, N=2 x 20 = 40, PMT = 0.08x1000x0.5=40, PV=-950. CPT I/Y = This is the semi-annual rate/YTM. Since this is a semi-annual payment bond, coupons are paid 2 times a year, m = 2. Bond equivalent yield = x 2 = or 8.53%(to 2 dp) Effective annual yield = ( )2 – 1 = = or 8.71% (to 2 dp)

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Annualizing YTM (4) Treasury bonds paying an 8% coupon rate with semi-annual payments currently sell at par value. What coupon rate would they have to pay in order to sell at par if they paid their coupons annually ? For the semi-annual bond, the effective annual yield to maturity = ( )2 – 1 = or 8.16%. If the annual coupon bonds are to sell at par, they must offer the same yield, which requires an annual coupon of 8.16%.

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Practice 6 (1) Chapter10: 4,5,6,12,13,15,16,20,27

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Homework 6 You are a US. Treasury bond dealer who trades a 4.75%, 3year, semi-annual coupon bond. Your required YTM is %. How should you quote your Asked price in percentage of par value as shown in Figure 10.1? Suppose today is Oct 23, A bond with a 10% coupon paid semiannually every Feb 15 and Aug15 is listed as selling at an ask price of 102:11. if you buy the bond from a dealer today, what price will you pay for it? the coupon period has 182 days.

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**Bond price, coupon rate & YTM (1)**

A $1,000 par value bond has coupon rate of 5% and the coupon is paid semi-annually. The bond matures in 20 years and has a yield to maturity of 10%. Compute the current price of this bond. FV=1000, PMT =25, I/Y=5, N=40. CPT, then PV. PV = Thus, price = $ < par value Show class how you get PMT = 25. Explanation: since coupons are paid semiannually, divide the annual rate of 5% by 2 = 2.5%. Thus the issuer pays 2.5% of par semi-annually. 2.5% of $1000 = 25. Thus PMT = 25 By similarly reasoning, with semi-annual payment, you must adjust the required rate of return to a semi-annual rate. That is, divide 10% by 2 and use the result for I/Y. therefore, I/Y = 5%. N = 40, because there are 40 periods. PV gives you how much investors are willing to pay, which is also the current price of the bond.

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**Bond price, coupon rate & YTM (2)**

Go back to the bond in the last problem. Suppose annual coupon rate = 10%. Verify that price = $1000 = par value Suppose annual coupon rate = 12% Verify that price = $1, > par value. It turns out that the following property is true. Illustrate the relationship between coupon rate, YTM, price and face value.

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**Bond price, coupon rate & YTM (3)**

Price < face value Bond is selling at a discount Coupon rate = YTM Price = face value Bond is selling at par Coupon rate > YTM Price > face value Bond is selling at a premium This property is true, whichever type of compounding is used. Rationale: Discount: When the coupon rate is lower than the ytm, the coupon payments alone will not provide investors as high a return as they could earn elsewhere in the market. To receive a fair return on such an investment, investors also need to earn price appreciation on their bonds. The bonds, therefore, would have to sell below par value to provide a “built-in” capital gain on the investment. Any discount from par value provides an anticipated capital gain that will augment a below-market coupon rate just sufficiently to provide a fair total rate of return. Essentially, investors will sell the bond until it reaches a price such that the combined coupon payments and capital gain provides the same level of total return as what is available in the market. Par: a bond will sell at par value when its coupon rate equals the market interest rate. In these circumstances, the investor receives fair compensation for the time value of money in the form of the recurring coupon payments. No further capital gain is necessary to provide fair compensation. Premium: if the coupon rate exceeds the ytm, the interest income by itself is greater than that available elsewhere in the market. Investors will bid up the price of these bonds above their par values.

