1

CHAPTER TWO: Time Value of Money and Term Structure of Interest

2

Yes ! is the expected rate of return, i.e., the mean of the discount rates for different terms

r

Let

No ! is the discount rate that cannot be used for so long period

?

01 )1(

Pr

CPV

n

tt

t

t

n

tt

t

tn

tt

t

r

C

r

C

11 )1()1(0)1(1

0

n

tt

t

r

CPNPV

r

Discounted Cash Flow Formula

3

Term Structure of Interest Rates

• Our objective is to value riskless cash flows.

• Given the rich set of fixed-income securities traded in the market, their prices provide the information needed to value riskless cash flows at hand.

4

Forms of Interest Rates

• In this market, this information on the time value of money is given in several different forms:– Spot interest rates– Price of discount bonds (e.g., zero-coupon bonds and

STRIPS)– Prices of coupon bonds– Yield-to-maturity (an average of spot interest rates)– Forward interest rates

• The form in which this information is expressed depends on the particular market.

5

Determination of Interest Rate

1. Capital production ability —— the more the capital’s expected return, the higher the interest rates and vice versa.

2. Uncertainty of capital production ability —— the more the uncertainty, the higher the risk premium required and the higher the interest rates and vice versa.

3. Time preference of consumption —— the stronger preference to current consumption, the higher the risk premium required and the higher the interest rates and vice versa.

4. Risk aversion —— the more the risk aversion, the higher the risk premium required and the lower the risk-free interest rates.

• Four basic factors

6

Theory of Real Interest Rates

• Real interest rates are determined by supply and demand of funds in the economy.

• 3 factors in determining real interest rates:– Aggregate endowments– Aggregate investment opportunities– Aggregate preferences for different

consumption path

7

• Consider a representative investor:– Has endowment of ( e0, e1)

– Faces a bond market with interest rate r.

8

• He maximizes his utility over his consumption now and later:

Where b is the bond holding, u’>0

and u”<0

0 1

0 0

1 1

max ( ) ( )

. .

(1 )

u c u c

s t c e b

c e r b

9

• The optimality condition is

0 1

1 0

0 0 0

1 0 0

'( ) (1 ) '( )

( )

'( ) "( )11 (1/ 1)

'( ) '( )

u c r u c

or for c c dc

u c c u c dcr

u c u c c

0 0

0 0

"( )1(1/ 1) [ ]

'( )

c u c dcr

u c c

Relative risk aversion coefficient

Thus, the real interest rate is given by

10

Nonlinear technology

Time 0

Time 1

0 1( , )c c

2u

1u

0e

1e -(1+RC)

Investment opportunity set

2 1u u

b

b(1+r)

11

Linear technology

Time 0

Time 1

0 1( , )c c

b

(1+r)b2u

1u

0e

1e

2 1u u

-(1+RC)

12

• More generally, consider consumption grow at random rate. Investors maximize their expected utility over many periods.

• Where is his holdings of discount bonds, is future endowments, is future consumption, both can be uncertain.

0

0 0 1 2

max [ ( )]

. . ...

(1 ) 1....

Tt

tt

T

tt t t

E u c

s t c e b b b

c e r b t T

1( , ...... )Tb b

1( , ...... )Te e

1( , ...... )Tc c

13

The Benchmark of Interest

— Yield to Maturity (YTM)

YTM varies with different financial instruments, because the exposure of financial instruments are quite different and the required risk premiums differ from each other.

Risk-free interest varies with terms . It’s called the term structure of interests.

— Risk-free interests

?

?

No!

Yes!

14

• Nominal and real interest rates— nominal interest rate = real interest rate+ inflation— real interest rate = pure time value+ risk premium

• Compound interest — interest earned on interest already earned

)1ln(* rr

— Continuously compoundingr — simple rate of return annuallym — times of interest payments annuallyr

m

1— compounding rate of interest payments annually

m

r

r

mm

m

1 11

m Let rr

me

m

m

r

lim 1 1 1

Continuously compounding

15

Financial Risks and Risk-free Security

— Default risk

— Liquidity risk

— Purchase power risk

— Interest risk

— Foreign exchange risk

— Other market risks

• Basic financial risks:

— Risk-free security:

• Substitute in reality: Treasury

16

— Treasury Yield Curve

• Treasury yield curve usually has three forms: upward, flat and downward.

ntrt ,,1,

• Zero-coupon rates set— Bills are zero coupon while notes and bonds have coupons. — Zero-coupon rates set can be obtained by conversion.

17

• Conversion example:

Treasury maturity par coupon rate current price

1 year

10%

0A

2 years

1,000

1,000

910.50

982.10

%83.950.910

50.91010001

r

22

221 )1(

1100

%83.91

100

)1(

1001000

1

10010.982

rrr

%08.112 r

B

18

— Shapes of Yield Curve

upward flat downward

• Some theories for the shapes of yield curve

— Unbiased expectations theory

— Liquidity preference theory

— Market segment theory

— Preferred habitat theory

19

115$%)151(100$ F

. One year’s risk-free rate

Forward Interest

— There is a no-dividend stock and its expected return is 15%. The current price is

A mini case:

100$0 S

%5fr . What is one year’s forward price of this stock?

