Understanding Interest Rates
Chapter 4
Present Value: Discounting the Future
• Present Value (PV) - the value today of a
payment that is promised to be made in
the future
Discounting the Future
First need to Understand Future Value (FV)
• Let i = 0.10 (10 percent)
• In one year: $100 +(.10 x $100) = $100 x (1+.10) = $110
• In two years: $110+(.10 x $110) = $110 x (1+.10) = $121
OR
$100 x (1+.10)2 = $121
• In three years: $121+(.10 x $121)= $121x(1+.10) = $133
OR
• In n years:
$100 x (1+.10)3 = $133
FV = $100 x (1+ .10)n
Future Value
Future value in n years of an investment of
PV today at interest rate i
(i is measured as a decimal, 10% = .10)
FVn = PV x (1+i)n
Calculate one plus the interest rate (measured
as a decimal) raised to the nth power and
multiply it by the amount invested (present
value).
Future Value
Note:
When computing future value, both n and i must be
measured in same time units - if i is annual, then n
must be in years.
The future value of $100 in 18 months at 5%
annual interest rate is:
FV = 100 *(1+.05)1.5
Future Value
• The future value of $100 in one month at a 5%
annual interest rate is:
FV = $100 *(1+.05)1/12 = $100.4074
• (1+.05)1/12 converts the annual interest rates
to a monthly rate.
• (1+.05)1/12 = 1.004074, which converted to
percentage is 0.4074% or 0.41%(rounded)
• Note: 0.05/12 = .004167
Basis Point
• Faction of a percentage point is called basis
point.
• A basis point is one-one hundredth of a
percentage point
• One basis point (bp) = 0.01 percent.
• On the previous slide: 0.41% is 41 basis
points.
Discounting the Future
Present Value (PV) Reverses the FV Calculation
FVn = PV x (1+i)n
PV = FVn /(1+i)n
Present Value
Present Value
Example 1:
Present Value of $100 received in 5 years
discounted at an interest rate of 8%.
PV = $100 / (1.08)5 = $68.05
Example2:
• PV of $20,000 received 20 years from now discounted at
8% is:
PV = $20,000 / (1+0.08)20 = $20,000/ 4.6609 = $4,291
• Discounted at 9%:
PV = $20,000 / (1+0.09)20 = $20,000/5.6044 = $3,568
Present Value
Example3:
• PV of $20,000 received 19 years from now discounted at
8% is:
PV = $20,000 / (1+0.08)19 = $20,000/ 4.3157 = $4,634
In general, present value is higher:
1.
The higher the future value of the payment (CF).
2.
The shorter the time period until payment (n).
3.
The lower the interest rate. (i)
CF
PV 
(1  i ) n
Mishkin Discusses Four Types of Credit
Market Instruments
• Simple Loan
• Fixed Payment Loan (I will just mention
this)
• Coupon Bond
- Special case: consol bond
• Discount Bond
We will focus on Bonds and look at three
types of bonds
•
Coupon Bonds: which make periodic interest
payments and repay the principal at maturity.
•
U.S. Treasury Bonds and most corporate bonds are coupon
bonds.
•
Discount or Zero-coupon bonds: which promise a
single future payment, such as a U.S. Treasury Bill.
•
Consols: which make periodic interest payments
forever, never repaying the principal that was
borrowed. (There aren’t many examples of these.)
Yield to Maturity -YTM
• The interest rate that equates the present value
(PV) of cash flow payments (CF) received from a
debt instrument with its value today
CF
PV 
n
(1  i )
Given values for PV, CF and n, solve for i.
Yield to Maturity - One Year Simple Loan
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 - YTM
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
Bonds: Our Objectives
• Bond price is a present value calculation.
• YTM is the interest rate.
• Supply and Demand determine the price of
bonds.
- We will also discuss loanable funds and
money supply/money demand.
• Why bonds are risky.
Coupon Bond Price
A simple contract-
Coupon Bond Price
 C
C
C
C  FV
PB  


 ......

1
2
3
n
n
(
1

i
)
(
1

i
)
(
1

i
)
(
1

i
)
(
1

i
)


Present Value of Coupon Payments
Present Value of Principal Payment
Present Value of Coupon Bond (PB) =
Present value of Yearly Coupon Payments (C)
+ Present Value of the Face Value (FV),
where: i = interest rate and n = time to maturity
Example: Price of a n-year Coupon Bond
Coupon Payment =$100, Face value = $1,000,
and n = time to maturity
 $100 $100
$100  $1000
PB  

 ......

1
2
n
n
(1  i)  (1  i)
 (1  i) (1  i)
Given values for i and n, we can determine
the bond price PB
Definition: Coupon Rate = Coupon Payment / Face Value
Price of a 10-year Coupon Bond
If n = 10, i = 0.10, C = $100 and FV = $1000.
 $100
$100
$100  $1000
PB  $1000 

 ......

1
2
10 
10
(
1

.
1
)
(
1

.
1
)
(
1

.
1
)
(
1

.
1
)


What’s the Coupon Rate?
Price of a 10-year Coupon Bond
If n = 10, i = 0.12, C = $100 and FV = $1000.
 $100
$100
$100 
$1000
PB  $887  

 ......

1
2
10 
10
(
1

.
12
)
(
1

.
12
)
(
1

.
12
)
(
1

.
12
)


What’s the Coupon Rate?
Coupon Bond - YTM
Suppose n = 10, PB = $950, C = $100 and FV = $1000.
 $100 $100
$100  $1000
PB  $950  

 ......

