54 Fixed-Income Bond Valuation- Prices and Yields

Bond Yield to Maturity

• A bond with a 12% annual coupon, 10 years to maturity, and selling at 88% of par has a yield to maturity over 14%.
• Calculation: PMT = 12; N = 10; PV = -88; FV = 100; CPT I = 14.3.
• An annual-pay, 4% coupon, 10-year bond has a yield to maturity of 5.2%.
• If the price of the bond is unchanged two years later, its yield to maturity is greater than 5.2%.
• This bond is priced at a discount because its 4% coupon is less than its 5.2% yield to maturity.
• As the bond gets closer to maturity, the discount will amortize toward par value, increasing its price if its yield remains unchanged.
• Therefore, if its price remains unchanged, its yield would have to increase.
• Initial price calculation: N = 10; I/Y = 5.2; PMT = 40; FV = 1,000; CPT PV = -908.23.
• Yield after two years with price unchanged: N = 8; PMT = 40; FV = 1,000; PV = -908.23; CPT I/Y = 5.446%.

Bond Valuation and Matrix Pricing

• An analyst wants to estimate the yield to maturity on a non-traded 4-year, annual pay bond rated A.
• Actively traded bonds with the same rating yield 3.2% for 3-year bonds and 5.0% for 6-year bonds.
• Using matrix pricing, the estimated YTM for the non-traded bond is closest to 3.8%.
• Interpolaion: 3.2% + [(4 – 3) / (6 – 3)] × (5.0% – 3.2%) = 3.8%.
• For option-free bonds, the value of a long-term bond is more sensitive to interest rate changes versus a short-term bond, all else being equal.
• Long-term, low-coupon bonds are more sensitive than short-term and high-coupon bonds.
• Prices are more sensitive to rate decreases than to rate increases because duration rises as yields fall.
• A bond with three years to maturity pays an annual coupon of 6%.
• Assuming a yield to maturity of 7%, the price as a percentage of par is closest to 97.38.
• Calculation: Present Value = 6/1.07 + 6/1.07^2 + 106/1.07^3 = 97.38 or I/Y = 7; FV = 100; N = 3; PMT = 6; CPT PV = $97.38.
• If a bond's quoted price is 105.22 and the accrued interest is $3.54, and the par value is $100, then the bond's clean price is $105.22.
• The clean price is the bond price without accrued interest, so it equals the quoted price.
• For a 20-year, $1,000 face value, 10% annual coupon bond, an investor requiring a 9% rate of return would value the bond at $1,091.
• Calculation: N = 20; I/Y = 9; PMT = 100 (0.10 × 1,000); FV = 1,000; CPT PV = 1,091.
• A year ago, a company issued a bond with a face value of $1,000 and an 8% coupon.
• Now the prevailing market yield is 10%.
• The bond is now traded at a market price of less than $1,000.
• A bond's price/value has an inverse relationship with interest rates.
• Since rates are increasing from the original 8% to 10%, the bond will sell at a discount, enabling investors to achieve the market yield.
• Assume a city issues a $5 million bond to build a hockey rink with an 8% semiannual interest and matures in 10 years.
• Current interest rates are 6%, and the present value of the bond is $5,743,874.

Bond Pricing

• Since current interest rates are lower than the coupon rate, the bond will be issued at a premium.
• Calculation: FV = $5,000,000; N = 20; I/Y = 3; PMT = (0.04)($5,000,000) = $200,000; Compute PV = $-5,743,874.
• For an option-free bond, as the yield to maturity increases, the bond price decreases at a decreasing rate.
• The relationship between price and yield for an option-free bond is inverse and convex toward the origin.
• Four years ago, Gamma Corporation issued a 20-year bond with a 6% annualized coupon to expand operations.
• The coupon is paid semiannually, and current yield is 5.8%.
• The price of the bond per $100 of principal is closest to $102.
• The bond now has 16 years remaining to maturity, so: PMT = 6% / 2 × $100 = $3.
• N = 16 × 2 = 32, I/Y = 5.8% / 2 = 2.9%, FV = $100, CPT PV = $102.07.
• A coupon bond that pays interest annually has a par value of $1,000, matures in 5 years, and has a yield to maturity of 10%.
• The bond is worth $1,075.82 today, given a 12% coupon rate.
• FV = 1,000, N = 5, I = 10, PMT = 120, CPT PV = 1,075.82.
• A 6-year $1,000 par bond is priced at $1,011 with a 7.5% coupon rate paid semiannually.
• Six-year bonds with comparable credit quality have a YTM of 6%.
• The investor should purchase this bond, because the bond is undervalued by $64.
• FV = 1,000, PMT = 37.5, N = 12, I/Y = 3%, CPT PV = -1,074.66, and 1,074.66 - 1,011 = 64.
• Matrix pricing is used primarily to price bonds that have low liquidity.
• Matrix pricing uses yields on similar issues that do trade to estimate yields on illiquid bonds.
• An investor buys a 25-year, 10% annual pay bond for $900 and will sell in 5 years, estimating a yield of 9%.
• The price for which the investor expects to sell this bond is closest to $1,091.
• N = 20, PMT = 100, FV = 1000, I/Y = 9, CPT PV = -1,091.29.
• A dealer is most likely to calculate accrued interest based on 30-day months and 360-day years to determine the full price of a corporate bond.
• Accrued interest for corporate bonds is typically calculated using 30/360, while, for government bonds, actual/actual is used.

