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Chapter 4
Understanding Interest Rates
Learning Objectives
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1. Detail terms present value and interest rate
2. Discern measurments of interest rates: YTM,
Current Yield, Yield on Discount basis
3. Illustrate inverse relationship of bond prices
and interest rates
4. Explain difference between nominal and real
interest rates
5. Explain difference between interest rates and
rates of return
Contents
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Present Value and Interest Rates
Subprime Mortgages and Interest Rates
Measuring Interest Rates
Bond Prices and Interest Rates
International Interest Rates
Nominal vs Real Interest Rates
Interest Rates vs Rates of Return
Present Value and Interest Rates
PV = FV / (1+i) ^ t
 A dollar paid to you 1 year from now is
worth less than one dollar paid to you
today
 Ex. PV of $250 paid in 2 years, i=15%
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PV = 250 / (1 + .15) ^ 2
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= $189.04
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Credit Market Instruments
1 – Simple Loan
 2 – Fixed-Payment Loan
 3 – Coupon Bond
 4 – Discount Bond
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Simple Loan
Lender provides borrower funds, borrower
repays at maturity date plus additional
interest amount
 $500 loan with 10% interest:
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1 year: $500 * (1 + 0.10) = $550
 2 years: $550 * (1 + 0.10) = $605
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P = F (1 + i) ^ t
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P = Total payment
F = Face Value
i = interest rate
t = number of periods
Coupon Bond
Face value amount is paid to issuer for
ownership of bond, then issuer pays
owner of bond fixed interest payment
every period until maturity, then pays face
value to owner
 Three pieces of information: Issuer of
bond, maturity date of bond, coupon rate
of bond
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Coupon Bond
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P = C * (1 – (1 / (1 + i) ^ t)) / r + F / (1 + i) ^ t
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P = bond price
C = Coupon rate
i = interest rate
t = number of periods
F = face value
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Ex. $1000, 5% coupon, annual bond with
15 year maturity. What would the price be
if the interest rate was 6.5%?
 A. $858.96
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Discount Bond
Like a coupon bond, but without coupons
 No interest payments, is issued at a
discount of face value
 Ex. Face value of $1000 may be bought
for $900 and in one years time the owner
will be paid $1000
 Examples include Canadian government
treasury bills and long term zero-coupon
bonds
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Discount Bond
Similar formula as Coupon Bond, just
missing the coupon section:
 P = F / (1 + i) ^ t
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P = price of bond
F = face value
i = interest rate
t = number of periods
$10,000, zero-coupon bond maturing in 7
years, interest rate is 4%. Selling price?
 A. $7,599.18
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Fixed Payment Loan
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Lender provides borrower funds, borrower
pays back lender in fixed payments every
period until the loan is paid off (ie.
Mortgage or auto loan)
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http://www.tdcanadatrust.com/mortgag
es/
Subprime Loans
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“A mortgage granted to a borrower
considered subprime, that is, a person
with a less-than-perfect credit report.
Subprime borrowers have either missed
payments on a debt or have been late with
payments. Lenders charge a higher
interest rate to compensate for potential
losses from customers who may run into
trouble or default.”
Subprime Meltdown
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Background:
U.S. housing bubble began in early 2000’s
Low interest rates and new “mania” for buying houses
caused housing prices to skyrocket
Some banks and newly-formed lending institutions took
advantage of the situation by offering long term
mortgages to people who could normally not afford them
– subprime borrowers
These borrowers relied on the capital value of their new
homes to pay off the mortgage, rather than their earning
power
Subprime Meltdown Outcome
The rapid rise in housing prices created a
bubble, which burst in late 2005
 Housing prices began to fall and,
eventually, many subprime borrowers
began to default on their loans
 This caused the collapse and bankruptcy
of many of these institutions, and has hurt
the financial industry substantially
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Subprime Meltdown
Subprime Meltdown Outcome
This meltdown has depressed the US
economy, causing the Federal Reserve to
lower it’s benchmark rate
 Has caused a weakening of the US dollar,
and billions of dollars of losses for
investment funds
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Subprime Meltdown International
All of this has caused Canada to be
affected as well
 Canadian dollar has risen against US
dollar and is now at parity
 Depressed US housing market and high
dollar has resulted in a crisis for the
forestry industry (ie. Canfor mill closures)
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Subprime Meltdown
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Initially, it was all caused by extremely low
interest rates in the early 2000’s in order to
stimulate the economy after the dot-com
crash
Measuring Interest Rates
Yield to Maturity
 Current Yield
 Yield on a Discount Basis
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Yield to Maturity
Interest rate that equates PV of cash flow
payments received with its value today
 Most accurate measure of interest rates
 1. Simple Loans
 2. Fixed-Payment Loans
 3. Coupon Bonds
 4. Perpetuity
 5. Discount Bonds
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Yield to Maturity
Simple Loans
 Q. Pete borrows $100 from Bob and Bob
wants back $110 from him next year.
YTM?
 A. PV = FV / ( 1 + i ) ^ t
 100 = 110 / (1+ i ) ^ 1
 r = 0.1 = 10%
 In a simple loan, i = YTM
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Yield to Maturity
Fixed Payment Loan
 Q. $100,000 loan, payments of $9439.29
for next 20 years. YTM?