1
2
10 
10
(1  i)  (1  i)
 (1  i) (1  i)
What’s the Coupon Rate?
What’s the YTM?
YTM = .1085 or 10.85%
Using the Approximation Formula
Previous example:
n = 10, PB = $950, C = $100 and FV = $1000.
Yearly Coupon Payment
Current Yield 
Price Paid
For our 10-year bond selling at $950:
 Coupon rate = 10%
 YTM = 10.85%
 Current Yield = 10.52%
• 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 – Special Case Coupon Bond
• Infinite maturity
• No face value.
• Fixed coupon payment of C forever.
• Pconsol = C/(1+i) + C/(1+i)2 + C/(1+i)3 + … + C/(1+i)t
• As t goes to infinity this collapses to:
PConsol = C / i => i = C / P
$2,000 = $100/.05
Discount or Zero Coupon Bond
•
Definition: A discount bond is sold at some price P, and pays
a larger amount (FV) after t years. There is no periodic
interest payment.
Let P = price of the bond, i= interest rate, n = years to
maturity, and FV = Face Value (the value at maturity):
P
FVn
(1 i ) n
Zero Coupon Bonds - Price
Examples: Assume i=4%
Price of a One-Year Treasury Bill with FV = $1,000:
1000
PB 
 $961.53
(1  0.04)
Price of a Six-Month Treasury Bill with FV = $1,000:
1000
PB 
 $980.58
1/ 2
(1  0.04)
Price of a 20-Year zero coupon bond at 8% and FV =
$20,000:
$20000
PB 
 $4,290.96
20
(1  0.08)
YTM - Zero Coupon Bonds
P
FV
(1i )
 FV 
 FV 
(1  i )  
 1 i  

 P 
 P 
n
n
(1 / n )
 FV 
i

 P 
(1 / n )
1
Zero Coupon Bonds - YTM
• For a discount bond with FV = $15,000 and P = $4,200,
and n = 20, the interest rate (or yield to maturity) would
be:
(1 / n )
FV


i
1

 P 
 15,000
i 

 4,200 
(1/ 20 )
1
i = 1.0657 -1=> i =6.57%
Note: This is the formula for compound annual rate of growth
Zero Coupon Bonds - YTM
• For a discount bond with FV = $10,000 and P = $6,491,
and n = 7, the interest rate (or yield to maturity) would be:
 10,000
i

 6,491 
(1 / 7 )
1
i = 1.06368 – 1 = .06368 or 6.368%
From a Coupon Bond to Zero Coupon Bonds
(called Strips)
 C
C
C
C  FV
PB  


 ......

1
2
3
n
n
(1  i)  (1  i)
 (1  i) (1  i) (1  i)
Create n+1 discount bonds
Current Yield
Two Characteristics of Current Yield
1. Is a better approximation of yield to maturity, nearer price is to
par (face value) and longer is maturity of bond
2. Change in current yield always signals change in same direction
as yield to maturity
Yield on a Discount Basis
SKIP
Distinction Between Interest Rates and
Return
The payments to the owner plus the change in value
expressed as a fraction of the purchase price
P -P
C
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
Holding Period Return
• the return from holding a bond and selling it before
maturity.
• the holding period return can differ from the yield
to maturity.
Holding Period Return
Example:
• You purchase a 10-year coupon bond, with a 10%
coupon rate, at face value ($1000) and sell one year
later.
RET(t t 1)
C ( Pt 1  Pt ) C  Pt 1  Pt
 

Pt
Pt
Pt
Current
Yield
Capital Gain
Holding Period Return
If the interest rate one year later is the same at
10%:
One year holding period return =
$100 $1000 $1000 $100


 .10
$1000
$1000
$1000
or 10.0%
Holding Period Return
If the interest rate one year later is lower at 8%:
One year holding period return =
$100 $1125 $1000 $225


 .225
$1000
$1000
$1000
or 22.5%
Holding Period Returns
If the interest rate in one year is higher at 12%:
One year holding period return =
$100 $893 $1000  $7


 .007
$1000
$1000
$1000
or -.70%
You need to know how to calculate the $1125 and $893
Key Conclusions From Table 2
• The return equals the yield to maturity (YTM)
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 the
holding period is less than the time to maturity
• The more distant a bond’s maturity, the greater
the size of the percentage price change associated
with an interest-rate change
Key Conclusions From Table 2
• 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, its return can be negative if
interest rates rise
Interest-Rate Risk
• Change in bond price due to change in interest
rate
• Prices and returns for long-term
bonds are more volatile than those for
shorter-term bonds
• There is no interest-rate risk for a bond whose
time to maturity matches the holding period
Reinvestment (interest rate) Risk
• If investor’s holding period exceeds the term to maturity
 proceeds from sale of bond are reinvested at new
interest rate
 the investor is exposed to reinvestment risk
• The investor benefits from rising interest rates, and suffers from
falling interest rates
Real and Nominal Interest Rates
• Nominal interest rate (i) makes no allowance
for inflation
• Real interest rate (r) 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 (πe)
• Ex post real interest rate is adjusted for actual changes
in the price level (π)
Real and Nominal Interest Rates
Fisher Equation:
i = r + πe
From this we get rex ante = i - πe
rex post = i - π
Inflation and Nominal Interest Rates
Mankiw
Inflation and Nominal Interest Rates
Real and Nominal Interest Rates
Real Interest Rate:
Interest rate that is adjusted for expected changes in the
price level
r = i - πe
1.
2.
Real interest rate more accurately reflects true cost of
borrowing
When real rate is low, greater incentives to borrow and less to
lend.
if i = 5% and πe = 3% then:
r = 5% - 3% = 2%
if i = 8% and πe = 10% then
r = 8% - 10% = -2%
A measure in inflationary expectations
i = r + πe
πe = i - r
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