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### Chapter 5

Bonds, Bond Valuation, and Interest Rates

Topics in Chapter

- Key features of bonds
- Bond valuation
- Measuring yield
- Assessing risk

Determinants of Intrinsic Value: The Cost of Debt

Net operating

profit after taxes

Required investments

in operating capital

−

Free cash flow

(FCF)

=

FCF1

FCF2

FCF∞

...

Value = + + +

(1 + WACC)1

(1 + WACC)2

(1 + WACC)∞

Weighted average

cost of capital

(WACC)

Market interest rates

Firm’s debt/equity mix

Cost of debt

Cost of equity

Market risk aversion

Firm’s business risk

Interest Rates & Interest-Bearing Securities

- Interest rates:
- Based on supply & demand for money
- Driven by risk factors
- Role of Federal Reserve
- Basis Point
- .01% or .0001

rd = r* + IP + DRP + LP + MRP

rd = Required rate of return on a debt security.

r* = Real risk-free rate.

IP = Inflation premium.

DRP = Default risk premium.

LP = Liquidity premium.

MRP = Maturity risk premium.

Risk & Term Structure of Interest RatesRisk & Term Structure

- r = r* + IP + DRP + LP + MRP
- r = nominal interest rate of a particular security (or required rate of return)
- r* = real risk-free interest ratetypically 1-4% depending on monetary policyassumes expected inflation = zero
- IP = Inflation premiumAve. inflation over life of bond
- DRP = Default risk premiumCompensation for possible defaultFunction of bond ratings

Risk & Term Structure

- r = r* + IP + DRP + LP + MRP
- LP = Liquidity Premium Compensation for possible difficulty selling bond quickly at fair market value
- MRP = Maturity Risk PremiumCompensation for possible loss in value due to increase in interest rates over maturity of bond. Affects longer maturities more than shorter.

Premiums Added to r* (real risk-free rate) for Different Types of Debt

- ST Treasury:
- only IP for ST inflation
- LT Treasury:
- IP for LT inflation, MRP
- ST corporate:
- ST IP, DRP, LP
- LT corporate:
- IP, DRP, MRP, LP

Inflation & Interest Rates

- Nominal Interest= 12%

- Inflation -1%

= Real Int. % =11%

If inflation =

& req’d real return =

Then Nominal rate =? =

12%

- 8%

=4%

- 8%

11%

=19%

Relationship b/w Nominal & Real Interest Rates, & Inflation

- Nom = Real + Inflation
- But, inflation not additive, it grows or compounds, so multiply
- Nom = (Real) x (Infl)
- And (1+Nom) = (1 + real) x (1 + infl)
- Is better determinant; known as Fisher effect

Estimating Inflation Premium (IP)

- Treasury Inflation-Protected Securities (TIPS) are indexed to inflation.
- IP for a particular length maturity can be approximated as the difference between the yield on a non-indexed Treasury security of that maturity minus the yield on a TIPS of that maturity.

Bond Spreads, the DRP, and the LP

- A “bond spread” is often calculated as the difference between a corporate bond’s yield and a Treasury security’s yield of the same maturity. Therefore:
- Spread = DRP + LP.
- Bond’s of large, strong companies often have very small LPs. Bond’s of small companies often have LPs as high as 2%.

Term Structure Yield Curve

- Term structure of interest rates: the relationship between interest rates (or yields) and maturities.
- A graph of the term structure is called the yield curve.

What factors can explain shape of this yield curve?

- Upward slope due to:
- Increasing expected inflation
- Increasing maturity risk premium
- What about liquidity & default risk?

Treasury vs. Corporate Yield Curves relationships

- Corp yield curves are higher than Treasuries, but not necessarily parallel.
- Spread b/w the two yield curves widens as corporate bond rating decreases due to:
- DRP & LP

Computing Yields

- Estimate the inflation premium (IP) for each future year. This is the estimated average inflation over that time period.
- Step 2: Estimate the maturity risk premium (MRP) for each future year.

Assume investors expect inflation to be 5% next year, 6% the following year, and 8% per year thereafter.

- Step 1: Find the average expected inflation rate over years 1 to n:

IP1 = 5%/1.0 = 5.00%.

IP10 = [5 + 6 + 8(8)]/10 = 7.5%.

IP20 = [5 + 6 + 8(18)]/20 = 7.75%.

Must earn these IPs to break even versus inflation; that is, these IPs would permit you to earn r* (before taxes).

Step 2: Find MRP based on this equation:

Assume the MRP is zero for Year 1 and increases by 0.1% each year.

MRPt = 0.1%(t - 1).

MRP1 = 0.1% x 0 = 0.0%.

