Publish on 28th October 2019
Category: Birds
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Chapter 4

Bond Valuation

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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.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.

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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 higher returns, on callable bonds.

4

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

V

B

=

$100

$1,000

.

.

.

+

$100

100

100

0

1

2

10

10%

100 + 1,000

V = ?

...

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

+

+

(1 + rd)1

(1 + rd)N

(1 + rd)N

5

10 10 100 1000N I/YR PV PMT FV-1,000

$ 614.46385.54$1,000.00

PV annuityPV maturity valueValue of bond

===

INPUTS

OUTPUT

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

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When rdrises,abovethe coupon rate, the bond’s value fallsbelowpar, so it sells at a discount.

10 13 100 1000N I/YR PV PMT FV-837.21

INPUTS

OUTPUT

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

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What would happen if inflation fell, and rddeclined to 7%?

If coupon rate > rd, price rises above 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 Value ($) vs. Years remaining to Maturity

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

9

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. Years remaining to Maturity

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At maturity, the value of any bond must equal its par value.The value of a premium bond would decrease to $1,000.The value of a discount bond would increase to $1,000.A par bond stays at $1,000 if rdremains constant.

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What’s “yield to maturity”?

YTM is the rate of return earned on a bond held to maturity. Also called “promised yield.”It assumes the bond will not default.

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

90

90

90

0

1

9

10

rd=?

1,000

PV1...PV10PVM

887

...

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10 -887 90 1000N I/YR PV PMT FV10.91

V

INT

M

B

=

(1 + rd)1

(1 + rd)N

...

+

INT

887

90

(1 + rd)1

1,000

(1 + rd)N

=

+

90

(1 + rd)N

+

+

+

+

INPUTS

OUTPUT

...

Find rd

(1 + rd)N

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Coupon Rate vs. Yield to Maturity

If coupon rate < rd, bond sells at a discount.If coupon rate = rd, bond sells at its par value.If coupon rate > rd, bond sells at a premium.If rdrises, 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% > rd= 7.08% bond’s value > par.

10 -1134.2 90 1000N I/YR PV PMT FV7.08

INPUTS

OUTPUT

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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

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Cap gains yield = YTM - Current yield= 10.91% - 10.15%= 0.76%.

Could also find values (prices) in Years 1 and 2, get difference, and divide by value in Year 1. Same answer.

YTM = Current yield + Capital gains yield.

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Semiannual Bonds

1. 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.

2Nrd/2OK INT/2 OKN I/YR PV PMT FV

INPUTS

OUTPUT

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2(10) 13/2 100/220 6.5 50 1000N I/YR PV PMT FV-834.72

INPUTS

OUTPUT

Value of 10-year, 10% coupon, semiannual bond if rd = 13%.

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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.

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10 -1135.9 50 1050N I/YR PV PMT FV3.765 x 2 = 7.53%

INPUTS

OUTPUT

Nominal Yield to Call (YTC)=?

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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 = 7.5%, not YTM = 8%.

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

Here: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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Estimating IP

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

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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%.

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Source: Fitch Ratings

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Bond Ratings and Bond Spreads(December 2013)

Interest rate (or price) risk for1-year and 10-year 10% bonds

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What is reinvestment rate risk?

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

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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.

What is reinvestment rate risk?

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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.

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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

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Bankruptcy

Two main chapters of Federal Bankruptcy Act:Chapter 11, ReorganizationChapter 7, LiquidationTypically, company (stock holders) wants Chapter 11, creditors may prefer Chapter 7.

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Bankruptcy

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.

Assignment

Review slides of chapter 7 from FINC3131Chapter 4 problems:1,2,3,4,5,6,7,8,9,10,11,12,13,16

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Bonds - Georgia College & State University