Bonds Solution

7/28/2019 Bonds Solution

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Chapter 5

Bonds, Bond Valuation, and

Interest Rates


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Topics in Chapter Key features of bonds

Bond valuation

Measuring yield

Assessing risk


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Value = + + +FCF1 FCF2 FCF

(1 + WACC)1 (1 + WACC)(1 + WACC)2

Free cash flow(FCF)

Market interest rates

Firms business riskMarket risk aversion

Firms debt/equity mixCost of debt

Cost of equity

Weighted average

cost of capital(WACC)

Net operatingprofit after taxes

Required investmentsin operating capital

=

Determinants of Intrinsic Value: The Cost of Debt

...


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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 .00014


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Risk & Term Structure of

Interest Ratesrd = 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.


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Risk & 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 rate

typically 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

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Risk & Term Structure

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

LP = Liquidity Premium

Compensation for possible difficulty sellingbond quickly at fair market value

MRP = Maturity Risk PremiumCompensation for possible loss in value dueto increase in interest rates over maturity ofbond.

Affects longer maturities more than shorter.

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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

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Inflation & Interest Rates

Nominal Interest= 12%

- Inflation -1%

= Real Int. % =11%

If inflation =

&reqd real return =

Then Nominal rate =? =

12%

- 8%

=4%

9

8%

11%

=19%


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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


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Estimating Inflation Premium (IP)

Treasury Inflation-Protected Securities(TIPS) are indexed to inflation.

IP for a particular length maturity canbe approximated as the differencebetween the yield on a non-indexed

Treasury security of that maturity minusthe yield on a TIPS of that maturity.


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Bond Spreads, the DRP, and

the LP A bond spread is often calculated as the

difference between a corporate bonds yield

and a Treasury securitys yield of the samematurity. Therefore:

Spread = DRP + LP.

Bonds of large, strong companies often have

very small LPs. Bonds of small companiesoften have LPs as high as 2%.


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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.


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Hypothetical Treasury Yield

Curve

0%

2%

4%

6%

8%

10%

12%

14%

1 3 5 7 9 11 13 15 17 19

Years to Maturity

InterestRa

te

MRP

IP

r*


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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?

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Treasury vs. Corporate YieldCurves relationships

Corp yield curves are higher thanTreasuries, but not necessarily parallel.

Spread b/w the two yield curves widens ascorporate bond rating decreases due to:

DRP & LP

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Computing Yields

Estimate the inflation premium (IP) foreach future year. This is the estimated

average inflation over that time period. Step 2: Estimate the maturity risk

premium (MRP) for each future year.

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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; thatis, these IPs would permit you to earn r* (beforetaxes).

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Step 2: Find MRP based onthis equation:

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%.

Assume the MRP is zero for Year 1 and

increases by 0.1% each year.


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

rRFt = r* + IPt + MRPt .

rRF = Quoted market interestrate 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%.


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Upward vs. Downward slopingyield 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 yrborrowings?

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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 allTreasury 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 thematurity risk premiums (MRPs) on the two

bonds; that is, what is MRP5MRP2?

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Interest Rates & Inflation Problem

Due to the recession, the rate of inflation expectedfor the coming year is only 3.5%. However, the

rate of inflation in Yr 2 and thereafter is expected tobe constant at some level above 3.5%. Assume thereal risk-free rate (r*) = 2% for all maturities, andthere 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?

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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 bondcontract)

Coupon payment= Coupon rate x face value=PMT

Market Rate of interest = Yield to Maturity = rate

used to discount bond CFs = I

**PV cash flow of bonds always opposite sign of PMT &FV!!!

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Key Features of a Bond

Par value: Face amount; paid atmaturity. Assume $1,000.

Coupon interest rate: Stated interestrate. Multiply by par value to getdollars of interest. Generally fixed.

(More)


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Key Features of a Bond

Maturity: Years until bond must berepaid. Declines.

Issue date: Date when bond wasissued.

Default risk: Risk that issuer will not

make interest or principal payments.

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Value of Financial Security

Value of any asset based on the net presentvalue of the expected future cash flows

discounted by the interest (discount) ratethat 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 28


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Value of a 10-year, 10%coupon bond if r

d= 10%

VB

=$100 $1,000

. . .+

$100

100 100

0 1 2 10

10%

100 + 1,000V = ?

...

= $90.91 + . . . + $38.55 + $385.54= $1,000.

++(1 + rd)1 (1 + rd)N (1 + rd)N


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10 10 100 1000N I/YR PV PMT FV

-1,000

$ 614.46

385.54

$1,000.00

PV annuityPV maturity value

Value of bond

==

=

INPUTS

OUTPUT

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


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When market interest rate (rd)rises abovecoupon rate, bonds value (PV or price)falls below par, so sells @ discount.

10 13 100 1000

N I/YR PV PMT FV

-837.21

INPUTS

OUTPUT

What would happen if expected inflationrose by 3%, causing r = 13%?


