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6.1 Application of the Time Value of Money Tool: Bond Pricing

6.1 Application of the Time Value of Money Tool: Bond Pricing. Bonds - Long-term debt instruments Provide periodic interest income – annuity series Return of the principal amount at maturity – future lump sum

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6.1 Application of the Time Value of Money Tool: Bond Pricing

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  1. 6.1 Application of the Time Value of Money Tool: Bond Pricing • Bonds - Long-term debt instruments • Provide periodic interest income – annuity series • Return of the principal amount at maturity – future lump sum • Prices can be calculated by using present value techniques i.e. discounting of future cash flows. • Combination of present value of an annuity and of a lump sum

  2. Table 6.1 Bond InformationonJuly 15, 2008

  3. 6.1 (A) Key Components of a Bond (continued): Figure 6.1 Merrill Lynch corporate bond • Par value : Typically $1000 • Coupon rate: Annual rate of interest paid. • Coupon: Regular interest payment received by holder per year. • Maturity date: Expiration date of bond when par value is paid back. • Yield to maturity: Expected rate of return based on price of bond

  4. 6.1 (A) Key Components of a Bond (continued) Example 1 Let’s say you see the following price quote for a corporate bond: Issue Price Coupon(%) Maturity YTM% Current Yld. Rating Hertz Corp. 91.506.35 15-Jun-2010 15.438 6.94 B • Price = 91.5% of $1000  $915; • Annual coupon = 6.35% *1000 $63.50 • Maturity date = June 15, 2010; • If bought and held to maturity  Yield = 15.438% • Current Yield = $ Coupon/Price = $63.5/$915  6.94%

  5. 6.1 (B) Pricing a Bond in Steps [Figure 6.2 ] Since bonds involve a combination of an annuity (coupons)and a lump sum (par value) its price is best calculated by using the following steps:

  6. 6.1 (B) Pricing a Bond in Steps (continued):Example 2 Year 0 1 2 3 18 19 20 $80 $80 $80 … $80 $80 $80 $1,000 Calculate the price of an AA-rated, 20-year, 8% coupon (paid annually) corporate bond (Par value = $1,000) which is expected to earn a yield to maturity of 10%. • Annual coupon = Coupon rate * Par value = .08 * $1,000 = $80 [+/-] [PMT] • YTM = r = 10% [I/Y] • Maturity = n = 20 [N] • Principal at maturity (par value) = FV = 1000 [+/-] [FV] • Price of bond = PV of coupons + PV of par value = 681.08 + 148.64 = 829.72 • CPT [PV]

  7. Example 2: Calculating the price of a corporate bond (continued) Present value of coupons = = = $80 x 8.51359 = $681.09 Present Value of Par Value = Present Value of Par Value = Present Value of Par Value = $1,000 x 0.14864 = $148.64 Price of bond = $681.09 + $148.64 = $829.73

  8. Example 2: Calculating the price of a corporate bond (continued) Method 2. Using a financial calculator Input: N I/Y PV PMT FV Key: 20 10 ? -80 -1000 Output 829.73

  9. 6.2 Semiannual Bonds and Zero-Coupon Bonds • Most corporate and government bonds pay coupons on a semiannual basis. • Some companies issue zero-coupon bonds by selling them at a deep discount. • For computing price of these bonds, the values of the inputs have to be adjusted according to the frequency of the coupons (or absence thereof). • For example, for semi-annual bonds, the annual coupon is divided by 2, the number of years is multiplied by 2, and the YTM is divided by 2. • The price of the bond can then be calculated by using the TVM equation, a financial calculator, or a spreadsheet.

  10. 6.2 Semiannual Bond ExampleFigure 6.4 Coca-Cola Semiannual Bond .