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Apply what we learnt A 10-year annual coupon bond was issued four years ago at par. Since then the bond’s yield to maturity decreased from 9% to 7%. Which of the following statements is true about the current market price of the bond? The bond is selling at a discount The bond is selling at par The bond is selling at a premium The bond is selling at book value Insufficient information Answer: C. Remember: YTM = discount rate = cost of debt capital = required rate of return The bond was issued at par. So it’s coupon rate = YTM of 9%. Now, YTM is 7% but coupon rate is still 9%. From the last table, we know that if coupon rate > YTM, then bond must be trading at a premium.

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Try one more One year ago Pell Inc. sold 20-year, $1,000 par value, annual coupon bonds at a price of $ per bond. At that time the market rate (i.e., yield to maturity) was 9 percent. Today the market rate is 9.5 percent; therefore the bonds are currently selling: at a discount. at a premium. at par. above the market price. not enough information. Answer: A Market rate = discount rate = required rate of return etc When the bond was issued, the YTM was 9%, but the bond was issued at a discount. So, the bond’s coupon rate must be less than 9%. Now, the YTM is even higher at 9.5%. If the coupon rate is less than 9%, it must be less than 9.5%. So the bond is trading at a discount.

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**Current yield (1) Annual coupon payment divided by bond price.**

Measures return from coupon payments. Shortcomings: Ignores capital gains or losses from bond sale. Ignores income from reinvestment of coupon payments.

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Current yield (2) Consider a 6%, 15-year, semi-annual payment bond with a par value of $1000. Compute the current yield if the yield to maturity is: (a) 7%, (b) 6%, (c) 5% Verify that Price = $ , current yield = 6.61% Price = $1000, current yield = 6% Price = $ , current yield = 5.43% A chance to compute current yield and to see the relationships. (a) N=30, PMT=0.5x0.06x1000=30, FV=1000, I/Y=7/2 = 3.5, CPT PV = , Current yield = (0.06x1000)/ = or 6.61% (b) Trivial. Since ytm=coupon rate, price=par value=1000. therefore current yield = coupon rate. (c) N=30, PMT=0.5x0.06x1000=30, FV=1000, I/Y=5/2 = 2.5, CPT PV = , Current yield = (0.06x1000)/ = or 5.43%

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**Current yield (3) Observe that… If bond is selling at… Coupon rate (%)**

YTM (%) Discount Price = 6 6.61 7 Par Price = 1000 Premium Price = 5.43 5

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**In general, we have the following relationship between coupon rate, current yield, & YTM**

If bond is selling at… Then we know that… Discount (below par) Coupon rate < Current yield < YTM Par Coupon rate = Current yield = YTM Premium (above par) Coupon rate > Current yield > YTM If bond is selling at a premium: Coupon rate > current yield because the coupon rate divides the coupon payments by par value rather than by the bond price. In turn, the current yield exceeds the yield to maturity because the yield to maturity accounts for the built-in capital loss on the bond; the bond bought today for more than par will eventually fall in value to par at maturity. If bond is selling at a discount, Coupon rate < current yield because the coupon rate divides the coupon payments by par value rather than by the bond price and now bond price is smaller than par value. In turn, the current yield is less than the yield to maturity because the yield to maturity accounts for the built-in capital gain on the bond; the bond bought today for less than par will eventually rise in value to par at maturity. If the bond is selling at par, coupon rate = current yield = ytm.

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Quick review A bond has a current yield of 9% and a yield to maturity of 10%. Is the bond selling above or below par value? Is the coupon rate of the bond more or less than 9%. BKM EOC Q16, 17 Using the relationship from the previous slide: bond is selling at a discount, i.e., below par value. Since the bond is at a discount, coupon rate is < current yield, therefore less than 9%.

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**Yield to call (1) Applicable only to callable bonds.**

What’s a callable bond? Bond that may be repurchased by the issuer at a specified price (“call price”) before the maturity date. Call period: The period of time during which the issuer can repurchase the bond. Time until call: The period of time before the issuer can start repurchasing the bond. Motive: if interest rates fall, issuer can repurchase the bonds and issue new bonds at lower coupon rate. This lowers interest payments. Mention time until call here because I need it later to compute YTC.