?

20

Position Immediate Cash Flow Cash Flow in the Future

Replicating Stock Using risk-free bond and forward contract

Suppose forward price F = $106 per share

Short sell $100 risk-free bond

Short sell one stock forward at $106 per share

Buy one stock at $100 per share

+$100 $105

0 106 – S1

$100 S1

Net Cash Flow 0 $1

ArbitrageStock forward price= $105 per share

21

Forward price of a risky asset is not the expectation of the future spot price of the asset.

Proposition!

22

The Forward Price for a Traded Asset

• The forward price for a traded asset without interim income is: F=SerT

• The forward price for a traded asset with deterministic dividend rate is: F=Se(r-q)T

• The above equation can be obtained through the following arbitrage strategy:– Buy spot e-qT of the asset and reinvest income fro

m the asset in the asset.– Short a forward contract on one unit of the asset.

23

The Forward Price for a Traded Asset

• The holding of the asset grows at rate q so that e-qT x eqT ,or exactly one unit of the asset, is held at time T. Under the terms of the forward contract, the asset is sold for F at time T, leading to the following cash flow:

0 TSe-qT

FSe-qT=Fe-rT

F=Se(r-q)T

24

• Forward interest rates are the expectation of future risk-free spot interest rates.

0 1 2 3 n

— Zero-coupon rates & forward interest rates

)1)(1()1( 2112

2 frr

)1()1)(1)(1()1( 132211 nnn

n fffrr

25

• Zero-coupon rates, discount factors & forward interest rates

vr ttt

1

1

v

rt

t

t

1

1f

v

vmj

j

j

11 ,

rt vt

t

1 11

rvt

t

t 1

1

v0 1

vv

f

m

jj

j

111

Discount factors tv

Forward rates jf

Zero-coupon rates tr

26

Valuation of FRA

• An FRA is equivalent to an agreement where interest at a predetermined rate, RK, is exchanged for interest at the market rate, R.– Reference rate R

– Interest rate RK to be earned

– Time period between T1 and T2

– Notional amount L

27

Valuation Rule of FRA

• FRA has the cash flow: L(R- RK)(T2-T1) at T2

• An FRA can be valued by assuming that the forward interest rate is certain to be realized.– The value of the FRA promising RK is:

– L(RF -RK)(T2-T1)P(0,T2)

– P(0,T2) is the price of zero discount bond maturing at T2 with notional 1.

• Is there anything special about this rule?

28

FRA: Cash Flow Decomposition

Buying an FRA

Floating rate deposit

Starting t1 ending t2

Fixed rate Loan

Starting t1 ending t2

= +

29

FRA: Cash Flow Decomposition

30

Swap Price

— Interest rate swap

0 2 n t1

10 2 n t

Cash Flow of Buyer

Cash Flow of Seller

31

— Interest Rate Swap• Quotation for LIBOR

32

— Pricing Par Bonds

0)1()1(1

0

n

n

tt r

Par

r

iParPNPV

Par i Par i Par i

P0

Par

0 1 2 … n tParPthenriIf 0,

33

— Zero-coupon pricing technique

0 1 2 …. n t

Par

Par

Par*i Par*i Par*i

Par

0 1 2 … n tPar*f1

Par*f2

Par*fn

Par

0 1 2 … n t

Par*i Par*i Par*i

Par*f1

Par*f2Par*fn

Investment Cash Flow

Financing Cash Flow

nn

n

tt

t r

Par

r

iParParNPV

)1()1(1

0

n

tt

t

nn

r

ri

1 )1(1

)1(1

1

34

— Further illustration of composition & decomposition

0 1 2 … n t

Par*f1

Par*f2

Par*fn

Par

Par

0 1 2 … n t

Par*f1

Par*f2

Par*fn

+ 0 1 2 … n t

Par

Par

= 0

NPV = 0

0 1 2 … n t

Par

Par

=

• Decomposition of finance cash flow

35

Swap as Sequence of FRA

• Calculate forward rates for each of the LIBOR rates that will determine swap cash flows.

• Calculate swap cash flows on the assumption that the LIBOR rates will equal the forward rate.

• Set the swap value equal to the present value of these cash flows.

36

Swap Decomposition of FRAs

10 2 n t

1

2

3

37

Currency Swap

0 1 2 …. n t

P1

P1

i1 i1 i1

0 1 2 …. n tP2

P2

i2 i2 i2

• Fixed interest rate currency swap

0 1 2 …. n t

P1

i1 i1 i1

P2

i2 i2 i2

38

Pricing Currency Swap as Sequence of Currency Forwards

• Currency forward contract can be priced as if the forward price of the underlying asset is realized.

• Forward price for a foreign currency can be thought of as a stock with price S and paying dividend with known rate of foreign currency interest rate rf

• Forward price of a foreign currency is

S*exp((rd-rf)T)

Where rd is the interest rate for domestic currency, and rf is the interest rate for foreign currency.

39

Summary of Chapter Two

1. Time Value of Money Term Structure of Interest

2. Risk-free Rates are Benchmark and Market Expectation

3. Forward Price is not the Expectation of Future Spot Price for Risky Assets

4. Forward Price for traded asset

5. Replication Composition & Decomposition