Modified
from
Chapter 4
Understanding
Interest Rates
Eco 2154
4.1
Here, we are
• Learning about the basic concepts for Bonds.
• There are different concepts of returns to a
bond.
• The most important concept is Yield-ToMaturity.
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4.2
1. Background Work: Present Value
A dollar tomorrow is less valuable than a dollar
today.
Interest rate is the discount rate which turns the
Future Value into the Present Value.
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4.3
Rate of Return(ROR or RET)
• ROR or RET is the return to a financial
investment for one period(year).
• This is a very general.
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4.4
Issue of Holding Period
• You may hold an asset for one-period or less.
-> Because the asset is a short-term bond; or
-> Because you just want to hold for part of the
entire duration of bond.
: The resultant ROR or RETs are called various
‘yields’
• You may hold an asset for the entire duration of
the multiple period.
: The accompanying ROR is called Yield-To-Maturity
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4.5
Yield to Maturity
• The yield to maturity is the interest rate that
equates the (entire) future value of cash flow
payments to be received from a debt
instrument with its (one) present value today.
• YTM is the effective annual(one year) rate of
return if the bond is held for the entire
duration of the maturity period.
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4.6
ROR and YTM
• The return of 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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4.7
What if you do hold the debt
instrument until it mature?
• You may hold the debt instrument of multiple
periods of duration only for one year.
• There are different ROR: They are not called
Yield-To-Maturity, but different Yields
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4.8
For the financial market,
• YTM carries a lot of information, reflecting the
investor’s long-term expectations of the financial
market, the monetary conditions, and the
macroeconomic conditions, and his planning.
• There are different YTM depending on the
duration of the bond: The different numbers of
YTM for the different durations are called ‘Term
Structure’.
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4.9
2. Four Types of Debt Instruments
: They have different formulas for Rate of Returns or Interest Rate
• Simple Loan
• Fixed Payment Loan
• Coupon Bond (most common): If the bonds
are held for the entire duration, the ROR is
called YTM.
• Discount Bond
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4.10
1) ROR on Simple Loan with One-period of duration
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 equ
Ecoals
2154 the
yield to maturity
4.11
2) ROR of Multi-period Fixed Same Payment
Loan :Annuity
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
n
1 + i (1 + i ) (1 + i )
(1 + i )
Eco 2154
4.12
3) YTM of Coupon Bonds:
Coupon + Lump sum re-payment
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
Eco 2154
4.13
Here, we will think about 3 possible
cases:
(1) You hold the bond for the entire duration of the
bond: Investment period = Duration of Bonds
(2) You hold the bond for only one year.
(3) You hold the bond for less than one year.
:(2) and(3) are possible because of ‘secondary
market of bonds’, where you can sell long-term
bonds before the maturity.
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4.14
**Approximation Formula of YTM of Coupon
Bond
Effective Rate of Returns = Yield to
Maturity: Actual Interest Rate
Coupon Payment  Annual Changes in Price
Average Price
Average Price = (Purchase Price + 100)/2;
Annual Changes in Prices = (Face Value – Purchase Price) /
Maturity Period
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4.15
Some Important Questions:
1) If Bond Price in the financial market goes up,
what will happen to YTM ?
2) In the financial market with substitutes for
bonds, can a bond sold at a higher price than
the Face Value?
3) Out of long-term and short-term bonds,
which one has more fluctuations in prices?
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4.16
*Coupon Bond
Discount or Premium?
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4.17
Coupon Bond—Yield to Maturity
(Cont’d)
The price of a coupon bond and the yield to maturity are
negatively related.
•
Bond’s Market Price < Face Value
“Bond has Discount”.
Then, YTM > Coupon Rate
•
Bond’s Market Price > Face Value
“Bond has Premium”.
Then, YTM < Coupon Rate
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4.18
Which YTM is going to be determined in the
financial market?
YTM
= Core of Rate of Returns as Equilibrium ROR in
the financial market
+ Risk Premium
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4.19
What if you hold a multiple period bond
for only one year?
ROR = Yield, but not YTM
The payments to the owner plus the change in value
expressed as a fraction of the purchase price
RET =
P
- Pt
C
+ t 1
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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4.20
What if you hold a multiple period bond
for less than one year?
ROR = Yield, but not YTM
You will either have the Current Yield or
The Capital Gains.
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4.21
4) Current Yield of
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 coupon bonds, this equation gives current yield
an easy-to-calculate approximation of yield to maturity
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4.22
5) Discount Bond
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.
Eco 2154
4.23
Yield on a Discount Basis
• Yield on a Discount Basis
i = (F – P)/P x 365/(days to maturity)
where: i = yield on a discount basis
F = face value
P = purchase price
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4.24
Example of Simple Loan
Short-term Bonds
• T-Bill
• CP
• CD
PV < FV at all times
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4.25
Example is T-Bill sold at Discount:
1) One Year T-Bill
Face Value = 100; Market Price = 95
If Maturity Period = 1 year
i= 5%
2) 3-Month T-Bill
Face Value = 100; Market Price =99
If Maturity Period = 3 months
i= 4%
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4.26
3. Few Things to Know
Rate of Return on a financial asset(bond) has
three components:
(1) ROR if the same money was invested on the
physical goods(factory, land, and gold, etc.)
(2) Compensation from inflation which erodes
the value of the bond (we can call it ‘Inflation
Risk/Certainty’)
(3) Other risk premiums
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4.27
Risk Premiums for a multiple period
bond
1) Default Risk
2) Liquidity Risk
3) Market Risk
4) Macro-economic Risk
Eco 2154
4.28
* Market Risk?
• Market Risk = risk coming from unexpected
changes in prices and returns in the financial
market
• Prices and returns for long-term
bonds are more volatile than those for
shorter-term bonds. Long-term bonds have
more Risks than short-term bonds
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4.29
Exmaple) Fluctuations of Bond Prices: Long-term and Shortterm Bonds
Suppose that YTMs of all bonds in the financial
market rise by 1%.
->Bond prices should fall to give a higher YTM.
• What is the required change in the market price of a
one-year bond?
• What is the required change in market price of a tenyear bond?
-> According to the Formula, the 10-year bond’s price
should change a lot more.
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4.30
Rate of Return
and Interest Rates (cont’d)
• 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.
Eco 2154
4.31
Rate of Return
and Interest Rates (cont’d)
Eco 2154
4.32
* Liquidity Risk:
• You need money or cash, but you may get
stuck with the bonds.
• In generalThe longer the term, the larger the
liquidity risk.
Eco 2154
4.33
* Macroeconomic Risk
• Economy Wide Issues
• Inflation Risk + Market Risk in a sense.
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4.34
2) Real versus 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.
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4.35
*Fisher Equation: If you are a money-lender, how
would you set the nominal interest rate?
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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4.36
There are two versions of Real Interest Rate
• Ex-ante Real Interest Rate
= Marginal Product of Capital
- More or stable in the short-run
• Ex-post Real Interest Rate
= Nominal Interest Rate – Actual Inflation Rate
- Actual Cost of borrowing money/funds
- Affects Investment
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4.37
• Ex-ante
i= ex-ante r + πe
here πe is the expected inflation rate.
• Ex-post
Ex-post r = i - π
here π is the actual inflation rate.
Eco 2154
4.38
Nominal Interest Rates and
Ex-post Real Interest Rate
Eco 2154
4.39