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Yield to call (2) Yield to call is calculated just like the yield to maturity, except: Time until call replaces time to maturity Call price replaces par value Fixed periodic coupon To compute YTC, we are assuming that the issuer will call back the bond immediately when it can do so, i.e., when the call period starts. YTC is also known as yield to first call because we assume the bond will be called as soon as it is first callable. YTC is the discount rate that solves the pricing equation, i.e., equate bond price to pv of coupons till call and call price. Time until call Yield to call

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Yield to call (3) A 20-year maturity bond with par value $1000 makes semi-annual coupon payments at a coupon rate of 8%. The bond is currently selling for $1,150 and is callable in 10 years at a call price of $1,100. Compute the bond equivalent yield to call and the effective annual yield to call. What is N? What is FV? Verify that Bond equivalent yield to call = 6.64% Effective annual yield to call = 6.75% N = time until call = 10 x 2 = 20 FV = call price = 1100. FV=1100, N=20, PMT=40, PV= CPT I/Y= semi-annual YTC = Bond equivalent YTC = x 2 = or 6.64% Effective annual YTC = ( )2 – 1 = or 6.75%

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**Realized compound yield (1)**

Compound rate of return based on coupon payments, reinvestment income and sale price during the holding period. Realized compound yield depends on: Reinvestment rate: interest rate at which coupon payments are reinvested. Holding period YTM at the end of holding period Reinvestment income refers to the income from reinvesting the coupon payments. In an economy with future interest rate uncertainty, the rates at which interim coupons will be reinvested are not yet known. Therefore, while realized compound yield can be computed after the investment period ends, it cannot be computed in advance without a forecast of future reinvestment rates. This reduces much of the attraction of the realized yield measure. We can compute the realized compound yield over one period or more than one period. When you compute the realized compound yield over more than one period, that is called horizon analysis. With a longer investment horizon, reinvested coupons will be a larger component of your final proceeds.

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**Realized Compound Yield (2)**

Realized compound yield, y Purchase price = what you paid for the bond Final proceeds = future value of coupon payments and reinvestment income + sale price n = length of holding period (could be in years, half-years, quarters etc). Length of holding period Derivation: Purchase price x (1 + y)m = Final proceeds (1 + y)m = (Final proceeds/ Purchase price) 1 + y = (Final proceeds/ Purchase price)1/m y = (Final proceeds/ Purchase price)1/m – 1

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**Realized Compound Yield (3)**

Suppose you buy a 30-year, 7.5%, annual payment coupon bond for $980 and plan to hold it for 20 years. Your forecast is that the bond’s YTM will be 8% when it is sold and that the reinvestment rate on the coupons will be 6%. Compute the annual realized compound yield. (Annual compounding) First, find the FV of coupons and re-investment income. This is just the FV of an annuity with payment equal to coupon payment. PV=0, PMT=0.075x1000=-75, I/Y=6, N=20, FV= Next, find the price at the end of year 20 (end of holding period). N=10, PMT=75, I/Y=8, FV=1000, CPT PV = Total proceeds = = Realized compound yield = ( /980)1/20 – 1 = or 6.9%

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**Realized Compound Yield (4)**

Five years ago, XYZ Inc issued a 5% semi-annual payment coupon bond with a maturity of ten years. You buy the bond now at a price of $ and plan to hold it for three years. You forecast that you can invest the coupon payments at a stated annual rate of 6.25% and that at the end of three years, the yield will be 7.75%. What is the bond equivalent realized compound yield? (i.e., annualize using simple interest) What is the effective annual realized compound yield? (i.e., annualize using compound interest) (Semi-annual compounding) Holding period (in 6-mth periods) = 3 x 2 = 6. Find the future value of coupons and reinvestment income. Reinvest at the semi-annual rate = 6.25/2 = PV=0, N=6, I/Y=3.125, PMT=-0.05 x 0.5 x 1000 = -25, CPT, FV= At the end of 3 years, the time to maturity is 2 years, therefore, bond price: FV=1000,N=2x2=4, PMT=25, I/Y=7.75/2=3.875, CPT, PV= You sell the bond at $ Total proceeds = = Compute semi-annual realized compound yield, y = ( /683.94)1/6 – 1 = = or 8.44% Assuming simple interest, annual realized compound yield = 8.44 x 2 = 16.88% Assuming compound interest, annual realized compound yield = (1.0844)2 – 1 = or 17.59%

