Chapter 4
Understanding
Interest Rates
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Present Value （現值）
• 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
(未來值折現)
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
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Simple Present Value
(簡單現值)
PV = today's (present) value
CF = future cash flow (payment)
i = the interest rate
CF
PV =
n
(1 + i )
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Time Line
Cannot directly compare payments scheduled in different points in the
time line
Year
PV
$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
•
•
•
•
Simple Loan(簡單放款)
Fixed Payment Loan(固定償還放款)
Coupon Bond(附息債券)
Discount Bond(貼現債券)(零息債券)
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Yield to Maturity
(到期收(殖)益率)
• 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
PV = amount borrowed = $100
CF = cash flow in one year = $110
n = number of years = 1
$110
(1 + i )1
(1 + i ) $100 = $110
$100 =
$110
$100
i = 0.10 = 10%
(1 + i ) =
For simple loans, the simple interest rate equals the
yield to maturity
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Fixed Payment Loan
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
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Coupon Bond
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
1+i (1+i ) (1+i )
(1+i ) (1+i ) n
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4-10
Table 1 Yields to Maturity on a 10%Coupon-Rate Bond Maturing in Ten Years
(Face Value = $1,000)
• When the coupon bond (附息債券) is priced at its face value
(面額), the yield to maturity equals the coupon rate (票面利率)
• The price of a coupon bond and the yield to maturity are
negatively related
• The yield to maturity（市場殖利率） is greater than the coupon
rate (票面利率) when the bond price (債券價格) is below its face
value (面值)
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Consol or Perpetuity (永續債券)
• A bond with no maturity date(無到期日) that does
not repay principal(本金) but pays fixed coupon(固
定票息) payments forever
P  C / ic
Pc  price of the consol
C  yearly interest payment
ic  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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4-12
Discount Bond (貼現債券)
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.
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4-13
Following the Financial News:
Bond Prices and Interest Rates
28
32
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4-14
由利率換算成價格
公式：(台灣票券市場)
P0 
F
1 i 
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天數
365
應用：
• 國庫券初級市場標售
價格
• 國庫券、商業本票、
銀行承兌匯票等工具
的次級市場買斷或賣
斷的價格(注意利息所
得稅的計算)
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票券利率另一常用公式
公式： (台灣票券市場)
F  P0 365
i

F
天數
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由利率換算成價格
公式：
天數
P  F  Fi
365
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應用在商業本票的發行成本
• 貼現息=發行金額×市場利率×(天數/365)
• 保證費=發行金額×保證費率×(天數/365)
• 簽證費=發行金額×簽證費率×(天數/365)
• 承銷費=發行金額×承銷費率×(天數/365)
• 發行成本=貼現息+保證費+簽證費+承銷費
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應用在附買回協定
附買回價格
=原約定價格 ×〔1+利率×(天數/365) ×(1-20%)〕
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Rate of Return
The payments to the owner plus the change in value
expressed as a fraction of the purchase price
RET =
P -P
C
+ 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
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Rate of Return and Interest
Rates
• The return equals the yield to maturity only if the
holding period(持有期間) equals the time to
maturity(到期期間)
• A rise in interest rates(利率上升) is associated with a
fall in bond prices(債券價格下降), resulting in a
capital loss(資本損失) if time to maturity is longer
than the holding period
• The more distant a bond’s maturity(距到期日越長),
the greater the size of the percentage
price change(價格變動百分比) associated with an
interest-rate change(利率變量)
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Rate of Return and Interest
Rates (cont’d)
• The more distant a bond’s maturity(距到期日越遠),
the lower the rate of return(報酬率下降越多) the
occurs as a result of an increase in the interest
rate(利率上升會導致)
• Even if a bond has a substantial initial
interest rate, its return can be negative if interest
rates rise (利率與報酬率成反比)
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4-22
Table 2 One-Year Returns on Different-Maturity
10%-Coupon-Rate Bonds When Interest Rates
Rise from 10% to 20%
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4-23
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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4-24
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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4-25
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.
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4-26
FIGURE 1 Real and Nominal Interest Rates
(Three-Month Treasury Bill), 1953–2008
Sources: Nominal rates from www.federalreserve.gov/releases/H15. The real rate is constructed using the
procedure outlined in Frederic S. Mishkin, “The Real Interest Rate: An Empirical Investigation,” CarnegieRochester 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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