  11. 6.2 Coca-Cola Semiannual Bonds at original issue (continued) Figure 6.5 Future cash flow of the Coca-Cola bond Using TVM Equation: 8.8% YTM at original issue • Using Financial Calculator: • 30 x 2 = 60 [N]  8.8 ÷ 2 = 4.4 [I/Y] at original issue • 42.50 [+/-] [PMT]  1000 [+/-] [FV] • CPT [PV] 968.48

  12. 6.2 Semiannual Bonds and Zero- Coupon Bonds (continued)

  13. 6.2 (A) Pricing Bonds after Original Issue The price of a bond is. a function of the remaining cash flows (i.e. coupons and par value) that would be paid on it until expiration As of August, 2008 the 8.5%, 2022 Coca-Cola bond has only 27 coupons left to be paid on it until it matures on Feb. 1, 2022 Figure 6.6 Remaining cash flow of the Coca-Cola bond

  14. EXAMPLE 3 - 6.2 (A) Pricing Bonds After Original Issue Four years ago, the XYZ Corporation issued an 8% coupon (paid semi-annually), 20-year, AA-rated bond at its par value of $1000. Currently, the yield to maturity on these bonds is 10%. Calculate the price of the bond today. Remaining number of semi-annual coupons = (20 - 4) * 2 = 32 coupons [N] Semi-annual coupon = (.08*1000)/2 = $40 [+/-] [PMT] Par value = $1000 = [+/-] [FV] Annual YTM = 10%  YTM/2  5% = [I/Y]  CPT  [PV] 841.97 Input: N I/Y PV PMT FV Key: 32 5 ? -40 -1000 Output 841.97

  15. 6.2 (A) Pricing Bonds after Original Issue (continued)

  16. 6.2 (A) Pricing Bonds after Original Issue (continued) Method 2: Using a financial calculator Input: N I/Y PV PMT FV Key: 32 5 ? -40 -1000 Output 841.97

  17. 6.2 (B) Zero-Coupon Bonds • Known as “pure” discount bonds and sold at a discount from face value • Do not pay any interest over the life of the bond. • At maturity, the investor receives the par value, usually $1000. • Price of a zero-coupon bond is calculated by merely discounting its par value at the prevailing discount rate or yield to maturity.

  18. 6.2 (C) Amortization of a Three-Year Zero-Coupon Bond w/ 8% Yield [Table 6.2] • The discount on a zero-coupon bond is amortized over its life. • Interest earned is calculated for each 6-month period. • for example .04*790.31=$31.62 • Interest is added to price to compute the ending price. • Zero-coupon bond investors have to pay tax on annual price appreciation even though no cash is received.

  19. Example 4: Price of and taxes due on a zero-coupon bond John wants to buy a 20-year, AAA-rated, $1000 par value, zero-coupon bond being sold by Diversified Industries Inc. The yield to maturity on similar bonds is estimated to be 9%. 4A - How much would he have to pay for it? 4B - How much will he be taxed on the investment after 1 year, if his marginal tax rate is 30%?

  20. Example 4A – Answer (continued) Method 1: Using TVM equation Bond Price = Par Value * [1/(1+r)n] Bond Price = $1000*(1/(1.045)40   Bond Price = $1000 * .1719287 = $171.93 Method 2: Using a financial calculator Input: N I/Y PV PMT FV Key: 404.5 ? 0-1000 Output 171.93

  21. Example 4B – Answer (continued) Calculate the price of the bond at the end of 1 year. Input: N I/Y PV PMT FV Key: 384.5 ? 0 -1000 Output 187.75 Taxable income = $187.75 - $171.93 = $15.82 Taxes due = Tax rate * Taxable income = 0.30*$15.82 = $4.75

  22. Example 4B (Answer) (continued) Alternately, we can calculate the semi-annual interest earned, for each of the two semi-annual periods during the year.  • $171.93 * .045 = $7.736 $171.93+7.736 = $179.667  Price after 6 months • $179.667 * .045 = $8.084 $179.667+8.084 = $187.75  Price at end of year • Total interest income for 1 year = $7.736 + $8.084 = $15.82  Tax due = 0.30 * $15.82 = $4.75

  23. 6.3 Yields and Coupon Rates • A bond’s coupon rate differs from its yield to maturity (YTM). • Coupon rate -- set by the company at the time of issue and is fixed (except for newer innovations which have variable coupon rates) • YTM is dependent on market, economic, and company-specific factors and is therefore variable.

  24. 6.3 (A) The First Interest Rate: Yield to Maturity • Expected rate of return on a bond if held to maturity. • The price that willing buyers and sellers settle at determines a bond’s YTM at any given point. • Changes in economic conditions and risk factors will cause bond prices and their corresponding YTMs to change. • YTM can be calculated by entering the coupon amount (PMT), price (PV), remaining number of coupons (n), and par value (FV) into the TVM equation, financial calculator, or spreadsheet.