MRP10 = 0.1% x 9 = 0.9%.

MRP20 = 0.1% x 19 = 1.9%.

Step 3: Add the IPs and MRPs to r*:

rRFt = r* + IPt + MRPt .

rRF = Quoted market interest

rate on treasury securities.

Assume r* = 3%:

rRF1 = 3% + 5% + 0.0% = 8.0%.

rRF10 = 3% + 7.5% + 0.9% = 11.4%.

rRF20 = 3% + 7.75% + 1.9% = 12.65%.

Upward vs. Downward sloping yield curves due to?

- Real risk-free rate = 3%
- Expected inflation for
- Year 1 =7%, Yr 2 = 5%; Yr 3 = 3%
- What are interest rates for 1, 2, & 3 yr borrowings?

Interest Rates & MRP problem

- Assume the real risk-free rate (r*) is 4% and inflation is expected to be 7 percent in Year1; 4% in yr 2; and 3% thereafter. Assume all Treasury Bonds are highly liquid and free of default risk. If 2-yr and 5-yr T-Bonds both yield 11%, what is the difference in the maturity risk premiums (MRPs) on the two bonds; that is, what is MRP5 – MRP2?

Interest Rates & Inflation Problem

- Due to the recession, the rate of inflation expected for the coming year is only 3.5%. However, the rate of inflation in Yr 2 and thereafter is expected to be constant at some level above 3.5%. Assume the real risk-free rate (r*) = 2% for all maturities, and there are no maturity premiums. If 3-year T-Bonds yield 3% (0.03) more than the 1-year T-Bonds, what rate of inflation is expected after year 1?

Coupon Bonds

- Bond = Debt = Borrowing
- Fixed Maturity (Maturity Date) = N
- Par Value=Face Value=Maturity Value=$1000=FV
- Coupon Rate=Stated Rate (locked in in bond contract)
- Coupon payment= Coupon rate x face value=PMT
- Market Rate of interest = Yield to Maturity = rate used to discount bond CF’s = I
- **PV cash flow of bonds always opposite sign of PMT & FV!!!

Bond Perspectives

Debt

Asset

Has $

Lender

Buyer or Investor

Bondholder

Creditor

Requires return to invest $ in bonds based on risk

Interest Received (earned) (Revenue) - pay tax on it

Capital Appreciation

- Needs $
- Borrower
- Issuer or seller
- Debtholder
- Cost of borrowing
- Interest Paid (Expense) – generates tax benefit (Svgs)
- Cost of Debt
- = Rd or Kd;
- After-tax cost = Rd (1-t)

Key Features of a Bond

- Par value: Face amount; paid at maturity. Assume $1,000.
- Coupon interest rate: Stated interest rate. Multiply by par value to get dollars of interest. Generally fixed.

(More…)

Key Features of a Bond

- Maturity: Years until bond must be repaid. Declines.
- Issue date: Date when bond was issued.
- Default risk: Risk that issuer will not make interest or principal payments.

Value of Financial Security

- Value of any asset based on the net present value of the expected future cash flows discounted by the interest (discount) rate that reflects risk factors
- Discount (interest rate) depends on:
- Riskiness of CFs reflected by DRP, MRP, LP
- General level of interest rates, which reflects inflation, supply & demand for $, production opportunities, time preferences for consumption

1

2

10

Value of a 10-year, 10% coupon bond if rd = 10%10%

...

V = ?

100

100

100 + 1,000

$100

$100

$1,000

V

=

.

.

.

+

+

+

B

(1 + rd)1

(1 + rd)N

(1 + rd)N

= $90.91 + . . . + $38.55 + $385.54

= $1,000.

PV maturity value

Value of bond

$ 614.46

385.54

$1,000.00

=

=

=

INPUTS

10 10 100 1000

N I/YR PV PMT FV

-1,000

OUTPUT

The bond consists of a 10-year, 10% annuity of $100/year plus a $1,000 lump sum at t = 10:10 13 100 1000

N I/YR PV PMT FV

-837.21

OUTPUT

What would happen if expected inflation rose by 3%, causing r = 13%?When market interest rate (rd)rises above coupon rate, bond’s value (PV or price) falls below par, so sells @ discount.

9 13 100 1000

N I/YR PV PMT FV

-846.05

OUTPUT

What happens if one year passes but the market i stays at 13%?8 13 100 1000

N I/YR PV PMT FV

-856.04

OUTPUT

What happens if a second year passes but the market i stays at 13%?1 13 100 1000

N I/YR PV PMT FV

-973.45

OUTPUT

What happens if 9 years pass but the market i stays at 13%?As a bond approaches maturity, it’s price approaches the face or maturity value of $1000

10 7 100 1000

N I/YR PV PMT FV

-1,210.71

OUTPUT

What would happen if inflation fell, and rd declined to 7%?If coupon rate > mrkt i% (rd), price rises above par, and bond sells at a premium.

