Investment Valuations

• Value of Investment = PV of expected future CFs

• Factors affecting value– Cash Flows

• Amount (size) and timing

– Discount Rate• Risk of investment

Cash Flows of A Bond

• Face (Par) Value– At maturity

• Coupons– At pre-set coupon payment dates

0 1 2 3 N

|------|-----|-----|-----|semi-annual

C C C C+Face value

Bond Rates and Yields

• Coupon rate– Used for computing coupon payments– Coupon payment = Face value * coupon rate per period

• Yield-to-Maturity (YTM)– Market interest rate– Used for computing bond value (price) or– Inferred from current market price

• Current yield – Current yield = Annual coupon payment / Price

Bond Valuation

• Bond value (price) = PV of CFs = PV of Coupon + PV of Par value– Coupon payments per period [PMT]– Par value [FV]– Time to maturity [N]– Discount rate (YTM) per period [I/Y]– Price (value) [PV]

An Example: Valuing a Bond

• You wish to value a bond with a 10% coupon rate per year payable semi-annually, 20 years to maturity, and a par value of $1000. The market interest rate is 10% per year.

• Period = semi-annual = 2 times per year– Coupon payments = 0.1 x $1000 / 2 = $50 every 6 months– Time to maturity = 20 x 2 = 40 periods– Interest rate per period = 10% (YTM) / 2 = 5% every 6 months– Par value = $1000 at maturity0 5% 1 2 3……….40 |-----|-----|-----|-----|-----| semi-annual $50 $50 $50 ……$50+$1000

• PMT 50, FV 1000, I 5%, N 40• CPT PV -1000• When Price = Par value, a bond is selling at par

A Discount Bond (Price < Face Value)

• Market interest rate increases to 12% per year– Interest rate per period = 12%/2 = 6% every 6 months

• All other factors remain the same0 6% 1 2 3……….40

|-----|-----|-----|-----|-----| semi-annual

$50 $50 $50 ……$50+$1000

• PMT 50, FV 1000, I 6%, N 40

• CPT PV -849.537

• When Price < Par value, a bond is selling at a discount

• Price < Par value YTM > Coupon rate

A Premium Bond (Price > Face Value)

• Market interest rate decreases to 8% per year– Interest rate per period = 8%/2 = 4% every 6 months

• All other factors remain the same0 4% 1 2 3……….40

|-----|-----|-----|-----|-----| semi-annual

$50 $50 $50 ……$50+$1000

• PMT 50, FV 1000, I 4%, N 40

• CPT PV –1197.9277

• When Price > Par value, a bond is selling at a premium

• Price > Par value YTM < Coupon rate

Bond Rates and Yields: Example

• A bond currently sells for $932.90. It pays $35 in coupon payments every 6 months, matures in 10 years, and has a face value of $1000. What are its coupon rate, current yield, and yield to maturity (YTM)?

• Coupon payments per year = $35 * x = $70• Coupon rate = $70 /$1000 = 7% per year• Current yield = $70 /$932.90 = 7.5% per year

Bond Rates and Yields: Example

• Computing YTM

• Period = semi-annual

• Time to maturity = 10 * 2 = 20 periods

0 1 2 3 ……… 20

|-----|-----|-----|------|-----|semi-annual

$35 $35 $35 $35 + $1000 (Inflow)

$932.90 (outflow)

• PMT 35, FV 1000, PV –932.90, N 20

• CPT I 3.9934% every 6 months (x 2 = 7.9869% per year)

Graphical Relationship Between Price and YTM

600

700

800

900

1000

1100

1200

1300

1400

1500

0% 2% 4% 6% 8% 10% 12% 14%

Interest Rate Risk

• Price Risk– Change in price due to changes in interest rates– Long-term bonds have more price risk than short-

term bonds• Reinvestment Rate Risk

– Uncertainty about future rates at which future cash flows can be reinvested

– Short-term bonds have more reinvestment rate risk than long-term bonds

Figure 6.2

Bond Pricing Theorem

The following statements about bond pricing are always true.

1. Bond prices and market interest rates move in opposite directions.

2. When a bond’s coupon rate = YTM, price = par value.– coupon rate > YTM price > par value (premium bond)– coupon rate < YTM price < par value (discount bond)

3. Longer maturity => greater % price change when interest rate (YTM) changes

4. Lower coupon rate => greater % price change when interest rate (YTM) changes

The Bond Indenture

• Contract between the company and the bondholders and includes– Basic terms

• Coupon rate, coupon payment dates, maturity dates, etc.– Total amount of bonds issued– May contain the following if applicable

• Description of property used as security• Sinking fund provisions• Call provisions

– Protective covenants

Bond Ratings

• Bond Rating Agencies– Independent, third party– Moody’s, Standard & Poor, D&B

• Investment Grade (BBB or higher)• Junk (BB or lower)

Types of Bonds

• Government Bonds• Municipal Bonds

– Tax-exempt– Comparing tax-exempt bonds and taxable bonds– After-tax yield = Taxable yield x (1 - Tax rate)

• Zero Coupon Bonds• Floating-rate Bonds• Others

Example: Taxable versus tax exempt bond

• A corporate (taxable) bond has a yield of 8% and a municipal bond has a yield of 6%

• If you are in a 40% tax bracket, which bond do you prefer?– After-tax return on the corporate bond = 8%(1 - .4) = 4.8%– After-tax return on the municipal bond = 6%

• At what tax rate would you be indifferent between the two bonds?

• AT return on the corporate bond = AT return on the muni• 8%(1 – T) = 6%• T = 25%

Determinants of Interest Rates (Bond Yields)

• Real Rate– Supply and demand for money

• Inflation Premium: real versus nominal rates – Fisher Effect:

• (1 + Nominal) = (1 + real) x (1 + inflation rate)

• Interest Rate Risk Premium– Maturity, coupon rate, call provisions

• Default Risk Premium– Bond rating, security, seniority, sinking fund

• Taxability Premium• Liquidity Premium

Fisher Effect Example

• If we require a 10% real return and we expect inflation to be 8%, what is the nominal rate?

• 1+ Nominal R = (1.1)(1.08) – 1 = .188 = 18.8%• Interest rates observed are nominal rates.• E.g. Return on T-note is 4.5%. Expected inflation rate is

3.5%. What is the implied real return?• (1.045) = (1 + real rate) (1.035)• Real rate = 0.00966 = 0.966%

Term Structure of Interest Rate (Yield curves)

• Relationship between time to maturity and yields• All else equal, i.e. pull out the effect of default risk,

different coupons, etc.• Yield curve – graphical representation of the term

structure– Normal (upward-sloping)

• long-term yields are higher than short-term yields– Inverted (downward-sloping)

• long-term yields are lower than short-term yields

• Today’s yield curve