Chapter 4 Interest Rates and Rates of Return

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Chapter 4Interest Ratesand Rates of Return
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Types of Debt Instruments

• Simple loans e.g., commercial paper
• Discount bonds e.g., Treasury bills, zero-coupon
  bonds
• Coupon bonds e.g., most Treasury bonds and
  corporate bonds
• Fixed-payment loans e.g., mortgages, installment
  debt, annuities
• Differ in how and when repayment occurs

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Simple Loans

• P is borrowed and F is repaid
• P is the principal
• F-P is the interest
• The interest rate on a simple loan is given by i
  (F-P)/P

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Discount Bonds

• Identical to simple loans except
• Sold at discount
• P is lent, F is repaid, interest F-P (increase
  in price from when sold until maturity date)
• Interest rate is given by i (F-P)/P

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Compounding

• A simple loan or a discount bond for n periods
• P is borrowed, and F is repaid n periods later.
• n is the term to maturity.
• The interest rate is given by i (F/P)1/n 1.
• Reason
• P becomes P(1i) one period hence, P(1i)2 two
  periods hence, , F P(1i)n n periods hence.
• Solving for i gives the formula above.

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EG Simple Loan/Discount Bond

• 1,000 is borrowed for 5 years (term to maturity)
• Repayment is 1610.51 five years from now
• Interest rate is 610.51/1000 61.051 percent per
  5-year period or 10 per year (1610.51/1000)1/5
  1 0.10

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Coupon Bonds

• P is borrowed
• A fixed coupon payment C is made every period
  until the maturity date
• At the maturity date, principal of F as well as a
  coupon payment of C is paid
• The coupon payments are considered to be interest

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Example Coupon Bond

• 1000 is borrowed
• Coupon payment is 100 each year
• Principal is 1000
• Term to maturity is 5 years

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Fixed-Payment Loan

• P is borrowed
• Every period, interest and some principal are
  paid
• Payment is a constant amount C each period
• At maturity, all of the principal has been repaid

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Example Fixed-Payment Loan

• 1000 is borrowed
• 263.80 in interest and principal is paid each
  year for 5 years
• The term to maturity is 5 years

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Comparison
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Pricing Future

• Future differ from present .
• Prices of future in terms of present are
  observed when they trade for each other in
  financial markets.
• Whenever such trades take place, the present
  traded the price of future in terms of
  present ? the future traded.
• So present and future are like everything else
  traded in markets.

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Present- Prices of Future

• Suppose that everyone could borrow and lend at
  the constant annual interest rate i.
• So
• 1 now yields (1i) 1 year hence
• 1 now yields (1i)2 2 years hence
• 1 now yields (1i)3 3 years hence
• 1 now yields (1i)n n years hence

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Reversing This

• 1/(1i) now yields 1 1 year hence
• 1/(1i)2 now yields 1 2 years hence
• 1/(1i)3 now yields 1 3 years hence
• 1/(1i)n now yields 1 n years hence
• Result The price of n years hence in terms of
  present 1/(1i)n
• The units of this price are now/ n years
  hence.

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Present Value

• Present value (PV) of any given future value (FV)
  is the amount that it exchanges for in the
  financial markets.
• PV (the price at which future n years hence
  trade in terms of present ) ? (FV n years
  hence).
• If the interest rate is constant at i, PV of FV1
  one year hence, FV2 2 years hence, FV3 3 years
  hence, FV1/(1i) FV2/(1i)2 FV3/(1i)3
  
• Note that FVs at different times are made
  comparable by multiplying by the prices 1/(1i)n,
  n 1, 2, 3, .

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Example

• Consider an investment that pays 100 next year,
  200 2 years hence, 300 3 years hence.
• For how much would it sell now if the interest
  rate is 5 percent per annum?
• Answer 100/1.05 200/1.052 300/1.053
  535.80.

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Pricing Simple Loansand Discount Bonds

• P PV of repayment F n periods hence.
• So P F/(1i)n
• Solving gives
• i (F/P)1/n - 1

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Pricing Coupon Bonds

• P PV of C for each of the next n periods PV
  of F n periods hence
• P C/(1i) C/(1i)2 C/(1i)n
• F/(1i)n
• PVs are additive just as the amounts spent on any
  other goods purchased in markets e.g., spent
  on apples and oranges

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Pricing Fixed-Payment Loans

• P PV of C for each of the next n periods
• P C/(1i) C/(1i)2 C/(1i)n

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Simpler Formulae

• Fixed-payment loan
• P (C/i)1-1/(1i)n
• Coupon bond
• P (C/i)1-1/(1i)n F/(1i)n

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Perpetuities

• Some British and other government debt is
  perpetual i.e., the maturity date is essentially
  infinity
• In that case, P C/i so that i C/P
• Perpetuities are commonly called consols