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9 13 100 1000

N I/YR PV PMT FV

-846.05

INPUTS

OUTPUT

What happens if one year passes but themarket i stays at 13%?


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8 13 100 1000

N I/YR PV PMT FV

-856.04

INPUTS

OUTPUT

What happens if a second year passes butthe market i stays at 13%?


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Bond Pricing in Excel

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Years to Mat: 10

Coupon rate: 10%

Annual Pmt: $100

Par value = FV: $1,000

Going rate, rd: 10%


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What would happen if inflationfell, and r

ddeclined to 7%?

If coupon rate > mrkt i% (rd), price risesabove par, and bond sells at a premium.

10 7 100 1000N I/YR PV PMT FV

-1,210.71

INPUTS

OUTPUT


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Bond Pricing in Excel

Years to Mat: 10

Coupon rate: 10%

Annual Pmt: $100

Par value = FV: $1,000

Going rate, rd: 7%

PV = ? $1210.71

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Summary of Bond price andinterest rate relationships

If market rate of interest increasesabove the stated (coupon) rate, then

bonds price falls and sells at discount If market rate of interest drops below

the stated (coupon) rate, then bonds

price increases and sells at a premium **INVERSE RELATIONSHIP b/w Market

i% and Bonds PRICE!***

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Bond prices & changinginterest rates

Suppose the bond was issued 20 yearsago and now has 10 years to maturity.

What would happen to its value overtime if required rate of return remainedat 10%, or at 13%, or at 7%?


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M

1,372

1,211

1,000

837

775

30 25 20 15 10 5 0

rd = 7%.

rd= 13%.

rd = 10%.

Bond Value ($) vs Yearsremaining to Maturity


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Bond Price Movements over time

At maturity, value of any bond mustequal 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.


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Whats market value of 10 year 10%coupon bond when market = 7%?

Bond sells at a premium::Price today = $1,210.71.

10 7 100 1000N I/YR PV PMT FV

?

INPUTS

OUTPUT

If you buy a 10% 10 year bond


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If you buy a 10%, 10 year bondtoday for $1,210.71, and hold it to

maturity, whats your rate of return?

Solve for i% = 7% = Yield to maturity(YTM)

10 (1210.71) 100 1000N I/YR PV PMT FV

?

INPUTS

OUTPUT


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Whats yield to maturity?

YTM is rate of return earned on a bond held tomaturity. Also called promised yield.

It assumes bond will not default. Includes both interest pmt component & cap gains

over bonds life

Interest rate equating bonds price today to NPV of

PMTs & FV. (Think market rate of interest)

Vs. Annualized Return which reflects only a one-year holding period


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YTM on a 10-year, 9% annual coupon,$1,000 par value bond selling for $887

90 9090

0 1 9 10rd=?

1,000PV1..

.PV10PVM

887 Find i % (rd)that works!

...


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10 -887 90 1000

N I/YR PV PMT FV

10.91

VINT M

B =(1 + r

d)1 (1 + r

d)N

... +INT

887 90

(1 + rd)1

1,000

(1 + rd)N

= +90

(1 + rd)N

++

++

INPUTS

OUTPUT

...

Find YTM (i% or rd)

(1 + rd)N


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YTM in Excel

Years to Mat: 10

Coupon rate: 9%

Annual Pmt: $90.00

Current price: $887.00

Par value = FV: $1,000.00

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Bond Prices & Int. Rates

If coupon rate < mrkt i % (rd), bondsells at a discount.

If coupon rate = i %, bond sells at itspar value.

If coupon rate > i%, bond sells at apremium.

If market i% rises, price falls.

Price = par at maturity.


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Find YTM if price were$1,134.20.

Sells at a premium. Because

coupon = 9% > mrkt i% =7.08%, bonds value > par.

10 -1134.2 90 1000

N I/YR PV PMT FV7.08

INPUTS

OUTPUT


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Definitions

Current yield = Interest Yield

Capital gains yield =Change in value

= YTM = +Exp totalreturn

ExpCurr yld

Exp capgains yld


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Definitions

Current yield =

Capital gains yield =

= YTM = +

Annual coupon pmtCurrent price

Change in priceBeginning price

Exp totalreturn

ExpCurr yld

Exp capgains yld


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9% coupon, 10-year bond, P =$887, and YTM = 10.91%

Current yield =

= 0.1015 = 10.15%.

$90$887

C ld C l


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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.

YTM = Current yield + Capitalgains yield.


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Semiannual Bonds

1. Multiply years by 2 to get periods = 2N.

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

3. Divide annual INT by 2 to get PMT =

INT/2.2N rd/2 OK INT/2 OK

N I/YR PV PMT FV

INPUTS

OUTPUT


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2(10) 13/2 100/2

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%.

S d h t F ti


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Spreadsheet Functionsfor Bond Valuation

PRICE

YIELD


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Call Provision

Issuer can refund if rates decline. Thathelps the issuer but hurts the investor.

Therefore, borrowers are willing to paymore, and lenders require more, on callablebonds.

