Valuing risky debt

The story teller makes no choice, soon you will not hear his voice. His job is to shed light and not to

master. – Garcia, Hunter

Debt & Interest RatesDebt & Interest Rates

Classical Theory of Interest Rates (Economics)Classical Theory of Interest Rates (Economics)

developed by Irving Fisher:developed by Irving Fisher:

Supply

Demand

$ Qty

r

Real r

Real Interest Rate = The theoretical rate (absent inflation) that you pay when you borrow money, as determined by supply and demand.

Debt & Interest RatesDebt & Interest Rates Nominal Interest Rate = The rate you actually pay when you borrow money.Nominal Interest Rate = The rate you actually pay when you borrow money.

Relationship between nominal rate, inflation, and real rate: Relationship between nominal rate, inflation, and real rate:

)1)(1()1( realnominal irr

Global Inflation RatesGlobal Inflation Rates

0.00

2.00

4.00

6.00

8.00

10.00

12.00

Ave

rag

e In

flat

ion

, %

Switzer

land

Nether

lands

USA

Canada

Sweden

Norway

Austra

lia

Denmar

kUK

Irelan

d

South

Afri

ca

Avera

ge

Germ

any (

ex 192

2/23)

Belgium

Spain

Franc

e

Japa

nIta

ly

Averages from 1900-2006

The Term Structure of Interest Rates

• Shows the relationship between interest rates (spot rates) and time to maturity.

• A graph of the term structure is known as the yield curve.

• The Term Structure tells us the cost of debt for The Term Structure tells us the cost of debt for various maturities.various maturities.

Term StructureTerm Structure

Spot RateSpot Rate - The actual interest rate today (t=0) - The actual interest rate today (t=0)

Forward RateForward Rate - The interest rate, fixed today, on a loan - The interest rate, fixed today, on a loan made in the future at a fixed time.made in the future at a fixed time.

Future RateFuture Rate - The spot rate that is expected in the future - The spot rate that is expected in the future

Yield To Maturity (YTM)Yield To Maturity (YTM) - The IRR on an interest bearing - The IRR on an interest bearing instrument instrument

YTM (r)

Year

1981 & 1987 Normal

19761 5 10 20 30

Spot rates• n-year spot rate = rate market uses to value a single payment n years hence

• Example:

Value of single payment of 10 in 3 years time:

10 , where r3 = 3-year spot rate

(1 + r3)3

• Value of 3-year bond with annual coupon of 10:

10 10 110

(1 + r1) (1 + r2)2 (1 + r3)3

• Think of a spot rate as the yield on a zero-coupon bond

+ +

What are forward rates?• Forward Rates: are rates from investing additional time periods.

- Forward rates are implicit in spot rates:

(1 + r2)2 = (1 + r1)(1 + f2)

• The forward rate for year 2 = f2 = (1 + r2)2 (1 + r1)1

- 1

Bond Values

• Bond prices are found by calculating the Bond prices are found by calculating the present value of the cash flows from the bond present value of the cash flows from the bond at the corresponding spot rate for each cash at the corresponding spot rate for each cash flow. flow. - Previously, we assumed a flat yield curve Previously, we assumed a flat yield curve

(constant spot rates in our bond calculations.)(constant spot rates in our bond calculations.)

nn

n

tt

t

tBond

r

Face

r

INTV

)1()1(1

Yield to Maturity

• Is the estimated IRR from investing in a bond and holding it to maturity. It is a complex average of the spot rates. Yields measure expected return only if coupons are reinvested to earn yield.

• Like IRRs, yields to maturity do not add up.• If know the yield to maturity, you can use it to

calculate bond values.

n

n

tt

tBond

y

Face

y

INTV

)1()1(1

ConvexityConvexity

• Convexity refers to the fact that bond price changes Convexity refers to the fact that bond price changes are not symmetric with changes in interest rates are not symmetric with changes in interest rates (yields).(yields).• As yields fall, prices rise at an increasing rate.As yields fall, prices rise at an increasing rate.• As yields rise, prices fall at a decreasing rate.As yields rise, prices fall at a decreasing rate.

ValueValue

YieldYield

3% 4% 5% 6% 7% 8% 9%

10%

11%

12%

12

35

710

2030

0.00

50.00

100.00

150.00

200.00

250.00

yield

maturity

Value of $100 invested at initial yield =6%

Value of investment in zero-coupon bond

3%

5%

7%

9%

11%

12

35

710

2030

50.00

75.00

100.00

125.00

150.00

175.00

yield maturity

6% coupon bond price

Coupon bonds

Classical Duration

n

ttClassical PVtD

1

)(%

• Classical Duration weighs the percentage of value Classical Duration weighs the percentage of value received by the time it is received.received by the time it is received.

• Where %PVWhere %PVtt = PV = PVtt / Bond Value / Bond Value

Duration is a measure of Interest Rate Risk.

Duration CalculationDuration Calculation

Year Ct PV(Ct) at 5.0%Proportion of Total Value

[PV(Ct)/V]Proportion of Total

Value Time

1 100 95.24 0.084 0.0842 100 90.7 0.08 0.163 1100 950.22 0.836 2.509

V = 1136.16 1 Duration= 2.753 years

1000 Face value 10% coupon bond with 3 years left to maturityand 5% yield.

DurationDuration

Year CF PV@YTM % of Total PV % x Year

1 68.75 65.54 .060 0.060

2 68.75 62.48 .058 0.115

3 68.75 59.56 .055 0.165

4 68.75 56.78 .052 0.209

5 1068.75 841.39 .775 3.875

1085.74 1.00 Duration 4.424

Example (Bond 1)Example (Bond 1)

Calculate the duration of our 6 7/8 % bond @ 4.9 % Calculate the duration of our 6 7/8 % bond @ 4.9 % YTMYTM

Duration Considers The Magnitude and Timing of Cash Flows

• What is the Duration of a zero coupon paying bond?What is the Duration of a zero coupon paying bond?• All else being equal, is Duration larger or smaller for All else being equal, is Duration larger or smaller for

long term versus short term bonds?long term versus short term bonds?• All else being equal, is Duration larger or smaller for All else being equal, is Duration larger or smaller for

bonds that pay a high coupon rate versus those that bonds that pay a high coupon rate versus those that pay a low coupon rate?pay a low coupon rate?

Modified Duration

• Modified Duration is often employed in estimating a change in bond prices for a change in yields.

Where:Where:

Dmodified = DClassical / (1+ y)

Change in bond price:

yDVV ModifiedinitialBondBond )()(

This is a linear approximation to actual changes.

1 2 3 4 5 6 7 8 9 10

pre

dic

ted

10

-ye

ar

10

pre

dic

ted

20

-ye

ar

20

pre

dic

ted

30

-ye

ar

30

-100.00

-50.00

0.00

50.00

100.00

150.00

200.00

250.00

yield

maturity

Actual vs duration-predicted value of $100 invested in zero-coupon purchased at 6% yield