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Bond Yields and Prices

• Simple loans/discount bonds P F/(1i)n
• Fixed-payment loans
• P C/(1i) C/(1i)2 C/(1i)n
• Coupon bonds
• P C/(1i) C/(1i)2 C/(1i)n
• F/(1i)n
• Consol P C/i
• i is called the yield to maturity or just yield
  for short

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Price Varies Inversely with Yield

• Increasing i reduces every PV in the right-hand
  side of these pricing equations and hence reduces
  P
• Discount Bond P F/(1i)n
• Fixed-Payment Bond P C/(1i) C/(1i)2
  C/(1i)n
• Coupon Bond P C/(1i) C/(1i)2
  C/(1i)n F/(1i)n
• Consol P C/i

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Prices and Face Values

• P and F need not be equal and indeed typically
  arent
• Issuers of bonds usually set C and F so that P
  approximates F initially
• Any subsequent changes in i lead P to differ from
  F

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Example

• 10-year bond, F 100K, C/F i 4/yr so that
  C 4K. Result P 100K F initially
• P (4K/.04)(1-1/1.0410)100K/1.0410 100K
• One year later i 5/yr and
• P (4K/.05)(1-1/1.059)100K/1.059 92.892K
• P lt F because new 9-year bonds with i 5/yr and
  F 100K must pay 5K/yr to have P 100K.
• An old 9-yr bond paying only 4K/yr is not as
  attractive and must sell for a lower P

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Properties of Coupon Bonds

• C/F is called the coupon rate
• If i coupon rate, P F
• If i gt coupon rate, P lt F
• If i lt coupon rate, P gt F
• Holding C and F constant,
• The lower i is, the higher P is
• The larger n is, the more i affects P
• For n-period discount bond, ?P/P ? n?i/(1i)

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Rate of Return

• On average, you obtain a rate of return equal to
  the yield to maturity if you hold a debt
  instrument until the maturity date
• If you sell it before then, you may receive a
  higher or lower rate of return
• The difference results from capital gains or
  losses

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Calculating the Rate of Return

• The one-period rate of return is calculated by
  summing the coupon payment and any capital
  gainswhether positive or negativeand dividing
  by initial price
• Let C be the coupon payment and P0 and P1 be the
  initial and final prices. The one-period rate of
  return is given by
• R C (P1 P0)/P0

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Reprising Previous Example

• R 4 (92.892-100)/100 -3.11
• Poor return after the fact could have earned 4
  by holding one-year bond
• 10-year bonds are riskier than one-year bonds for
  those holding for only one year
• Rate of return on bond is -3.11 whether it is
  sold or not (opportunity cost same)

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Facts about Rate of Return

• Rate of return (or return for short) yield to
  maturity only when a bond is held to maturity
• When interest rates rise, return falls and vice
  versa
• Reason bonds prices fall resulting in capital
  losses
• The effect is larger, the longer the term to
  maturity is

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Real and Nominal Interest Rates

• Most bonds repay fixed amounts of money in the
  future and are bought with money now.
• No one wants money for its own sake but only for
  the goods that it buys.
• Money now buys goods only now, and money in the
  future buys goods only then.

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The Economic Reality

• Savers paying money for a bond now are really
  giving up goods now in return for goods they will
  buy with the money that they will receive in the
  future.
• Borrowers receiving money for a bond now are
  really receiving goods now in return for the
  goods they wont buy with the money they must
  repay in the future.

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Pricing Future Goodsin Terms of Present Goods

• Present and future goods are being implicitly
  exchanged in financial markets at a price.
• The price of goods 1 year hence in terms of
  present goods is 1/(1r), where r is the one-year
  real interest rate.
• So
• present goods (goods 1 year hence)/(1 r)

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Real Interest Rates

• The real interest rate is the additional goods
  that borrowers must pay, and are willing to pay,
  in the future for being able to have goods now.
• Real interest rates can be calculated to a first
  approximation by subtracting the expected
  inflation rate ?e from the nominal interest rate
  i
• r i ?e
• The real interest rate fluctuates around a
  constant value over time.

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Example

• Nominal interest rate 5/yr expected inflation
  rate 3/yr real interest rate 2/yr.
• Saver gets 5 more money next year but expects
  prices to be 3 higher. Result saver expects to
  receive 2 more goods next year.
• Borrower repays 5 more money next year but
  expects money to cost 3 fewer goods. Result
  borrower expects to repay 2 more goods next year.

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Inflation and Nominal Interest Rates Move Together
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Fluctuation of the Real Interest Rate