INTRODUCTION TO FINANCIAL ENGINEERING

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CHAPTER TWO Time Value of Money and Term
Structure of Interest
2
Discounted Cash Flow Formula
?
No ! is the discount rate that cannot be
used for so long period
Let
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.

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Determination of Interest Rate

• Four basic factors

• Capital production ability the more the
  capitals expected return, the higher the
  interest rates and vice versa.
• Uncertainty of capital production ability the
  more the uncertainty, the higher the risk premium
  required and the higher the interest rates and
  vice versa.
• 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.
• Risk aversion the more the risk aversion, the
  higher the risk premium required and the lower
  the risk-free interest rates.

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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, ugt0
• and ult0

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• The optimality condition is

Thus, the real interest rate is given by
Relative risk aversion coefficient
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Nonlinear technology
Time 1
b(1r)
-(1RC)
b
Time 0
Investment opportunity set
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Linear technology
Time 1
(1r)b
-(1RC)
b
Time 0
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.

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The Benchmark of Interest
Yield to Maturity (YTM)
?
No!
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 interests
?
Yes!
Risk-free interest varies with terms . Its
called the term structure of interests.
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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

Continuously compounding
simple rate of return annually
times of interest payments annually
compounding rate of interest payments
annually
Let
Continuously compounding
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Financial Risks and Risk-free Security

• Basic financial risks

• Default risk
• Liquidity risk
• Purchase power risk
• Interest risk
• Foreign exchange risk
• Other market risks

Risk-free security

• Substitute in reality Treasury

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Treasury Yield Curve

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

• Zero-coupon rates set

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

• Treasury maturity par coupon rate current
  price

1 year
0
A
1,000
910.50
10
2 years
1,000
982.10
B
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Shapes of Yield Curve
downward
upward
flat

• Some theories for the shapes of yield curve

Unbiased expectations theory Liquidity
preference theory Market segment theory
Preferred habitat theory
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Forward Interest
A mini case

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

. One years risk-free rate

. What is one years
forward price of this stock?
?
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Suppose forward price F 106 per share
Replicating Stock Using risk-free bond and
forward contract
Position Immediate Cash Flow Cash Flow
in the Future
Short sell 100 risk-free bond
? 105
100
Short sell one stock forward at 106 per share
0
106 S1
Buy one stock at 100 per share
? 100
S1
Net Cash Flow
0
1
Stock forward price 105 per share
Arbitrage
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Proposition!
Forward price of a risky asset is not the
expectation of the future spot price of the asset.
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The Forward Price for a Traded Asset

• The forward price for a traded asset without
  interim income is FSerT
• The forward price for a traded asset with
  deterministic dividend rate is FSe(r-q)T
• The above equation can be obtained through the
  following arbitrage strategy
• Buy spot e-qT of the asset and reinvest income
  from the asset in the asset.
• Short a forward contract on one unit of the
  asset.

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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

Se-qTFe-rT
FSe(r-q)T
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Zero-coupon rates forward interest rates

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

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

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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

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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?

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FRA Cash Flow Decomposition
Floating rate deposit Starting t1 ending t2
Buying an FRA
Fixed rate Loan Starting t1 ending t2


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FRA Cash Flow Decomposition
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Swap Price
Interest rate swap
Cash Flow of Buyer
Cash Flow of Seller
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Interest Rate Swap

• Quotation for LIBOR

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Pricing Par Bonds
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Zero-coupon pricing technique
Investment Cash Flow




Financing Cash Flow

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Further illustration of composition
decomposition
NPV 0

• Decomposition of finance cash flow

0


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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
37
Currency Swap

• Fixed interest rate currency swap

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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
• Sexp((rd-rf)T)
• Where rd is the interest rate for domestic
  currency, and rf is the interest rate for foreign
  currency.

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Summary of Chapter Two

• Time Value of Money ? Term Structure of Interest
• Risk-free Rates are Benchmark and Market
  Expectation
• Forward Price is not the Expectation of Future
  Spot Price for Risky Assets
• Forward Price for traded asset
• Replication ? Composition Decomposition