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**Yield to Maturity vs. Realized Compound Yield (1)**

Consider a 7% annual payment bond with two years to maturity. The YTM is 8% right now. Compute the realized compound yield if the reinvestment rate is (a) 7%, (b) 8%, (c) 9%. Verify that the realized compound yield is: 7.97% < YTM 8% = YTM 8.03% > YTM Current price: FV=1000, I/Y=8, PMT=70, N=2, CPT PV = Solve for FV of coupon and reinvestment income for each part: PV=0, PMT=-70, N=2, I/Y=7, CPT FV=144.90 PV=0, PMT=-70, N=2, I/Y=8, CPT FV=145.60 PV=0, PMT=-70, N=2, I/Y=9, CPT FV=146.30 At maturity, receive par value of $1000. Compute realized compound yield: Y = ( ( )/ )1/2 – 1 = or 7.97% Y = (( )/ )1/2 – 1 = or or 8% Y = (( )/ )1/2 – 1 = or 8.03%

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**Yield to Maturity vs. Realized Compound Yield (2)**

In general, if you hold a bond to maturity, then: RCY < YTM if reinvestment rate < YTM RCY = YTM if reinvestment rate = YTM RCY > YTM if reinvestment rate > YTM RCY = realized compound yield

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**Holding period return (HPR)**

Rate of return over a single investment (holding) period. Has two components: Price change = ending price – beginning price Coupon payment Just a quick review of the HPR. HPR measures return over a single period. Also, HPR assumes coupon payment is received at the end of the period (so no reinvestment of coupon).

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HPR problem Suppose you buy a bond of General Electric at a price of $990. The bond pays coupons semi-annually, has an annual coupon rate of 6%, a face value of $1,000 and will mature in six months’ time. You intend to hold the bond till it matures. What is the 6-month HPR? Hold this investment for 6-months, until maturity. Semiannual coupon = 0.06 x 1000 x 0.5 = 30 6-month HPR = (1000 – )/990 = 40/990 = or 4.04% for 6-months

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HPR vs. YTM (1) When the YTM stays the same during the holding period, HPR = YTM. If the YTM changes after you bought the bond, then HPR will be different from the initial YTM. If YTM falls after you bought the bond, HPR > initial YTM. If YTM rises after you bought the bond, HPR < initial YTM. * This property is true only if the end of the holding period is NOT the maturity date. If YTM falls, the bond price rises and there will be a capital gain. This will increase the overall return so that HPR > initial YTM. If YTM rises, the bond price falls and there will be a capital loss. This will decrease the overall return so that HPR < initial YTM. The property is true only if the end of the holding period is NOT the maturity date.

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HPR vs. YTM (2) Consider a 30-year bond paying an annual coupon of $80 and selling at $1000. The bond’s initial YTM is 8%. Suppose you buy the bond now and hold it for one year. Compute the 1-year HPR if YTM at the end of the year is (a) 8%, (b) 7%, (c) 9%. Verify that: If year-end YTM is 8%, then HPR = 8% If year-end YTM is 7%, then HPR = 20.28% If year-end YTM is 9%, then HPR = -2.2% Use the problem here to illustrate the points made on the previous slide. Implicit assumption is that you sell after one year. Note that coupon rate = 80/1000 = 0.08 or 8% Year-end price N=29, PMT=80, I/Y=8, FV=1000, CPT PV= Actually, by noting that coupon rate = YTM at year end, you can immediately conclude that the end year price must be 1000. HPR = (1000 – 1000) + 80/1000 = 0.08 or 8% b) N=29, PMT=80, I/Y=7, FV=1000, CPT PV= HPR = (( – 1000) + 80)/1000 = or 20.28% c) N=29, PMT=80, I/Y=9, FV=1000, CPT PV= HPR = (( – 1000) + 80)/1000 = or -2.2% The capital loss outweighs the coupon income so that the overall HPR is negative, i.e., a loss.

