Transcript Chapter 4 Understanding Interest Rates

Chapter 4
Understanding
Interest 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
• Why?
– A dollar deposited today can earn interest and
become $1 x (1+i) one year from today.
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Discounting the Future
L et i = .10
In one year $100 X (1+ 0.10) = $110
In tw o 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)
In n years
$100 X (1 + i )
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n
3
Simple Present Value
P V = to d ay's (p resen t) valu e
C F = fu tu re cash flo w (p aym en t)
i = th e in terest rate
PV =
CF
(1 + i )
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n
Time Line
•Cannot directly compare payments scheduled in different points in the
time line
Year
PV
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$100
$100
$100
$100
0
1
2
n
100
100/(1+i)
100/(1+i)2
100/(1+i)n
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Four Types of Credit Market
Instruments
•
•
•
•
4-6
Simple Loan
Fixed Payment Loan
Coupon Bond
Discount Bond
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Yield to Maturity
• The interest rate that equates the
present value of cash flow payments
received from a debt instrument with
its value today
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Simple Loan
P V = am o u n t b o rro w ed = $ 1 0 0
C F = cash flo w in o n e year = $ 1 1 0
n = n u m b er o f years = 1
$110
$100 =
(1 + i )
1
(1 + i ) $ 1 0 0 = $ 1 1 0
(1 + i ) =
$110
$100
i = 0 .1 0 = 1 0 %
F o r sim p le lo an s, th e sim p le in terest ra te eq u als th e
yield to m atu rity
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Fixed Payment Loan
T he sam e cash flow paym ent every period throughout
the life of the loan
L V = loan value
FP = fixed yearly paym ent
n = num ber of years until m aturity
LV =
FP
1+ i
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
FP
(1 + i )
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2

FP
(1 + i )
3
 ...+
FP
(1 + i )
n
Coupon Bond
U sing the sam e strategy used for the fix ed-paym ent loan:
P = price of coupon bond
C = yearly coupon paym ent
F = face value of the bond
n = years to m aturity date
P =
C
1+ i
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
C
(1+ i )
2

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C
(1+ i )
3
. . . +
C
(1+ i )
n

F
(1 + i )
n
Table 1 Yields to Maturity on a 10%Coupon-Rate Bond Maturing in Ten Years
(Face Value = $1,000)
• 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
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Consol or Perpetuity
• A bond with no maturity date that does not repay
principal but pays fixed coupon payments forever
P  C / ic
Pc  price of the consol
C  yearly interest
payment
i c  yield to maturity
of the consol
can rewrite above equation
as this : ic  C / Pc
For coupon bonds, this equation gives the current yield, an
easy to calculate approximation to the yield to maturity
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Discount Bond
For an y one year discount bo nd
i =
F-P
P
F = Face value of the discoun t bo nd
P = current price of the discoun t bo nd
T h e yield to m aturity equals the increase
in price over the year divided by the initial price.
A s w ith a co upon bond , th e yield to m aturity is
negativ ely related to the current b ond p rice.
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The Distinction Between Interest
Rates and Returns
• Rate of
Return:
T h e p ay m en ts to the ow ner plus the change in value
ex pressed as a fraction o f the purchase price
RET =
C
Pt
+
Pt 1 - Pt
Pt
R E T = retu rn fro m holding the bond from tim e t to tim e t + 1
Pt = price of bond at tim e t
Pt 1 = price of the b on d at tim e t + 1
C = coupon p aym ent
C
Pt
Pt 1 - Pt
Pt
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= current yield = ic
= rate of capital gain = g
The Distinction Between Interest
Rates and Returns (cont’d)
• The return equals the yield to maturity only if the
holding period equals the time to maturity
• A rise in interest rates is associated with a fall in
bond prices, resulting in a capital loss if time to
maturity is longer than the holding period
• The more distant a bond’s maturity, the greater the
size of the percentage price change associated with
an interest-rate change
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The Distinction Between Interest
Rates and Returns (cont’d)
• The more distant a bond’s maturity, the lower the
rate of return the occurs as a result of an increase
in the interest rate
• Even if a bond has a substantial initial interest
rate, its return can be negative if interest rates
rise
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Table 2 One-Year Returns on DifferentMaturity 10%-Coupon-Rate Bonds When
Interest Rates Rise from 10% to 20%
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Interest-Rate Risk
• Prices and returns for long-term bonds are
more volatile than those for shorter-term
bonds
• There is no interest-rate risk for any bond
whose time to maturity matches the holding
period
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The Distinction Between 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
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Fisher Equation
i  ir  
e
i = nom inal interest rate
ir = real interest rate

e
= expected inflation rate
W hen the real interest rate is low ,
there are greater incentives to borrow a nd few er incentives to lend.
T he real inter est rate is a better indicator of the in centives to
borrow and lend.
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Figure 1 Real and Nominal Interest Rates
(Three-Month Treasury Bill), 1953–2011
Sources: Nominal rates from www.federalreserve.gov/releases/H15 and inflation from
ftp://ftp.bis.gov/special.requests/cpi/cpia.txt. The real rate is constructed using the procedure outlined in Frederic S.
Mishkin, “The Real Interest Rate: An Empirical Investigation,” Carnegie-Rochester Conference Series on Public Policy 15
(1981): 151–200. This procedure involves estimating expected inflation as a function of past interest rates, inflation, and
time trends and then subtracting the expected inflation measure from the nominal interest rate.
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