Bond Value and Yield

• A 10-year, semiannual, 8% coupon bond with a $1,000 face value has a value of $875.38 if similar bonds yield 10%.
• N = 10 × 2 = 20; PMT = $80 / 2 = $40; I/Y = 10 / 2 = 5%; FV = 1,000; Compute the bond's value PV = -875.38.
• Given a required yield to maturity of 6%, a semiannual pay coupon bond with an 8% coupon and 15 years until maturity has an intrinsic value of $1,196.
• Use semiannual payments to solve: I = 6/2 = 3%; PMT = 80/2 = $40; N = 15 × 2 = 30; FV = $1,000; CPT PV = $1,196.
• Interest rates have fallen over the seven years since a $1,000 par, 10-year bond was issued with a 7% coupon.
• The present value of this bond, if the required rate of return is currently 4.5%, assuming annual payments, is $1,068.72.
• A bond offers a 12% coupon paid semiannually and has 15 years left to maturity.
• Assuming a par value of $1,000 and a yield to maturity of 16%, the price of the bond is closest to $777.
• Semiannual coupon payment = $1,000 × (0.12 / 2) = $60; FV = 1,000; PMT = 60; N = 15 × 2 = 30; I/Y = 16 / 2 = 8; CPT PV = -774.84
• The probable change in price of a 30-year semiannual 6.5% coupon, $1000 par value bond yielding 8% if the yield decreases to 7% is $107.31.
• Price at 8%: N = 60, FV = $1,000, I = 4%, PMT = $32.50, CPT PV = $830.32. Price at 7%: N = 60, FV = $1,000, I = 3.5%, PMT = $32.50, CPT PV = $937.64. Change in price is $937.64 - $830.32 = $107.31.
• A zero-coupon bond matures three years from today, has a par value of $1,000, and a yield to maturity of 8.5% (assuming semi-annual compounding).
• Current value of this issue: $779.01 (Bond Value = $1,000 / 1.0425^6 = $779.01), N = 6; I/Y = 4.25; PMT = 0; FV = 1,000; CPT PV = 779.01.
• A 10-year, 5% bond is issued at a price to yield 5.2%.
• If the yield on this bond decreases by 100 basis points three months after issuance, the price is below par at issuance but above par three months later.
• A bond issued at a yield higher than its coupon will be priced below par, or at a discount.
• Three months later, the yield has declined to 4.2%, and the bond trades at a premium to par, reflecting that the coupon is now higher than the yield.
• Ron Logan, CFA, purchased $50 million in 6.0% coupon Southwest Manufacturing bonds at par three years ago.
• Today, the bonds are priced to yield 6.85%.
• Southwest bonds are trading at a discount, and the yield to maturity has increased since purchase.
• The yield on the bonds has increased, so the value of the bonds has fallen below par.

Zero Coupon Bonds

• Parsons Inc. is issuing an annual-pay bond that will pay no coupon for the first five years and then pay a 10% coupon for the remaining five years to maturity.
• The 10% coupon interest for the first five years will be paid (without additional interest) at maturity.
• If the annual YTM on this bond is 10%, then the price of the bond per $1,000 of face value is closest to $814.
• A coupon bond can be viewed as a collection of zero-coupon bonds, making it the most accurate statement regarding zero-coupon bonds and spot interest rates.
• Spot rates are defined as interest rates used to discount a single cash flow to be received in the future.
• Considere a 10-year, 6% coupon, $1,000 par value bond paying annual coupons with a 10% yield to maturity.
• The change in the bond price from a 400 basis point increase in yield is closest to $170.
• New bond price will be $171.51 lower.
• An investor buys a 20-year, 10% semi-annual bond for $900 and wants to sell the bond in 6 years.
• If estimated yields will be 10%, the estimate of the future price is $1,000, since at 10% coupon rate/yield the bond will sell at par value.
• A 7% callable semiannual-pay bond with a $1,000 face value has 20 years to maturity.
• With a yield to maturity of 8.25% and the yield to call is 9.25%, the value of the bond is closest to $879: N = 20 × 2 = 40; I/Y = 8.25/2 = 4.125; PMT = 70/2 = 35; FV = 1,000; Compute PV = 878.56.
• The price of a bond is equal to the present value of future cash flows discounted at the yield to maturity.
• A 20-year, 10% semi-annual bond for $900 will have an estimated future price of 1,079 (N = (20 – 6)(2) = 28; PMT = (1,000 × 0.10) / 2 = 50; I/Y = 9/2 = 4.5; FV = 1,000; CPT PV = 1,079)
• A bond is issued at a discount if the current yield is lower than the yield to maturity.
• Accrued interest equals interest earned from the previous coupon to the sale date.

