 # Copyright 2015 by Diane S. Docking 1 Bond Valuation.

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Copyright 2015 by Diane S. Docking 1 Bond Valuation

Copyright 2015 by Diane S. Docking Learning Objectives To understand the cash flow characteristics of a bond. how the price of a bond is determined. why the price of a bond changes. that the price/yield curve of an option-free bond is convex. that the two characteristics of a bond that affect its price volatility are its coupon and its maturity. why the yield to maturity is used as a measure of a bond’s return. the importance of the reinvestment rate in realizing the yield to maturity. 2

Copyright 2015 by Diane S. Docking 3 Bond Valuation Bonds are simply valued as the PV of future cash flows As market interest rates ↑ (↓), bond prices ↓ (↑)

Bond Valuation The present value of a bond (V b ) can be written as: Par = the par or face value of the bond, usually \$1,000 INT = the annual interest (or coupon) payment T = the number of years until the bond matures r = the annual interest rate (often called yield to maturity (ytm)) Assumes semi-annual interest payments. The present value of a bond (V b ) can be written as: Par = the par or face value of the bond, usually \$1,000 INT = the annual interest (or coupon) payment T = the number of years until the bond matures r = the annual interest rate (often called yield to maturity (ytm)) Assumes semi-annual interest payments. Copyright 2015 by Diane S. Docking 4

Example : Bond Price A \$1,000 face value bond, 6% coupon, 3-year maturity is available for purchase today. Interest is paid semi-annually.  If current market rates are 6%, what should be the price of this bond?  If current market rates are 8%, what should be the price of this bond? Copyright 2015 by Diane S. Docking 5

6 Solution to Example: Bond Price 30 1,000 = PV Copyright 2015 by Diane S. Docking = 30/(1.03) 1 29.13 = 30/(1.03) 2 28.28 837.49 \$1,000.00 = 30/(1.03) 3 1 0 2 34 5 6 30 27.45 26.65 25.88 25.12 = 30/(1.03) 4 = 30/(1.03) 6 = 1,000/(1.03) 6 = 30/(1.03) 5

This problem can more easily be solved using the financial calculator: Copyright 2015 by Diane S. Docking 7 FV = \$1,000 N = 3 years x 2 = 6 PMT = 1,000 x 6%/2 = \$30 I/Y = 6%/2 = 3% CPT PV = \$1,000 Bond is selling at a PAR value. Solution to Example: Bond Price (cont.)

8 Solution to Example: Bond Price 30 1,000 = PV Copyright 2015 by Diane S. Docking = 30/(1.04) 1 28.85 = 30/(1.04) 2 27.74 790.31 \$947.58 = 30/(1.04) 3 1 0 2 34 5 6 30 26.67 25.64 24.66 23.71 = 30/(1.04) 4 = 30/(1.04) 6 = 1,000/(1.04) 6 = 30/(1.04) 5

This problem can more easily be solved using the financial calculator: Copyright 2015 by Diane S. Docking 9 FV = \$1,000 N = 3 years x 2 = 6 PMT = 1,000 x 6%/2 = \$30 I/Y = 8%/2 = 4% CPT PV = \$947.58 Bond is selling at a DISCOUNT to face value. Solution to Example: Bond Price (cont.)

10 Copyright 2015 by Diane S. Docking

11 Copyright 2015 by Diane S. Docking FV = 1,000 N = 25 x 2 = 50 PMT = 1,000 x 10%/2 = \$50 I/Y = 8%/2 = 4% PV = \$1,214.82 Price Change when interest rates go from 10% to 8 % = +\$214.82 FV = 1,000 N = 25 x 2 = 50 PMT = 1,000 x 10%/2 = \$50 I/Y = 12%/2 = 6% PV = \$842.38 Price Change when interest rates go from 10% to 12 % = −\$157.62

12 Copyright 2015 by Diane S. Docking

13 Sensitivity of Bond Prices to Interest Rate Movements Price-Sensitive Bonds 1. _______ relationship between interest rates and prices of bonds 2. _______ maturity—more price variation for a change in interest rates 3. _______ coupon rate bonds are more price sensitive (the principal is a greater % of current value) 4. _______________ bonds most sensitive 5. Price sensitivity is _______ for declining rates than for increasing rates

Impact of Maturity on Price Volatility Absolute Value of Percent Change in a Bond’s Price for a Given Change in Interest Rates Absolute Value of Percent Change in a Bond’s Price for a Given Change in Interest Rates Time to Maturity Short time to maturity – low volatility Volatility Long time to maturity – high volatility Copyright 2015 by Diane S. Docking 14

Impact of Coupon Rates on Price Volatility Bond Value Bond Value Interest Rate Low-Coupon Bond High-Coupon Bond Copyright 2015 by Diane S. Docking 15

Impact of r on Price Volatility Bond Price Interest Rate How does volatility change with interest rates? Price volatility isinversely relatedto the level of theinitial interestrate r Copyright 2015 by Diane S. Docking 16

Example 1: Bond Valuation BBB Manufacturers has outstanding bonds with a \$1,000,000 face value. The coupon rate on the bonds is 5%, interest is paid semi-annually, and the bonds mature in 10 years.  If current market rates are 7%, what should be the price of these bonds?  If current market rates are 3%, what should be the price of these bonds? Copyright 2015 by Diane S. Docking 17

Solution to Example 1: Bond Valuation If current market rates are 7%, what should be the price of these bonds? If current market rates are 3%, what should be the price of these bonds? Copyright 2015 by Diane S. Docking 18 FV = \$1,000,000 N = 10 years x 2 = 20 PMT = 1,000,000 x 5%/2 = \$25,000 I/Y = 7%/2 = 3.5% CPT PV = \$857,875.97 FV = \$1,000,000 N = 10 years x 2 = 20 PMT = 1,000,000 x 5%/2 = \$25,000 I/Y = 3%/2 = 1.5% CPT PV = \$1,171,686.39 Bonds are selling at a DISCOUNT to face value. Bonds are selling at a PREMIUM to face value.

