# Chapter 4 � Understanding Interest Rates

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```					Chapter 4 – Understanding
Interest Rates
Chapter 4 – Understanding Interest Rates

   Measuring interest rates
   Yield on a discounts
   The distinction between interest rates and returns
   The distinction between real and nominal rates
Measuring Interest Rates
   Present Value
   A dollar paid to you one year from now is less valuable
than a dollar paid to you today
   We can deposit money into a savings account and earn
interest
Simple Loan

   The lender provides the borrower with an amount
of funds (principal)
   The borrower repays the principal and an interest
payment at maturity
   Interest Rate:

Interest Payment
i
PrincipalAmount
Discounting the Future
Let i = .10
In one year \$100 x (1+ 0.10) = \$110
In two years \$110 x (1 + 0.10) = \$121
or 100 x (1 + 0.10) 2
In three years \$121 x (1 + 0.10) = \$133
or 100 x (1 + 0.10)3
In n years
\$100 x (1 + i) n
Discounting the Future

   The process of calculating today’s value of dollars
   If you are promised \$100 of cash flow for certain
10 years from now, it would not be as valuable
today because you could invest it and earn more
than the initial \$100.
Simple Present Value

PV = today's (present) value
CF = future cash flow (payment)
i = the interest rate
CF
PV =          n
(1 + i )
Four Types of Credit Market
Instruments
   Simple Loan
   Fixed Payment Loan (Fully Amortized Loan)
   The lender provides the borrower with funds that must be repaid
by making the same payment every period
   Coupon Bond
   Pays a fixed amount every year and then a specified final amount
(face or par value) is repaid at maturity
   Discount Bond
   Is bought below face value, at maturity receive the face value
Yield to Maturity

   The interest rate that equates the present value of
cash flow payments received from a debt
instrument with its value today
   It is considered the most accurate measure of
interest rates
Simple Loan

   For the case of the simple loan, calculating the
yield to maturity is the same as the present value

CF
PV =          n
(1 + i )
Simple Loan – Yield to Maturity
PV = amount borrowed = \$100
CF = cash flow in one year = \$110
n = number of years = 1
\$110
\$100 =
(1 + i )1
(1 + i ) \$100 = \$110
\$110
(1 + i ) =
\$100
i = 0.10 = 10%
For simple loans, the simple interest rate equals the yield to maturity
Fixed Payment Loan

   These loans are mortgages and other installment
type loans
   The borrower makes the same payment to the bank
every month
   To calculate yield to maturity we equate today’s
value of the loan to all future payments
Fixed Payment Loan – Yield to
Maturity
The same cash flow payment every period throughout
the life of the loan
LV = loan value
FP = fixed yearly payment
n = number of years until maturity
FP     FP       FP              FP
LV =             2
      3
 ...+
1 + i (1 + i) (1 + i)          (1 + i) n
Coupon Bond

   Not very popular today
   Got their name because the bond holder would have to
mail in a coupon to receive their payment
   The bond is calculated by adding up the present value
of all coupon payments and the face value
   A coupon bond is describe by it’s coupon rate
   i.e. if you get \$100 coupon and it has a \$1,000 face value it is
considered a 10% (100/1,000) coupon bond
Coupon Bond – Yield to Maturity

Using the same strategy used for the fixed-payment loan:
P = price of coupon bond
C = yearly coupon payment
F = face value of the bond
n = years to maturity date
C    C      C               C      F
P=         2
    3
. . . +      n
     n
1+i (1+i) (1+i)            (1+i) (1+i )
   When the coupon bond is priced at its face value, the yield to
maturity equals the coupon rate
   The price of a coupon bond and the yield to maturity are
negatively related
   The yield to maturity is greater than the coupon rate when the
bond price is below its face value
Consol or Perpetuity

    A bond with no maturity date that does not repay principal but pays
fixed coupon payments forever

Pc  C / ic
Pc  price of the consol
C  yearly interest payment
ic  yield to maturity of the consol

Can rewrite above equation as ic  C / Pc

For long term coupon bonds, this equation gives current yield,
an easy-to-calculate approximation of yield to maturity
Discount Bond