  25. 6.3 (B) The “Other” Interest Rate: Coupon Rate • The coupon rate on a bond is set by the issuing company at the time of issue • It represents the annual rate of interest that the firm is committed to pay over the life of the bond. • If the rate is set at 7%, the firm is committing to pay .07*$1000 = $70 per year on each bond, • It is paid either in a single check or two checks of $35 paid six months apart.

  26. 6.3 (C) Relationship of Yield to Maturity and Coupon Rate • An issuing firm gets the bond rated by a rating agency such as Standard & Poor’s or Moody’s. • Then, based on the rating and planned maturity of the bond, it sets the coupon rate to equal the expected yield as indicated in the Yield Book (available in the capital markets at that time) and sells the bond at par value ($1000). • Once issued, if investors expect a higher yield on the bond, its price will go down and the bond will sell below par or as a discount bond and vice-versa. • Thus, a bond’s YTM can be equal to (par bond), higher than (discount bond) or lower than (premium bond) its coupon rate.

  27. 6.3 (C) Relationship of Yield to Maturity and Coupon Rate (continued) Table 6.3 Premium Bonds, Discount Bonds, and Par Value Bonds

  28. 6.3 (C) Relationship of Yield to Maturity and Coupon Rate (continued) Figure 6.8 Bond prices and interest rates move in opposite directions.

  29. Example 5: Computing YTM Last year, The ABC Corporation had issued 8%coupon (semi-annual), 20-year, AA-rated bonds (Par value = $1000) to finance its business growth. 5A - If investors are currently offering $1200 on each of these bonds, what is their expected yield to maturity on the investment? 5B - If you are willing to pay no more than $980 for this bond, what is your expected YTM? • Remaining number of coupons = 19 * 2 = 38 • Semi-annual coupon amount =( 0.08 * $1000 ) ÷ 2 = $40

  30. Example 5A - Answer PV = $1200 (current market price) TI-BAII+: 38 [N]  1200 [PV]  40 [+/-] [PMT]  1000 [+/-] [FV]  CPT  [I/Y] 3.097 Thus, annual YTM = 3.097 x 2 = 6.19 Note: This is a premium bond, so it’s YTM (6.19%) < Coupon rate (8%)

  31. Example 5B - Answer (continued) PV = $980 TI-BAII+: 38 [N]  980 [PV]  40 [+/-] [PMT]  1000 [+/-] [FV]  CPT  [I/Y] 4.1048 Thus, the annual YTM = 4.1048 x 2 = 8.2 Note: This would be a discount bond, so it’s YTM (8.2%) > Coupon rate (8%)

  32. 6.4 Bond Ratings • Ratings are produced by Moody’s, Standard and Poor’s, and Fitch • Range from AAA (top-rated) to C (lowest-rated) or D (default). • Help investors gauge likelihood of default by issuer. • Assist issuing companies establish a yield on newly-issued bonds. • Junk bonds: is the label given to bonds that are rated below BBB. These bonds are considered to be speculative in nature and carry higher yields than those rated BBB or above (investment grade).  • Fallen angels: is the label given to bonds that have had their ratings lowered from investment to speculative grade. • Credibility Issue: 2008 Financial Crisis revealed serious corruption and incompetency of the rating agencies, which led DoJ to sue them recently [2013.02]

  33. 6.5 Some Bond History and More Bond Features (continued) Example 6: Calculating Yield to Call Two years ago, the Mid-Atlantic Corporation issued a 10% coupon (paid semi-annually), 20-year maturity, bond with a 5-year deferred call feature and a call penalty of one coupon payment in addition to the par value ($1000) if exercised. If the current price on these bonds is $1080, what is its yield to call?