Bond Pricing in Excel

- PV = ? $1210.71

Summary of Bond price and interest rate relationships

- If market rate of interest increases above the stated (coupon) rate, then bond’s price falls and sells at discount
- If market rate of interest drops below the stated (coupon) rate, then bond’s price increases and sells at a premium
- **INVERSE RELATIONSHIP b/w Market i% and Bond’s PRICE!***

Bond prices & changing interest rates

- Suppose the bond was issued 20 years ago and now has 10 years to maturity. What would happen to its value over time if required rate of return remained at 10%, or at 13%, or at 7%?

Bond Value ($) vs Years remaining to Maturity

rd = 7%.

1,372

1,211

rd = 10%.

M

1,000

837

rd= 13%.

775

30 25 20 15 10 5 0

Bond Price Movements over time

- At maturity, value of any bond must equal its par value.
- Value of a premium bond decreases to $1,000.
- Value of a discount bond increases to $1,000.
- A par bond stays at $1,000 if mrkt i% (rd)remains constant.

10 7 100 1000

N I/YR PV PMT FV

?

OUTPUT

What’s market value of 10 year 10% coupon bond when market = 7%?Bond sells at a premium::

Price today = $1,210.71.

10 (1210.71) 100 1000

N I/YR PV PMT FV

?

OUTPUT

If you buy a 10%, 10 year bond today for $1,210.71, and hold it to maturity, what’s your rate of return?Solve for i% = 7% = Yield to maturity (YTM)

What’s “yield to maturity”?

- YTM is rate of return earned on a bond held to maturity. Also called “promised yield.”
- It assumes bond will not default.
- Includes both interest pmt component & cap gains over bond’s life
- Interest rate equating bond’s price today to NPV of PMTs & FV. (Think market rate of interest)
- Vs. Annualized Return which reflects only a one-year holding period

1

9

10

rd=?

...

90

90

90

1,000

PV1

.

.

.

PV10

PVM

Find i % (rd) that “works”!

887

YTM on a 10-year, 9% annual coupon, $1,000 par value bond selling for $887Find YTM (i % or rd)

INT

INT

M

...

V

=

+

+

+

B

(1 + rd)N

(1 + rd)1

(1 + rd)N

1,000

90

90

...

887

=

+

+

+

(1 + rd)1

(1 + rd)N

(1 + rd)N

INPUTS

10 -887 90 1000

N I/YR PV PMT FV

10.91

OUTPUT

Bond Prices & Int. Rates

- If coupon rate < mrkt i % (rd), bond sells at a discount.
- If coupon rate = i %, bond sells at its par value.
- If coupon rate > i%, bond sells at a premium.
- If market i% rises, price falls.
- Price = par at maturity.

10 -1134.2 90 1000

N I/YR PV PMT FV

7.08

OUTPUT

Find YTM if price were $1,134.20.Sells at a premium. Because coupon = 9% > mrkt i% = 7.08%, bond’s value > par.

Definitions

Current yield = “Interest Yield”

Capital gains yield =Change in value

= YTM = +

Exp total

return

Exp cap

gains yld

Exp

Curr yld

Definitions

Annual coupon pmt

Current price

Current yield =

Capital gains yield =

= YTM = +

Change in price

Beginning price

Exp total

return

Exp

Curr yld

Exp cap

gains yld

YTM = Current yield + Capital gains yield.

Cap gains yield = YTM - Current yield

= 10.91% - 10.15%

= 0.76%.

Could also find values in Years 1 and 2,

get difference, and divide by value in

Year 1. Same answer.

2N rd/2 OK INT/2 OK

N I/YR PV PMT FV

OUTPUT

Semiannual Bonds1. Multiply years by 2 to get periods = 2N.

2. Divide nominal rate by 2 to get periodic rate = rd/2.

3. Divide annual INT by 2 to get PMT = INT/2.

20 6.5 50 1000

N I/YR PV PMT FV

-834.72

INPUTS

OUTPUT

Value of 10-year, 10% coupon, semiannual bond if rd = 13%.Spreadsheet Functions for Bond Valuation

- PRICE
- YIELD

Call Provision

- Issuer can refund if rates decline. That helps the issuer but hurts the investor.
- Therefore, borrowers are willing to pay more, and lenders require more, on callable bonds.
- Most bonds have a deferred call and a declining call premium
- Yield to call: yearly rate of return earned on a bond until it’s called

Callable Bonds and Yield to Call

- A 10-year, 10% semiannual coupon,$1,000 par value bond is selling for$1,135.90 with an 8% yield to maturity.It can be called after 5 years at $1,050.