Bond Investments

• Consider two 7% annual-pay, option-free bonds: Bond R (4 years to maturity, priced to yield 6%) and Bond S (7 years to maturity, priced to yield 6%), with a $1,000 par value.
• With a 50 basis point parallel upward shift in interest rates, the value of the two-bond portfolio is $2,044.
• New value for Bond R is $1,017: N = 4; PMT = 70; FV = 1,000; I/Y = 6.50%; CPT PV = 1,017.
• New value for Bond S at $1,027: N = 7; PMT = 70; FV = 1,000; I/Y = 6.50%; CPT PV = 1,027.
• The new value is $2,044 (1,017 + 1,027).
• A $1,000-face value, 12-year, 8%, semiannual coupon bond with a YTM is 10.45% will change in value by $23.06 for a decrease in yield of 38 basis points.
• With YTM = 10.45% (I/Y = 5.225%), PMT = 40, N = 24, FV = 1,000, PV = $834.61, With YTM = 10.07% (I/Y = 5.035), PV = $857.67.
• A bond that pays an annual coupon of 5% and has three years left until maturity will have its price change 0.84% as a percentage of par, given that the term structure of interest rates is flat at 6%.
• An investor purchased a 6-year annual interest coupon bond one year ago, with a 10% coupon rate and $1,000 par value.
• The yield to maturity was 8% at purchase, so this bond was worth $1,092.46 (N = 6, PMT = (0.10)(1,000) = 100, i = 8, FV = 1,000, CPT PV = 1,092.46)
• A 5-year bond with a 10% coupon has a present yield to maturity of 8%.
• If rates remain constant one year out, then the price of the bond will lower.
• Assume that a bond sells at a value of more than its face value (premium), such that as time passes, the bond value will converge upon the face value.

Municipal Bonds

• Assume that a city issues a $5 million bond to build a new arena, and the bond pays 8% semiannual interest and matures in 10 years.
• Current interest rates are 9%; the present value of the bond is $4,674,802, and 7-year value will be $4,871,053.
• Since current interest rate is above the coupon rate the bond will be priced at a discount.
• Austin Traynor is considering buying a $1,000 face value, semi-annual coupon bond with a quoted price of 104.75 and accrued interest since the last coupon of $33.50.
• Seller to receive $1,081.00 at the settlement date, ignoring transaction costs.
• The full price is equal to the flat (clean) price plus interest accrued ($1,047.50 + $33.50).
• Bond with a yield to maturity of 7% with a periodicity of 4 that has a face value of $100,000 and matures in 13 years where each coupon payment is $1,800.
• The current price is closest to $101,698 (N = 13 × 4 = 52; FV = 100,000; PMT = 1,800; I/Y = 7 / 4 = 1.75; CPT PV = 101,698).
• Yields have not changed since bond was issued at discount, the price least likely to remain constant; it will follow the constant yield price trajectory and approach par value as maturity approaches.
• Georgia Corporation has $1,000 par value bonds with 10 years remaining to maturity.
• With a coupon of 7.5% which is paid semi-annually.
• If current yield to maturity on similar bonds is 8.2%, what is the current value of the bonds: Bond Payment = ($1,000)(0.075 / 2) = $37.50, FV = $1,000; PMT = $37.50; N = 10 × 2 = 20; I/Y = 8.2 / 2 = 4.1%; CPT PV = -952.85.
• A new-issue, 15-year, $1,000 face value 6.75% semi-annual coupon bond is priced at $1,075, so it is a premium bond where the required market yield is less than 6.75%.
• The market yield is lower than the coupon on premium bonds.

Bond Pricing Formulas

• 10 year zero bond formula PV = 1000/(1 + .048)^20; semi annual; yield 9.6%; Value = 390; Rate divided by 2
• A 10%, 10-year bond sold to yield 8% will decrease in price after following constant yield price trajectory
• Price at Issuance: N = 10; FV = 1,000; PMT = 100; I = 8; CPT PV = 1,134
• Price after on year: N = 9; FV = 1,000; PMT = 100; I = 8; CPT PV = 1,125
• An investor purchased a $1,000 5% coupon bond with quarterly coupon payments that matures in 12 years and has a yield to maturity of 7.0%, the price the investor pays is closest to.
• N = 12 × 4 = 48, FV = 1,000, PMT = 50/4 = 12.5, I/Y = 7.0/4 = 1.75; CPT PV = -838.53
• A $1,000 par, semiannual-pay bond is trading for 89.14, has a coupon rate of 8.75%, and accrued interest of $43.72.
• The flat price of the bond is $891.40.