   U.S. Treasury Bills
   Yield to Maturity is calculated similarly to the
simple loan
   Thus, the interest rate is calculated by
CF
PV =
(1 + i )n
   Where PV is the purchase price today and CF is the face value
Discount Bond

For a one year discount bond
CF
PV 
1 i
PV  PV *(i )  CF
i     CF   - PV  / PV
where CFis the face value and PFis the price today
Discount Bond – Yield to Maturity

For any one year discount bond
F-P
i=
P
F = Face value of the discount bond
P = current price of the discount bond
The yield to maturity equals the increase
in price over the year divided by the initial price.
As with a coupon bond, the yield to maturity is
negatively related to the current bond price.
Bond Prices

   The bond price and interest rate are negatively
related
   As the interest rate increases
   Future coupon and final bond payments are worth less
when discounted back to the present
   The opportunity cost of buying the bond increases, thus
you pay less for the bond
Yield on a Discount Basis
Less accurate but less difficult to calculate
F-P         360
idb =     X
F    days to maturity
idb = yield on a discount basis
F = face value of the Treasury bill (discount bond)
P = purchase price of the discount bond
Uses the percentage gain on the face value
Puts the yield on an annual basis using 360 instead of 365 days
Always understates the yield to maturity
The understatement becomes more severe the longer the maturity
Yield on a Discount Basis

   Notice it uses the percent change of the face
value, YTM uses percent change of the price
F-P         360
idb =     X
F    days to maturity
   Secondly, it uses only 360 days not 365 days
   Thus it understates the interest rate
   The percent change is also smaller (price < face value)
   The denominator is larger under the yield on a discount
Rate of Return and Interest Rates

   The return equals the yield to maturity only if the holding period
equals the time to maturity
   An increase in interest rates is associated with a fall in bond
prices, resulting in a capital loss only if the time to maturity is
longer than the holding period
   i.e. don’t buy the bond when interest rates are expected to increase

   The more distant a bond’s maturity, the greater the size of the
percentage price change associated with an interest-rate change
Rate of Return

The payments to the owner plus the change in value expressed as
C    P -P
a fraction of the purchase price: RET =    + t 1 t
Pt      Pt
RET = return from holding the bond from time t to time t + 1
Pt = price of bond at time t
Pt 1 = price of the bond at time t + 1
C = coupon payment
C
= current yield = ic
Pt
Pt 1 - Pt
= rate of capital gain = g
Pt
Rate of Return and Interest Rates

   The more distant a bond’s maturity, the lower the rate of
return that occurs as a result of an increase in the interest
rate
   Even if a bond has a substantial initial interest rate, it’s
return can be negative if the interest rate rises
   If the bond is sold before the maturity date, the return on
the bond is different than the YTM
C    C       C                C          F
P=          2
     3
. . . +        n

1+i (1+i ) (1+i )            (1+i )     (1+i ) n
Interest-Rate Risk
   Prices and returns for long-term bonds are more
volatile than those for shorter-term bond
   Interest rate increases – hold short term bonds
   Interest rate decreases – hold long term bonds

   There is no interest-rate risk for any bond whose
time to maturity matches the holding period
Real and Nominal Interest Rates

   Nominal interest rate makes no allowance for inflation
   Real interest rate is adjusted for changes in price level
so it more accurately reflects the cost of borrowing
   Ex ante real interest rate is adjusted for expected
changes in the price level
   Ex post real interest rate is adjusted for actual changes
in the price level
Fisher Equation

i  ir   e
i = nominal interest rate
ir = real interest rate
 e = expected inflation rate
When the real interest rate is low, there are greater incentives
to borrow and fewer incentives to lend. The real interest rate is a better
indicator of the incentives to borrow and lend.
Returns

   So far we have only considered nominal returns
   A big reason for an increase in interest rates are due to
higher levels of inflation
   Recently, the U.S. has experienced negative real
interest rates.
   Only way to avoid inflation risk is to purchase U.S.
Treasury Bonds that are indexed to inflation.
   TIPS – Treasury Inflation Protected Securities

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 views: 10 posted: 1/25/2009 language: English pages: 32