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Understanding interest rates
The Economics of Money, Banking, and Financial Markets Mishkin, 7th ed.
Chapter 4
Present value
The present value of a given future payment is smaller when: the interest rate is higher, and/or the payment is further in the
future.
rate.interest annual simple a is where
(1) )1(
years) in received ($
i
i
FVnFVPV
n
Simple loan of $1 at 10% interest
Year 1 2 3 n $1.10 $1.21 $1.33 $1x(1 + i)n
$1PV of future $1 =
(1 + i)n
Present value
Types of credit market instruments
Simple loan Fixed-payment loan Coupon bond Discount (zero coupon)
bond
Yield to maturity Yield to maturity for a debt
instrument = the interest rate at which the present value of the payment stream equals the current price of the instrument, i.e., interest rate that equates today’s value with present value of all future payments
Yield to maturity – simple loan (i =10%)
$100 = $110/(1 + i )
$110 – $100 $10i = = = 0.10 = 10%
$100 $100
ni)1(
payment FuturePrice
Yield to maturity – fixed-payment loan (i =12%)
2532 )(1
$126
)(1
$126
)(1
$126
1
$126000$1
:example
maturity n payment fixed FP loan value LV where
iiii
(2) )(1
FP
)(1
FP
)(1
FP
1
FPLV
32
niiii
Yield to maturity – coupon bond
101032
32
)(1
$1000
)(1
$100
)(1
$100
)(1
$100
1
$100000$1
C/F)10% rate(coupon :example
bondcoupon of price P
maturity maturity toyield
valueface F payment coupon C :where
(3) )(1
F
)(1
C
)(1
C
)(1
C
1
CP
iiiii
ni
iiiii nn
Yield to maturity – coupon bond
When bond is sold at face value, the yield to maturity = coupon rate
Price of bond and yield to maturity are inversely related Yield to maturity is greater than the coupon rate when
bond price is below the face value
Consol (perpetuity)
maturity toyield
paymentyearly C
consol theof price P :where
(5) P
Cor (4),
CP
i
ii
Yield to maturity – discount bond
%1.11111.0900
9001000then
year, one $1000, F and $900 Pfor if
(6) P
PF then ,1for
maturity maturity, toyield
bonddiscount theof pricecurrent P
bonddiscount theof valueface F :where
)(1
FP
i
in
ni
i n
Overview Current bond prices and interest
rates (yields to maturity) are inversely related for all types of bonds.
Bond Page of the Newspaper (a)
Both coupon bondsbond:Maturity>10yrsnote: Maturity<10yrs
Change on bid price from yesterday
For long-term bonds, the YTM is almost the
same as current yield, but this is not true for short-term
bonds. Differences represent dealer’s profits
Current yield=C/P
10 55%. coupon rate
asked price
=11.125
105 1732
Bond Page of the Newspaper (b), (c)
discount bondsMaturity<1yr
Change on asked yield from yesterday
Both discount yields=((F-P)/F)*(360/days to Mat.)
Coupon rate 5 5/8
Current yield=C/P
8 4%. coupon rate
asked price
=8 5
8102.75
138*$1,000
Mature at 2031
True discount yield=((F-P)/P)*(365/days to Mat.)
Other measures of interest rates
Current yield: ic = C/P ...(7) P=price of the coupon bond, C=yearly payment Only accurate for consols A reasonable approximation to the yield to
maturity for bonds with a maturity of 20+ years.
Relatively poor approximation to the yield to maturity for short-term bonds.
A better approximation when the bond price is close to the face value.
Changes in current yield are always in the same direction as changes in the yield to maturity.
Other measures of interest rate
idb: yeild on a discount basis, F: face value
P: purchase price of the discount bond Always understates the yield to maturity since:
this formula uses F instead of P(<F) in denominator a year (=365 days) has more than 360 days
Changes in the same direction as the yield to maturity. (larger days to maturity has smaller i-idb )
(8) maturity todays
360
F
PFidb
Interest rates and rate of return
Rate of return includes capital gains and losses as well as current interest receipts
gain capital of rate the: yield,current :
1 , at time bond theof price :,
paymentcoupon : return, of rate : where
(10) (9),
:year onefor bond theholding fromReturn
1
11
gi
ttPP
CRET
giP
PP
P
C
P
PPCRET
c
tt
ct
tt
tt
tt
One-year returns on 10%-coupon-rate bond
Maturity and the Volatility of Bond Returns
Key Findings from Table 21. Only bond whose return = yield is one with maturity = holding
period2. For bonds with maturity > holding period, i P implying capital loss3. Longer is maturity, greater is % price change associated with
interest rate change4. Longer is maturity, more return changes with change in interest
rate5. Bond with high initial interest rate can still have negative return if i
Conclusion from Table 2 Analysis1. Prices and returns more volatile for long-term bonds
because have higher interest-rate risk2. No interest-rate risk for any bond whose maturity equals
holding period
Bond prices and rate of return
Rate of return = initial yield to maturity if a bond is held to maturity
Bond prices decline as interest rates rise, leading to capital losses
Long-term bonds exhibit more variability in price in response to interest-rate changes
Longer-term bonds experience more capital gains and losses when interest rates change (interest-rate risk)
A negative return can occur for bonds when the interest rate rise.
Real and nominal interest rates
Fisher equation:
inflation of rate expected the:
rateinterest real the:
rateinterest nominal the: where
(12)
or (11),
e
r
er
er
i
i
ii
ii
U.S. Real and Nominal Interest Rates
Financial innovation In January 1997, the U.S. Treasury
introduced indexed bonds. Interest and principal payments are adjusted for changes in the price level.