Most bonds have a deferred call and adeclining call premium

Yield to call: yearly rate of return earned on

a bond until its called

C ll bl B d d Yi ld t


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Callable Bonds and Yield toCall

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.


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10 -1135.9 50 1050

N I/YR PV PMT FV

3.765 x 2 = 7.53%

INPUTS

OUTPUT

Nominal Yield to Call (YTC)


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If you bought bonds, would you bemore 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, henceYTC = 7.53%, not YTM = 8%.

I t t ll bl


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Investor returns on callablebonds

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.


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Whats a sinking fund?

Provision to pay off a loan over its liferather than all at maturity.

Similar to amortization on a term loan. Reduces risk to investor, shortens

average maturity.

But not good for investors if ratesdecline after issuance.

Si ki f d ll


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Sinking funds are generallyhandled in 2 ways

Call x% at par per year for sinkingfund purposes.

Call if rd is below the coupon rate and bondsells at a premium.

Buy bonds on open market.

Use open market purchase if rd is abovecoupon rate and bond sells at a discount.

Bond Ratings % defaulting within:


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Bond Ratings % defaulting within:

S&P and Fitch Moodys 1 yr. 5 yrs.

Investment grade bonds:AAA Aaa 0.0 0.0

AA Aa 0.0 0.1

A A 0.1 0.6BBB Baa 0.3 2.9

Junk bonds:

BB Ba 1.4 8.2B B 1.8 9.2

CCC Caa 22.3 36.9

Source: Fitch Ratings

Bond Ratings and Bond


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Bond Ratings and BondSpreads (YahooFinance, March 2009)

Long-term Bonds Yield (%) Spread (%)

10-Year T-bond 2.68

AAA 5.50 2.82AA 5.62 2.94

A 5.79 3.11

BBB 7.53 4.85BB 11.62 8.94

B 13.70 11.02

CCC 26.30 23.62


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Bond Ratings Median Ratios(S&P)

Interestcoverage

Return oncapital

Debt tocapital

AAA 23.8 27.6% 12.4%AA 19.5 27.0% 28.3%

A 8.0 17.5% 37.5%

BBB 4.7 13.4% 42.5%BB 2.5 11.3% 53.7%

B 1.2 8.7% 75.9%

CCC 0.4 3.2% 113.5%

Other Factors that Affect Bond


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Other Factors that Affect BondRatings

Provisions in the bond contract

Secured versus unsecured debt

Senior versus subordinated debt Guarantee provisions

Sinking fund provisions

Debt maturity

(More)


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Other factors

Earnings stability

Regulatory environment Potential product liability

Accounting policies

I t t t ( i ) i k f 1


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

i % 1-year Change 10-year Change

5% $1,048 $1,386

10% 1,0004.8%

1,00038.6%

15% 9564.4%

74925.1%

Interest rate risk: Rising mrkt i %

(rd) causes bonds price to fall.


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0

500

1,000

1,500

0% 5% 10% 15%

1-year

10-year

rd

Value

What is reinvestment rate


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What is reinvestment raterisk?

The risk that CFs will have to bereinvested at future lower rates,

reducing income. Illustration: Suppose you just won

$500,000 playing the lottery. Youll

invest the money and live off interest.You buy a 1-year bond with a YTM of10%.


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

If rates fall to 3%, income will dropfrom $50,000 to $15,000. Had youbought 30-year bonds, income would

have remained constant.


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The Maturity Risk Premium

Long-term bonds: High interest rate risk, lowreinvestment 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 theMRP is more affected by interest rate riskthan by reinvestment rate risk.


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Other types of Bonds

Zero coupon:

Pays no coupon & sells @ disct below par

Convertible: To stock @fixed price @ bondholders option

Income:

Pays interest only if interest earned by issuer;wont bankrupt co.

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Other types of Bonds

Revenue:

Interest paid from revenue generated by projectbeing financed by bonds

Floating rate:

Adjusts coupon rate periodically based onmarket interest rates

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Bankruptcy

Two main chapters of FederalBankruptcy Act:

Chapter 11, Reorganization Chapter 7, Liquidation

Typically, company wants Chapter 11,

creditors may prefer Chapter 7.


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If company cant meet its obligations, it filesunder Chapter 11. That stops creditors fromforeclosing, taking assets, and shutting downthe business.

Company has 120 days to file areorganization plan.

Court appoints a trustee to supervisereorganization.

Management usually stays in control.


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Company must demonstrate in itsreorganization plan that it is worth

more alive than dead. Otherwise, judge will order liquidation

under Chapter 7.

If the company is liquidated


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If the company is liquidated,heres the payment priority:

Past due property taxes Secured creditors from sales of secured assets. Trustees 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


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In a liquidation, unsecured creditors generallyget zero. This makes them more willing toparticipate in reorganization even though

their claims are greatly scaled back. Various groups of creditors vote on the

reorganization plan. If both the majority ofthe creditors and the judge approve,

company emerges from bankruptcy withlower debts, reduced interest charges, and achance for success.

Bonds Solution

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