EMBA 514Debt Valuation

and Interest Rates

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Example Bond

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Key Terminology for Bonds

1. Par or Face value Usually $1,000 per bond for corporates Principal amount underlying the loan, usually repaid with a

bullet maturity (interest-only payments during life of bond)

2. Coupon rate Nominal annual interest rate used to determine the cash

flow (coupon PMTs) on the bond Established contractually in advance of issuance Coupon rate x Face Value = total coupon PMT per year Coupons are typically paid twice per year

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Key Terminology for Bonds

3. Yield-to-maturity Interest rate used to discount the cash flows on the bond

to determine today’s price (‘rate’ in Excel) This is the market interest rate that changes continuously,

reflecting investors’ rate of return requirements YTM is the discount rate that equates the present value of

the bond’s cash flows with its price analogous to the Internal Rate of Return

Depends largely on maturity and default risk

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Key Terminology for Bonds

4. Maturity Number of years until the face value is returned to the

investor, along with the final coupon payment.

5. Default risk Reflected by credit rating Key determinant of the yield-to-maturity and price

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What’s the value of a 10-year, annual-pay coupon bond, if YTM = 10%, and coupon rate = 10%?

YTM YTM

$1,

1

000

11 10 10 . . . +

$100

1+YTM

100 100

0 1 2 1010%

100 + 1,000PV = ?

...

++++

$100

PMT = .10 x 1,000 =100; FV = 1,000; rate= .10; n = 10; ?PV 1,000

PMT = CR x Face Value 6

Relationship Between Price and YTM

1,210.71

1,000.00

837.21

7 10 13 YTM %

Price

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Bond value over time

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 (YTM ) remained at 10%, changed to 13%, or 7%?

Premium bond’s price (7% YTM )declines toward

$1,000 (face value) at maturity Discount bond’s price increases toward $1,000

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M

Bond Value ($)

Years remaining to Maturity

1,372

1,211

1,000

837

775

30 25 20 15 10 5 0

YTM = 7%.

YTM = 13%.

YTM = 10%.

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Solve for the YTM on a 10-year, 9% annual-pay coupon, that sells for $887 today

90 90 90

0 1 9 10YTM=?

1,000PV1 . . .PV10

PVM

887 Find YTM that ‘works’

...

PMT = 90; PV = -887; FV = 1,000; n = 10; ?rate 10.91% 10

If YTM > coupon rate, bond sells at a discount from face value.

If YTM = coupon rate, bond sells at its face value ($1,000).

If YTM < coupon rate, bond sells at a premium to face value.

If YTM rises, price falls.

Summary of price and YTM relationships

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Two forms of risk related to changing interest rates1. Price risk2. Reinvestment risk

YTM assumes the bond is held to maturity (no price risk) all coupons are reinvested at the YTM (no

reinvestment risk)

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Price risk

Exposure is to rising interest rates and inflation, since the investor is locked into a fixed rate of return on the bond

Price inflation will reduce the purchasing power of future cash flows on the bond

As a result the bond will sell for a lower price

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Maturity % Loss =

Bond Maturity and Price Risk for a 6% coupon bond, when YTM increases from 6% to 8%

1 981.48 1.85%

2 964.33 3.57

5 920.15 7.99

10 865.80 13.42

20 803.64 19.64

30 774.84 22.52

Price @8% priceOld

priceOldpriceNew

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M

Bond Value ($)

Years remaining to Maturity

1,372

1,211

1,000

837

775

30 25 20 15 10 5 0

YTM = 7%.

YTM = 13%.

YTM = 10%.

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Price risk increases with maturity Price risk is greater for longer maturities because the

investor is committed to the coupon rate for a longer period of time, and thereby suffers greater opportunity costs when interest rates rise.

Price risk is greater for lower coupon bonds since these have longer effective maturities.

This maturity effect is evident in the present value formula, where for larger ‘n’, the PV is more sensitive to changes in ‘i’

ni

FVPV

)1(

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Reinvestment rate risk

The risk that coupon PMTs will have to be reinvested in the future at lower rates, reducing income and ‘realized yield’ below the promised YTM.

This risk is greatest for higher coupon (more PMT to reinvest) and shorter maturities relative to the investment

period (have to reinvest face value when bond matures)

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Semiannual Bonds1. Multiply N by 22. Divide nominal YTM by 2 to get

semiannual discount rate.3. Divide coupon rate by 2 to get semiannual

PMT.

2n YTM/2 OK CR/2 OK N Rate PV PMT FV

INPUTS

OUTPUT

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Find the value of a 10-year, 8% coupon,semiannual bond if YTM = 6%.

