Chapter 7: Bonds and Their Valuation
I. Basic valuation model
A. Preliminary ideas
1. Market value of any investment asset = present value of cash flows
2. Interest rates cash flows discounted by = asset’s required return
3. Required return function of expected rate of inflation and perceived risk of asset
4. Higher perceived risk → higher required return → lower asset market value
B. Notation
1. CFt = cash flow expected at end of year t
2. N = relevant time period
3. rd = appropriate required return = discount rate
4. V0 = value of asset at time zero
C. Basic valuation result:
II. Bond Valuation
A. Introduction
1. Bond is long-term debt instrument that pays a specific amount of periodic interest over specified period of time.
a. Treasury bonds c. Municipal bonds
b. Corporate bonds d. Foreign bonds
2. Bond’s principal = par value = face value = amount borrowed by company and the amount owed to bondholder on maturity date
3. Bond’s maturity date = time at which bond becomes due and principal must be repaid
4. Bond’s coupon rate = is the specified interest rate (or dollar) amount that must be periodically paid
5. Bond’s current yield = annual interest income divided by current price of bond or
6. Bond’s yield to maturity = interest earned on a bond from time it acquired until maturity date of bond
7. Some other possible features
a. Floating rate bond = bond whose interest rate fluctuates with shifts in general level of interest rates
b. Zero coupon bond = bond that pays no annual interest but is sold at discount below par
c. Original issue discount (OID) bond = bond originally offered at price below its par value
d. Call provision = provision in bond that gives issuer the right to redeem bonds prior to normal maturity date
e. Sinking fund provision = provision that requires issuer to retire portion of bond issue each year
f. Convertible bond = bond that is exchangeable at option of the holder for the issuing firm’s stock
g. Warrant = long-term option to buy stated number of common stock shares at specified price
h. Putable bond = bond with provisions that allow its investor to sell it back to the company prior to maturity at prearranged price
i. Income bond = bond that pays interest only if it is earned
j. Indexed (purchasing power) bond = bond that has interest payments based on an inflation index to protect holder from inflation
B. Notation
1. Let INT = annual interest paid in dollars
2. Let M = face value = par value in dollars
3. Let N = number of years to maturity
4. Let rd = required return on the bond
5. Let VB = value of bond at time zero
6. Note: the coupon rate = INT/M
C. Market value of bond:
or
D. Example: N=3, rd=12%, INT = 100, M = $1,000
1. Math
2. Microsoft Excel Spreadsheet:
a. +PV(rate,Nper,Pmt,FV,type)
rate = 0.12
Nper = 3
Pmt = 100
FV = 1000
Type = 0 or omitted
b. +-1*PV(0.12,3,100,1000,0) = $951.96
E. First key result: VB is inversely related to rd
1. Note: In bond valuation formula: rd is in denominator
↑rd → ↓VB
↓ rd → ↑VB
2. Numerical example: N = 10, INT = $100, M =$1,000
a. Note: coupon rate = $100/$1000=0.10 =10%
b. Table: Various rd and associated B0
rd / VB5% / $1,386.09
6% / $1,294.40
7% / $1,210.71
8% / $1,134.20
9% / $1,064.18
10% / $1,000.00
11% / $941.11
12% / $887.00
13% / $837.21
14% / $791.36
15% / $749.06
i. Note: rd and VB are inversely related
ii. If rd = coupon rate → VB = M
Bond Pricing Principle 1: Par bonds - - If a bond’s price equals its face value, then its yield equals its coupon rate.
iii. If rd < coupon rate → VB > M → bond sold at premium
Bond Pricing Principle 2: Premium bonds - - If a coupon bond has a price higher than its face value, its yield to maturity is less than the coupon rate.
iv. If rd > coupon rate → VB < M → bond sold at discount
Bond Pricing Principle 3: Discount bonds - - If a coupon bond has a price lower than its face value, its yield to maturity is greater than the coupon rate.
3. Assessing a bond’s riskiness
a. Interest rate risk = risk of a decline in a bond’s price due to an increase in interest rates
b. Reinvestment rate risk = risk that a decline in interest rates will lead to a decline in income from a bond portfolio
4. As bonds are sold closer to their maturity date, VB converges toward M. Assume a ten-year bond, INT = $100, M = $1,000 is issued at year 0. The table below shows the prices the bond will be resold at for various years.
Year Bond Sold / rd= 12.00% / rd= 10.00% / rd= 8.00%0 / $887.00 / $1,000.00 / $1,134.20
1 / 893.44 / 1,000.00 / 1,124.94
2 / 900.65 / 1,000.00 / 1,114.93
3 / 908.72 / 1,000.00 / 1,104.13
4 / 917.77 / 1,000.00 / 1,092.46
5 / 927.90 / 1,000.00 / 1,079.85
6 / 939.25 / 1,000.00 / 1,066.24
7 / 951.96 / 1,000.00 / 1,051.54
8 / 966.20 / 1,000.00 / 1,035.67
9 / 982.14 / 1,000.00 / 1,018.52
10 / 1,000.00 / 1,000.00 / 1,000.00
F. Next key result: As interest rates change, the percentage change in bond prices is larger the maturity of the bond
1. Example: Assume interest rates fell from 12% to 8%. Table below shows increase in bond price and % change in price for various maturities.