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Bond prices over time Suppose the yield to maturity is the same from the time you purchased a bond to the bond’s maturity. If you bought the bond at …. Bond price over time Premium Price falls and approaches par value. Par Price remains at par value till maturity. Discount Price rises and approaches par value. The analysis assumes: (a) YTM stays the same, (b) payments of cash flows are guaranteed. Premium bonds: As the premium bond approaches maturity, the above-market coupon payments become fewer. The PV of coupons become a less important component of the bond price, while the PV of face value becomes a more important component of the bond price. As a result, bond price falls as maturity nears. Another way to say this is that the above-market coupons are responsible for producing the premium in the bond price. As maturity nears, there are less of these above-market coupons, so the premium becomes smaller, i.e., the bond price falls and approaches par value. Discount bonds: As the discount bond approaches maturity, the below-market coupon payments become fewer. The PV of coupons become a less important component of the bond price, while the PV of face value becomes a more important component of the bond price. As a result, bond price rises as maturity nears. Another way to say this is that the below-market coupons are responsible for producing the discount in the bond price. As maturity nears, there are less of these below-market coupons, so the discount becomes smaller, i.e., the bond price rises and approaches par value. Example: Suppose YTM=6% always. Consider 8% annual payment coupon bond with three years to maturity. Bond price = 80/ /(1.06)2 + 80/(1.06) /(1.06)3 = PV of 3 coupons = , PV of par value = Wt of PV of coupons = / = 0.20 or 20% Wt of PV of par = / =0.80 or 80% After one year, time to maturity is 2 years. Bond price = 80/ /(1.06) /(1.06)2 = PV of 2 coupons = , PV of par value = Wt of PV of coupons = / = 0.14 or 14% Wt of PV of par = / =0.86 or 86% ================================================================ Consider 4% annual payment coupon bond with three years to maturity. Bond price = 40/ /(1.06)2 + 40/(1.06) /(1.06)3 = PV of 3 coupons = , PV of par value = Wt of PV of coupons = / = 0.11 or 11% Wt of PV of par = / = 0.89 or 89% Bond price = 40/ /(1.06) /(1.06)2 = PV of 2 coupons =73.34, PV of par value = 890 Wt of PV of coupons = 73.34/ = 0.08 or 8% Wt of PV of par = / =0.92 or 92%

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**Figure 9.6 Premium and Discount Bonds over Time**

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Quick review (1) Consider a bond with a 10% coupon rate and with YTM of 8%. If the bond’s YTM remains constant, then in one year, will the bond price be Higher Lower Unchanged None of the above BKM EOC Q4 Based on the preceding discussion, answer is (b) lower.

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**Quick review (2) Which of the following statements is correct?**

If market rates do not change, the price of a bond selling at a premium increases over time. If interest rates are greater than zero, it is possible for a zero-coupon bond to sell at a premium (i.e. for more than par value). If a bond’s yield to maturity is greater than its coupon rate, the bond will sell at a discount. If market rates do not change, the price of a bond selling at a discount decreases over time. All of the above statements are false. Answer: C.

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**Default risk What is default? Default risk**

Failure of the bond issuer to make the promised coupon &/or par value payments to the bond holder. Default risk To the bond holder, default risk is the uncertainty in cash flows arising from the possibility that the issuer can fail to make promised payments. With the exception of the U.S. government, all issuers have default risk. Some issuers have greater default risk than others.