Find the semiannual bond’s Effective Annual yield:

Or in Excel use the Effect function: Effect(.06,2)

PMT = .08/2 x 1,000 = 40; n = 10x2 = 20; rate = .06/2 = .03; FV = 1,000

?PV 1,148.77

0609.103.112

06.1 2

2

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See Excel Bond Pricing Functions

YTM; Price; Accrued Interest Slide 2 example =PRICE(“11/30/2004”, “7/1/2017”, .07125, .0056,100, 2)

purchase maturity coupon YTM FV freq

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YTM 5.56%

Price $114.00

Accrued Interest $2.95

Total dollars $116.95

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Treasury “Yield Curve” February 2013

Treasury Yield Curves 2006 and 2013

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Pure expectations theory of the yield curve Attributes the relationship between yields on

different maturity securities entirely to difference in expectations of future interest rates. Long-term interest rates are an average of expected

short-term interest rates. If this were not true, investors would buy the maturity

offering the higher expected yield, driving price up and yield down to equilibrium.

Thus, if PET is correct, you can use the current yield curve to forecast expected future interest rates.

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Implications of Pure Expectations Theory

Over any defined holding period, if PET forecast holds there is no sequence of investments that will produce lower borrowing costs than any other sequenceStrategically selecting maturities is a bet against

the forecast embedded in today’s yield curve

Long-term rates will be less volatile than short-term rates

The term structure of interest rates can be used as an indicator of the market’s forecast for the economy

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Liquidity premium theory Extends the pure expectations theory by

incorporating investor expectations of price risk (liquidity) in establishing market rates The pure expectations theory assumes that securities that

differ only in terms of maturity are perfect substitutes LPT claims if the expected return on a series of short-term

securities equals the expected return on a long-term security, investors will prefer the short-term securities

Investors with uncertain holding periods will prefer the short-term securities, since price volatility is lower (i.e. more liquid)

Borrowers in general will prefer longer-term securities, since they can fix their borrowing cost

Therefore borrowers must entice investors by offering higher rates on longer-term securities

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Liquidity premium theory

Long-term rates are an average of expected short-term rates, and liquidity premiums

The forward rate equals the expected rate plus a liquidity premium

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Yield curves and liquidity premiums

Calculated forward rates will exceed expected rates

The difference will be the amount of the liquidity premium

Pure Expectations

Including a liquidity premium

I

nte

rest

rat

eMaturity

The size of the liquidity premium will be positively

related to maturity

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Liquidity Premium Theory

LPT provides an explanation for the normally upward sloping yield curve. PET cannot explain this fact.

Implication of LPT for maturity selection Borrowing using long maturities will be more

expensive than a series of short maturities over long horizons

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http://www.clevelandfed.org/research/data/yield_curve/

Real versus Nominal Interest Rates

r = Real interest rate

h = Inflation premium

R = any Nominal rate

The nominal return includes the real return and an inflation premium:

Fisher relation: (1 + R) = (1 + r)(1 + h)

R = r + h + (r x h)

1)1(

)1(

h

Rr

R = r + h + MRP + DRP + LP

Here we add additional risk premia: R = Nominal required rate of return on a

debt security. r = Real short-term risk-free rate. h = Inflation premium.MRP = Maturity Risk premium (price risk)Stop here for YTM on Treasury Security

For corporate debt add:DRP = Default risk premium by credit rating LP = Liquidity premium

Nominal Interest Rates on Bonds

Bond Ratings Provide a Measureof Default Risk

Investment Grade Junk Bonds

Moody’s Aaa Aa A Baa Ba B Caa C

S&P AAA AA A BBB BB B CCC D

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BBB 10-year Credit SpreadsAbove Treasury 2003-13

High-yield credit spreads 1997-2013

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http://research.stlouisfed.org/fred2/graph/?s%5B1%5D%5Bid%5D=BAMLH0A0HYM2

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Interest Rate Swaps

An agreement to exchange a fixed (floating) rate payment for a floating (fixed) rate over some time periodProvides flexibility to alter debt agreements

without refunding or renegotiationMost important interest rate risk management

tool for financial institutions Counterparty is a large financial

intermediary (bank/investment bank) that makes a market in swaps

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Interest Rate Swaps

Interest payments are calculated off of ‘notional principal’; no principal is exchanged

Collateral is often required to guarantee performance

See Excel example

Interest Rate Swap

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LIBOR + 0.5%

DealerA BLIBOR LIBOR

6.25%5.80% 5.70%

Swap

Duration measures the interest rate risk of a security Duration measures how price sensitive a

security is to changes in interest rates. The greater (shorter) the duration, the greater

(lesser) the price sensitivity.

Duration incorporates the timing and size of all cash flows on the security.

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Duration Defined Measuring Duration

Macaulay’s Duration is a weighted average of the time until the expected cash flows from a security will be received

Macaulay’s Duration

)t(xSecurity theof Price

r)+(1CF

=Dt

tn

1=t

Weight x TimeThe weights are the present values of each of the

cash flows divided by the total present value (price)39

Duration Example What is the duration of a bond with a $1,000

face value, 6% annual coupon, 4 years to maturity and a 8% YTM?