Example 2: Bond Valuation Mary bought a bond when it was issued by Mattress Co. 14 years ago. The bond, which has a \$1,000 face value and a coupon rate of 10%, matures in 6 years. Interest is paid semi- annually.  If the yield on similar risk investments is 14%, what is the current market value (price) of the bond?  Suppose the yield on similar risk investments is only 8%. What is the current market value (price) of the bond? Copyright 2015 by Diane S. Docking 19

Solution to Example 2: Bond Valuation If the yield on similar risk investments is 14%, what is the current market value (price) of the bond? Suppose the yield on similar risk investments is only 8%. What is the current market value (price) of the bond? Copyright 2015 by Diane S. Docking 20 FV = \$1,000 N = 6 years x 2 = 12 PMT = 1,000 x 10%/2 = \$50 I/Y = 14%/2 = 7% CPT PV = \$841.15 FV = \$1,000 N = 6 years x 2 = 12 PMT = 1,000 x 10%/2 = \$50 I/Y = 8%/2 = 4% CPT PV = \$1,093.85 Bond is selling at a DISCOUNT to face value. Bond is selling at a PREMIUM to face value.

Bond Prices Summary Premium bond: if Coupon > market rate; then Price > Par Discount bond: if Coupon < market rate; then Price < Par Par bond: if Coupon = market rate; then Price = Par Premium bond: if Coupon > market rate; then Price > Par Discount bond: if Coupon < market rate; then Price < Par Par bond: if Coupon = market rate; then Price = Par Copyright 2015 by Diane S. Docking 21

Copyright 2015 by Diane S. Docking 22 Finding Bond Yields (market rates): Yield to Maturity The Yield to Maturity (YTM) – is the average rate of return you earn per year if you buy a bond and hold it until it matures. Example: A \$1,000 face value bond, 6% coupon, 3-year maturity is available for purchase today at a price of \$852.48. Interest is paid semi-annually. What is the bond’s YTM? PV = -\$852.48 FV = \$1,000 N = 3 years x 2 = 6 PMT = 1,000 x 6%/2 = \$30 Cpt I/Y = 6% x 2 = 12%

Copyright 2015 by Diane S. Docking 23 Finding Bond Yields (market rates): Holding Period Yield The Holding Period Yield (HPY) – is the average rate of return you earn per year if you buy a bond and then sell it before it matures. Example: A \$1,000 face value bond, 6% coupon, 3-year maturity is available for purchase today at a price of \$852.48. Interest is paid semi-annually. You purchase the bond and sell it 1 year later for \$900? During the year you received 2 interest payments. What is your holding period yield? PV = -\$852.48 FV = \$900 N = 1 years x 2 = 2 PMT = 1,000 x 6%/2 = \$30 Cpt I/Y = 6.2222% x 2 = 12.44%

Copyright 2015 by Diane S. Docking 24 Realized Rates of Returns Rate of Return: we can decompose returns into two pieces: where = current yield, and = capital gains.

Copyright 2015 by Diane S. Docking 25 Example: Determining Realized Rate of Return Union Corporation’s 30-year bonds currently pay an annual interest payment of \$100.00 per every \$1,000 face value. Bonds are currently selling at par. Assume you purchase \$10,000 of Union bonds at today’s market price. Time passes and at the end of 1 year, the bond’s are selling for 105% of par. If you sell the bonds in one year, what is your annual rate of return on this investment?

Copyright 2015 by Diane S. Docking 26 Solution to Example: Determining Realized Rate of Return FV = 10,500 PV = 10,000 Pmt = 1,000 n = 1  i = 15%

Copyright 2015 by Diane S. Docking 27 Example 2: Determining Realized Rate of Return Union Corporation’s 30-year bonds currently pay an annual interest payment of \$100.00 per every \$1,000 face value. Bonds are currently selling at par. Assume you purchase \$10,000 of Union bonds at today’s market price. Time passes and at the end of 1 year, the bond’s are selling for 94% of par. If you sell the bonds in one year, what is your annual rate of return on this investment?

Copyright 2015 by Diane S. Docking 28 Solution to Example 2: Determining Realized Rate of Return FV = 9,400 PV = 10,000 Pmt = 1,000 n = 1  i = 4%

Copyright 2015 by Diane S. Docking 29 Example 3: Determining Realized Rate of Return 1. Union Corporation’s 30-year bonds currently pay an annual interest payment of \$100.00 per every \$1,000 face value. Bonds are currently selling at par. Assume you purchase \$10,000 of Union bonds at today’s market price. Time passes and at the end of 2 years, the bond’s are selling for 105% of par. If you sell the bonds in two years, what is your annual rate of return on this investment? 2. Union Corporation’s 30-year bonds currently pay an annual interest payment of \$100.00 per every \$1,000 face value. Bonds are currently selling at par. Assume you purchase \$10,000 of Union bonds at today’s market price. Time passes and at the end of 2 years, the bond’s are selling for 94% of par. If you sell the bonds in two years, what is your annual rate of return on this investment?

Copyright 2015 by Diane S. Docking 30 Solution to Example 3: Determining Realized Rate of Return 1. 2. FV = 10,500 PV = 10,000 Pmt = 1,000 n = 2  i/y = 12.3546% FV = 9,400 PV = 10,000 Pmt = 1,000 n = 2  i/y = 7.1029%