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**Measuring default risk**

Credit rating agencies measure default risk of: Large corporate bond issues Municipal bond issues International sovereign bond issues Well-known credit rating agencies: Moody’s Standard & Poor’s (S&P for short) Duff & Phelps Fitch Each agency assigns a letter grade to reflect level of default risk. On the next slide, we look at the letter grades assigned by Moody’s and S&P, the two most prominent rating agencies.

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**S&P, Moody’s credit rating scheme**

The top rating is AAA or Aaa. Moody’s modifies each rating class with a 1,2, or 3 suffix (e.g., Aaa1, Aaa2, Aaa3) to provide a finer gradation of ratings. The other agencies use a + or – modification. Those rated BBB or above (S&P, Duff & Phelps, Fitch) or Baa and above (Moody’s) are considered investment grade bonds, while lower-rated bonds are classified as speculative grade or junk bonds. Certain regulated institutional investors such as insurance companies have not always been allowed to invest in speculative grade bonds.

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**Investment vs. speculative grade**

Bonds fall into one of the two broad groups: Investment grade bond: Rated BBB and above by S&P, or Rated Baa and above by Moody’s Speculative grade or junk bond: Rated BB or lower by S&P, or Rated Ba or lower by Moody’s or Unrated bond We only define these two groups using S&P and Moody’s ratings.

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**Determinants of credit rating**

To assign rating, credit rating agencies look at level and trend of issuer’s financial ratios. Key ratios: Coverage: times-interest-earned Leverage: debt-to-equity, debt-to-assets Liquidity: current, quick Profitability: ROA Cash flow-to-debt Give some examples of the ratios in each group. Relate these back to FI3300.

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**Protection Against Default**

Bond indenture specifies restrictions on issuer to protect bondholder against default. Common restrictions: Sinking fund Subordination of further debt Dividend restrictions Collateral Bond indenture: the document defining the contract between the bond issuer and the bondholder. The restrictions are also called protective covenants.

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**Yield to Maturity and Default Risk (1)**

Promised/stated YTM: YTM you get if issuer makes all promised payments. Maximum possible YTM. Expected YTM: YTM you get if you consider the possibility of default. Consider the following example. Because companies can default on payments, we distinguish between promised YTM and the expected YTM.

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**Yield to Maturity and Default Risk (2)**

A firm issued a 9% semi-annual payment coupon bond 20 years ago. The bond now has 10 years left until its maturity date, but the firm is having financial difficulties. Investors believe that the firm will be able to make good on the remaining interest payments but that at the maturity date, the firm will be forced into bankruptcy, and bondholders will receive only 70% of par value. The bond is selling at $750. Calculate the promised (bond equivalent) YTM and the expected (bond equivalent) YTM. Verify that the promised YTM is 13.7% and the expected YTM is 11.6%. Key point is to realize that , for stated YTM, FV is $1000, for expected YTM, FV is 0.7 x 1000 = 700. Stated YTM: N=20, PMT=45, FV=1000, PV=-750. CPT, I/Y = % (semi-annual rate). The annual YTM (bond equivalent yield) = x 2 = or 13.7% p.a. Expected YTM: N=20, PMT=45, FV=700, PV=-750. CPT, I/Y = % (semi-annual rate). The annual YTM (bond equivalent yield) = x 2 = or 11.6% p.a.

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Summary Compute the price of (a) fixed-coupon bond, b) zero-coupon bond. Bond price is inversely related to interest rates. Different measures of bond returns: yield to maturity, current yield, yield to call, realized compound yield, holding period return. Relationships between different measures of bond returns. Evolution of bond prices over time if YTM remains the same. Credit/ default risk as a source of risk in bond investing. Determinants of bond safety and rating.

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Practice 6 (2) Chapter10: 7,8,9,10,14,18,21,22,26,30.

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