# years tillmaturity when bond resold / VB if
rd = 0.12 / VB if
rd = 0.08 / % change
in VB
2 / $966.20 / $1,035.67 / 7.19%
5 / 927.90 / 1,079.85 / 16.38
8 / 900.65 / 1,114.93 / 23.79
12 / 876.11 / 1,150.72 / 31.34
15 / 863.78 / 1,171.19 / 35.59
18 / 855.01 / 1,187.44 / 38.88
21 / 848.76 / 1,200.34 / 41.42
24 / 844.31 / 1,210.58 / 43.38
27 / 841.15 / 1,218.70 / 44.89
30 / 838.90 / 1,225.16 / 46.04
2. Example: Assume interest rates increased from 12% to 15%. Table below shows decrease in bond price and % change in price for various maturities.
# years tillmaturity when bond resold / VB if
rd = 0.12 / VB if
rd = 0.15 / % change
in VB
2 / $966.20 / $918.71 / -4.91%
5 / 927.90 / 832.39 / -10.29
8 / 900.65 / 775.63 / -13.88
12 / 876.11 / 728.97 / -16.80
15 / 863.78 / 707.63 / -18.08
18 / 855.01 / 693.60 / -18.88
21 / 848.76 / 684.38 / -19.37
24 / 844.31 / 678.31 / -19.66
27 / 841.15 / 674.32 / -19.83
30 / 838.90 / 671.71 / -19.93
G. The yield to maturity
1. yield to maturity (YTM) = rate of return investors earn if they buy a bond at a specific price and hold it till maturity. (Assumes bond issuer makes all scheduled interest and principal payments as promised.)
2. the yield to maturity = the discount rate used to determine bond’s value. If price of bond = par value → yield to maturity = discount rate = coupon rate
3. If INT = 100, M = 1000, N = 10, and rd = 8.766%, then VB=$1,080.20
Then YTD = rd = 8.766%
4. Microsoft Excel function: internal rate of return
a. +IRR(values, guess)
i. values = sequence of cash flows in order, starting with year 0. Includes both positive and negative cash flows.
ii. guess= fraction to begin iteration to solution
iii. Example: Spreadsheet looks like:
A / B1 / Year / Cash Flow
2 / 0 / -$1,080.20
3 / 1 / 100
4 / 2 / 100
5 / 3 / 100
6 / 4 / 100
7 / 5 / 100
8 / 6 / 100
9 / 7 / 100
10 / 8 / 100
11 / 9 / 100
12 / 10 / 1,100
+IRR(B2:B12, 0.10) = 8.76%
b. Another Excel function: RATE(nper,pmt,pv,fv,type,guess)
i. guess = fraction to initiate algorithm
ii. RATE(10,100,-1080.20,1000,0,0.08)=8.76%
H. Semiannual interest and bond values
1. Convert annual interest to semiannual interest by dividing INT by 2
2. Multiply N by 2
3. Divide rd by 2
4.
5. INT = 80, M = $1,000, N =12 years, rd= 10%. INT paid semiannually.
I. Time line examples
1. Annual interest
2. Semiannual interest
J. Yield to call
1. = rate of return earned on a bond if it is called before its maturity date
2. Ex: Assume rd = 5%, N = 14, INT = 100, M = $1,000
a. Determine VB
Year / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 12 / 13 / 14Cash Flow / $100 / $100 / $100 / $100 / $100 / $100 / $100 / $100 / $100 / $100 / $100 / $100 / $100 / $1,100
b. Now assume bond can be called 9 years after being issued
i.
ii. Call price = often set equal to par value plus one year’s interest
iii. YTC = yield to call = rate of return earned on bond if it is called before its maturity date
iv. Assume bond called after nine years. Initial price = VB = $1,494.93
Year / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9Cash Flow / $100 / $100 / $100 / $100 / $100 / $100 / $100 / $100 / $1,100
(1)
YTC = 4.21%
(2) RATE(9,100,-1494.93,1100,0,0.03)=4.21%
III. Corporate bonds
A. Types of corporate bonds
1. Mortgage bond = bond backed by fixed assets.
a. First mortgage bonds are senior in priority to claims of second mortgage bonds.
b. All mortgage bonds subject to indenture = formal agreement between issuer and bondholders
2. Debenture = long-term bond not secured by mortgage on specific property
3. Subordinated debenture = bond having claim on assets only after senior debt has been paid off in event of liquidation
B. Bond ratings
1. Moody’s Investor Service (Moody’s) and Standard & Poor’s Corporation (S&P)
Moody’s and S&P Bond RatingsInvestment Grade / Junk Bonds
Moody’s / Aaa / Aa / A / Baa / Ba / B / Caa / C
S&P / AAA / AA / A / BBB / BB / B / CCC / C
2. Junk bond = high-risk, high-yield bond
3. Factors that affect bond ratings
a. Various ratios i. Antitrust
b. Mortgage provisions j. Overseas operations
c. Subordinate provisions k. Environmental factors
d. Guarantee provisions l. Product liability
e. Sinking fund m. Pension liabilities
f. Maturity n. Labor unrest
g. Stability o. Accounting policies
h. Regulation
Chapter 7: Bonds and Their Valuation. Page 9