If you bought bonds, would you be more likely to earn YTM or YTC?

- Coupon rate = 10% vs. YTC = rd = 7.53%. Could raise money by selling new bonds which pay 7.53%.
- Could thus replace bonds which pay $100/year with bonds that pay only $75.30/year.
- Investors should expect a call, hence YTC = 7.53%, not YTM = 8%.

Investor returns on callable bonds

- In general, if a bond sells at a premium, then coupon > market rate, so a call is likely.
- So, investors expect to earn:
- YTC on premium bonds.
- YTM on par & discount bonds.

What’s a sinking fund?

- Provision to pay off a loan over its life rather than all at maturity.
- Similar to amortization on a term loan.
- Reduces risk to investor, shortens average maturity.
- But not good for investors if rates decline after issuance.

Sinking funds are generally handled in 2 ways

- Call x% at par per year for sinking fund purposes.
- Call if rd is below the coupon rate and bond sells at a premium.
- Buy bonds on open market.
- Use open market purchase if rd is above coupon rate and bond sells at a discount.

What factors affect default risk and bond ratings?

- Financial ratios
- Debt ratio
- Coverage ratios, such as interest coverage ratio or EBITDA coverage ratio
- Profitability ratios
- Current ratios

(More…)

Other Factors that Affect Bond Ratings

- Provisions in the bond contract
- Secured versus unsecured debt
- Senior versus subordinated debt
- Guarantee provisions
- Sinking fund provisions
- Debt maturity

(More…)

Other factors

- Earnings stability
- Regulatory environment
- Potential product liability
- Accounting policies

Interest rate (or price) risk for 1-year and 10-year 10% bonds

Interest rate risk: Rising mrkt i % (rd) causes bond’s price to fall.

10-year

i %

Change

Change

1-year

5%

$1,048

$1,386

38.6%

4.8%

10%

1,000

1,000

25.1%

4.4%

15%

956

749

What is reinvestment rate risk?

- The risk that CFs will have to be reinvested at future lower rates, reducing income.
- Illustration: Suppose you just won $500,000 playing the lottery. You’ll invest the money and live off interest. You buy a 1-year bond with a YTM of 10%.

Year 1 income = $50,000. At year-end get back $500,000 to reinvest.

- If rates fall to 3%, income will drop from $50,000 to $15,000. Had you bought 30-year bonds, income would have remained constant.

The Maturity Risk Premium

- Long-term bonds: High interest rate risk, low reinvestment rate risk.
- Short-term bonds: Low interest rate risk, high reinvestment rate risk.
- Nothing is riskless!
- Yields on longer term bonds usually are greater than on shorter term bonds, so the MRP is more affected by interest rate risk than by reinvestment rate risk.

Other types of Bonds

- Zero coupon:
- Pays no coupon & sells @ disct below par
- Convertible:
- To stock @fixed price @ bondholder’s option
- Income:
- Pays interest only if interest earned by issuer; won’t bankrupt co.

Other types of Bonds

- Revenue:
- Interest paid from revenue generated by project being financed by bonds
- Floating rate:
- Adjusts coupon rate periodically based on market interest rates

Bankruptcy

- Two main chapters of Federal Bankruptcy Act:
- Chapter 11, Reorganization
- Chapter 7, Liquidation
- Typically, company wants Chapter 11, creditors may prefer Chapter 7.

If company can’t meet its obligations, it files under Chapter 11. That stops creditors from foreclosing, taking assets, and shutting down the business.

- Company has 120 days to file a reorganization plan.
- Court appoints a “trustee” to supervise reorganization.
- Management usually stays in control.

Company must demonstrate in its reorganization plan that it is “worth more alive than dead.”

- Otherwise, judge will order liquidation under Chapter 7.

If the company is liquidated, here’s the payment priority:

- Past due property taxes
- Secured creditors from sales of secured assets.
- Trustee’s costs
- Expenses incurred after bankruptcy filing
- Wages and unpaid benefit contributions, subject to limits
- Unsecured customer deposits, subject to limits
- Taxes
- Unfunded pension liabilities
- Unsecured creditors
- Preferred stock
- Common stock

In a liquidation, unsecured creditors generally get zero. This makes them more willing to participate in reorganization even though their claims are greatly scaled back.

- Various groups of creditors vote on the reorganization plan. If both the majority of the creditors and the judge approve, company “emerges” from bankruptcy with lower debts, reduced interest charges, and a chance for success.

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