  34. Example 6 Answer – Yield to Call • Remaining number of coupons until first call date = 5 (5-year deferred call feature) – 2 (2 years passed) = 3 years = 3 x 2 = 6 [N] • Semi-annual coupon = $50[+/-] [PMT] • Call price = Par value + penalty (one coupon in this case) = 1000 + 50 = $1050[+/-] [FV] • Current market price = $1080[PV] • CPT  [I/Y] 4.2131 Therefore, the annual Yield to Call (YTC) = 4.2131 x 2 = 8.43%

  35. 6.6 U.S. Government Bonds • Include bills, notes, and bonds sold by the Department of the Treasury • State bonds, issued by state governments • Municipal bonds issued by county, city, or local government agencies. • Treasury bills, are zero-coupon, pure discount securities with maturities ranging from 1-, 3-, and 6-months up to 1 year. • Treasury notes have between two to 10 year maturities. • Treasury bonds have greater than 10-year maturities, when first issued.

  36. 6.6 U.S. Government Bonds (continued) Table 6.6 Government Notes and Bonds, Prices as of April 8, 2008

  37. 6.6 (A) Pricing a U.S. Government Bond • Similar to the method used for pricing corporate bonds and can be done by using TVM equations, a financial calculator or a spreadsheet program. • For example, let’s assume you are pricing a 7-year, 6% coupon (semi-annual) $100,000 face value Treasury note, using an expected yield of 8%: Figure 6.11 U.S. Government Treasury note cash flows. TI-BAII+: 7 x 2 = 14 [N]  8 ÷ 2 = 4 [I/Y]  10000 x 0.06 ÷ 2 = 3000[+/-] [PMT]  10000[+/-] [FV]  CPT  [PV] 89436.88

  38. 6.6 (B) Pricing a Treasury bill Price of T-bill = Face value * [1 - BDY * DTM ÷ 360] DTM = Days to maturity Bank discount yield (BDY)is a special discount rate used in conjunction with treasury bills under a 360 day-per-year convention (commonly assumed by bankers). Bond equivalent yield (BEY)is the APR equivalent of the bank discount yield calculated by adjusting it as follows: BEY= 365 * BDY________ 360 - (DTM * BDY)

  39. 6.6 (B) Pricing a Treasury bill (continued) Table 6.7 Selected Historical Treasury Bill Bank Discount Rates

  40. Example 7: Calculating the price and BEY of a Treasury bill Calculate the price and BEY of a treasury bill which matures in 105 days, has a face value of $10,000 and is currently being quoted at a bank discount yield of 2.62%. Price of T-bill = Face value * [1 - BDY * DTM ÷ 360] = 10000 x [ 1 – (0.0262) x 105 ÷ 360 ] = 10000 x 0.9923583 = $ 9,923.58 BEY= 365 * BDY________ 360 - (DTM * BDY) = [ 365 x 0.0262 ] ÷ [ 360 – ( 105 x 0.0262 ) ] = 9.563 ÷ 357.249 = 0.0268 = 2.68%

  41. Additional Problems 1:Pricing a semi-annual bond Last year, The Harvest Time Corporation sold $40,000,000 worth of 7.5% coupon, 15-year maturity, $1000 par value, AA-rated; non-callable bonds to finance its business expansion. Currently, investors are demanding a yield of 8.5% on similar bonds. If you own one of these bonds and want to sell it, how much money can you expect to receive on it?

  42. Additional Problems 2:Yield-to-Maturity Joe Carter is looking to invest in a four-year bond that pays semi-annual coupons at a coupon rate of 5.6 percent and has a par value of $1,000. If these bonds have a market price of $1,035, what yield to maturity is being implied in the pricing?

  43. Additional Problems 3:Zero Coupon Bond Krypton Inc. wants to raise $3 million by issuing 10-yearzero couponbonds with a face value of $1,000. Their investment banker informs them that investors would use a 9.25% percent discount rate on such bonds. At what price would these bonds sell in the market place assuming semi-annual compounding? How many bonds would the firm have to issue to raise $3 million?

  44. Additional Problems 4:Tax on zero-coupon bond income Let’s say that you buy 100 of the 7411 bonds that were issued by Krypton Inc. as described in Problem 3 above for $404.85. At the end of the year, how much money will the bond be worth, and how much tax will you be assessed assuming that you have a marginal tax rate of 35%?

  45. Additional Problems with AnswersProblem 5 Price, and BEY, on a Treasury bill: Calculate the price, and BEY of a treasury bill which matures in 181 days, has a face value of $10,000, and is currently being quoted at a bank discount yield of 2.32%.

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