Time Cash Flow PV Weight Weight*Time1 60.00 55.56 5.95% 0.062 60.00 51.44 5.51% 0.113 60.00 47.63 5.10% 0.154 1,060.00 779.13 83.44% 3.34

933.76 100.00% 3.66Mod Dur = 3.39Function 3.39

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Duration and Price Volatility

Comparing Price SensitivityThe greater the duration, the greater the price

sensitivity

ii)(1

Duration sMacaulay' -

P

P

i)(1

Duration sMacaulay'Duration Modified

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Duration and Price Volatility

Comparing Price SensitivityWith Modified Duration, we have an estimate

of price volatility:

i *Duration Modified - P

P Pricein Change %

Compare the actual price of the 4-year 6% bond with the Duration estimate if the YTM decreases by 1% to 7%.

Actual: 966.13

Duration estimate: 933.76(1.0339)=965.41

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Effect of contract terms on Duration Duration depends on:

1. The coupon rate.

2. The time to maturity.

3. The yield-to-maturity.

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Pictorial look at duration Cash flows of a seven year 12% bond discounted at

12%. Shaded area of each box is PV of cash flow

Distance (x-axis) is a measure of time

Time

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Effects of the coupon

Duration is the distance to the fulcrum (5.1 years)

DurationHigh C, Lower Duration

Low C, Higher Duration

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Effect of maturity and yield on duration

Duration increases with increased maturity

Effect of yield

yield, weight on earlier payments , fulcrum shifts left

yield, weight on earlier payments , fulcrum shifts right

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Excel Function

MDURATION("1/1/2000","1/1/2007",0.12, 0.12,1)

(purchase date, maturity, annual coupon rate, YTM, frequency of coupon)

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Maturity and Fixed vs. Floating Debt Considerations

Exposure to fluctuating interest rates

Exposure to fluctuating credit spreads

Market timing

Impact on investments

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Refinance Risk

If financing LT assets with ST debt, must refinance during life of assets Interest rate risk if rates rise Underinvestment if credit markets won’t refinance

risk is greatest for firms with less certain cash flows (growth/cyclical/high risk)

Suggest growth/cyclical/riskier firms should use LT debt, however, lenders prefer to have renegotiation option with these types of firms

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Repurchase risk

If financing ST assets with LT debt, must repurchase debt when assets mature (assuming no reinvestment), or carry excess cash Interest rate risk if rates fallOverinvestment in cash if debt not

repurchased

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Maturity Matching

Borrowers can mitigate interest rate risk and investment risk by matching the maturity (measured in cash flow timing) of assets and debt.

Lenders talk about measuring the ‘financing need’ and matching that need

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Fixed versus floating rate debt

For a given maturity, firms whose cash flows are more highly correlated with interest rates can reduce cash flow variability by using floating rate debt

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Credit Spread Risk

If firm possesses inside information regarding credit risk, and believes risk will fall in the future, then use more ST debtDon’t lock in high credit spread long-term Improved cash flows will mitigate refinance risk

Shorter maturity credit spreads are lower, but spreads are extremely volatile

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Evidence

Survey data on 392 financial execs shows: ‘matching assets and liabilities’ is the most

important factor affecting debt maturity ‘cost of refinancing in bad times’ is 2nd most

important factor Debt maturity increases as credit rating falls,

until the rating becomes speculative (BB or lower).

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Evidence

The 3rd most important factor determining maturity is execs claiming they issue ST debt when “ST rates are low compared to LT rates”, or “we are waiting for LT rates to come down”

ST borrowings, including borrowings tied to floating interest rates, are higher when the term spread—the difference between 10-year and 1-year interest rates—is high relative to the long-term mean

Demand for interest rate swaps that swap fixed payments for floating payments increases when the term spread increases

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Summary of Debt Maturity and Fixed/Floating Considerations

Interest rate risk Maturity matching

Impact on investments Bias toward LT/Fixed for growth/cyclical/high risk

Credit spread risk ST if lower risk than market expects, and vice versa

Market timing Assumes have superior forecast of short-term rates Generally cheaper to borrow S.T. or Floating Rate due to term premium

Cash flow cyclical with economy Appropriate to use more Floating Rate

Call provisions Issuer (borrower) can refund the bond if rates

decline. That helps the issuer but hurts the investor. Pay the ‘call premium’ to refund

Therefore, borrowers are willing to pay a higher YTM, and investors require higher YTM, on callable bonds.

Most callable bonds have deferred callability (3-5 years).

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Convertibility Debt convertible into specified number of shares of stock = the‘conversion ratio’ Market value includes ‘straight bond value’ plus the value of the call option on the stock Allows borrower the ability to issue stock at a less dilutive price if the stock price increases Provides downside protection to stock investor

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Convertibility

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