 LV = FP / (1 + i) + FP / (1 + i)^2 + ... +
FP / (1 + i) ^ n
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LV = Loan Value
FP = Fixed Yearly Payment
n = number of years until maturity
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A. 0.7 = 7%
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Yield to Maturity
Coupon Bond
 PP = C ((1-(1/(1+i)^n)/i) + FV/ (1+i)^n
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PP = Value of Bond
C = Coupon Payment
i = YTM
N = number of periods
FV = Face value of bond
YTM
Price of 8% coupon bond is $1122, FV of
$1000, 12 years to maturity, paid annually,
what is YTM?
 PP = C ((1-(1/(1+i)^n)/i) + FV/ (1+i)^n
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PP = Value of Bond
C = Coupon Payment
i = YTM
N = number of periods
FV = Face value of bond
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A. 0.65 = 6.5%
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Yield to Maturity
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Perpetuity
Coupon bond with no maturity date (goes on
forever)
P=C/i
P = Price of perpetuity
C = Yearly Payment
i = YTM
Q. Bond with price of $2000 and pays $100
annually forever, what is YTM?
Answer: 100 / 2000 = 0.05 = 5%
Yield to Maturity
Discount Bond
 i = (F – P) / P
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F = Face Value
P = Current Price
Ex. Treasury Bills that pays a FV of $1000
in 1 year. Purchase Price is $900. YTM?
 A. i = (1000 – 900) / 900 = 0.111 = 11.1%
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Yield to Maturity
Yield to Maturity for bonds is the actual
interest rate
 Bond prices and interest rates are
negatively related
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Current Yield
Yield of a bond at a point in time
 Does not reflect total return
 Used as approximation to describe interest
rates for long term bonds
 CY = C / P
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C = Coupon payment
P = Current price
Current Yield
Q. What is the current yield of a $1000,
8% coupon bond maturing in 3 years and
selling for a price of $1200?
 A. CY = C / P
 = 80 / 1200
 = 0.0666 = 6.66%
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Yield on a Discount Basis
Most accurate measure of interest rates
for short term bonds
 Yield of bills with a maturity of less than
one year
 i = (F – P) / P * 365 / Days to Maturity
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i = Yield on a discount basis
F = Face Value
P = Purchase Price
Yield on a Discount Basis
 Q.
91 day treasury bill selling for $988
with FV of $1000
 A. i = (F – P) / P * 365 / Days to Maturity
 i = (1000 – 988) / 988 * 365 / 91 =
0.0487 = 4.87%
Central Banks and Interest
Rates
Very important in determining interest
rates
 Central banks determine lending rates and
heavily influence market interest rates
 Lending rate is the rate that the central
bank will lend to other banks, thereby
insuring that all lending rates (inter-bank or
otherwise) are above this rate
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Central Banks
Can be government controlled, or
independent of government (Bank of
Canada, Federal Reserve)
 Determine countries monetary policy
 Sole issuer of the countries' currency
 Supervises banking industry
 “Lender of last resort” and governments
banker
 Manages gold reserves
 Sets interest rate
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International Interest Rates
Canada
 United States
 United Kingdom
 Europe
 Japan
 Australia
 China
 India
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-
4.5%
4.75%
5.75%
4.0%
0.5%
6.5%
7.02%
7.75%
Canada
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Determined by the Bank of Canada
United States
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Determined by the Federal Reserve
United Kingdom
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Determined by the Bank of England
2006
2004
2002
2000
1998
1996
1994
1992
1990
1988
1986
1984
1982
1980
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Interest Rate
Japan
Determined by Bank of Japan
Japan Interest Rates
10
9
8
7
6
5
4
3
2
1
0
Japan
Island nation with no natural resources
and very high population, and yet has the
second largest economy in the world with
the highest life expectancy, and very
technologically advanced
 For past 15 years has been in prolonged
stagnation due to many factors
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Japan
As a result, the interest rate has been very
low (0.1% for past 5 years) in order to
stimulate the economy
 Extremely low compared to Canada
(4.5%)
 Low interest rate has resulted in negative
inflation
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Interest Rates vs Returns
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Rate of return on a bond not necessarily the yield to
maturity of the bond
Rate of return is dependant on behaviour of market
interest rates over the holding period of the bond
If a bond is held for its entire life, then yield to maturity is
equal to the return of the bond
RET = (C / Pt) + (Pt+1 – Pt) / Pt
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RET = Return
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Pt = Initial Price of Bond
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Pt+1 = Selling Price of Bond
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C = Coupon Payment
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Interest Rates vs Returns
Q. What is the rate of return on a bond
bought for $988 and sold one year later for
$1200? The coupon rate is 5%, and face
value is $1000.
 A. (C / Pt) + (Pt+1 – Pt) / Pt
= ( 50 / 988) + (1200 – 988) / 988
= 0.0506 + 0.2146
= 0.2652 = 26.5%
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Volatility of Bonds
Longer term = larger risk/more volatility
 Known as interest rate risk
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Real vs Nominal Rates
So far, i = nominal interest rate
 Real interest rate is a more accurate rate
that is adjusted for inflation
 i = real interest rate + expected inflation
 OR
 Real interest rate = i – expected inflation
 i = nominal rate
 Actual inflation is not known until later
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Real vs Nominal Rates
When real interest rate is low, there are
greater incentives to borrow and fewer to
lend
 Measured by indexed bonds
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Interest and principal payments are adjusted
for price levels by indexing to the Consumer
Price Index (CPI)
Real vs Nominal Rates
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Real rate is lower than nominal rate
because of inflation
Learning Objectives
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1. Detail terms present value and interest rate
2. Discern measurements of interest rates: YTM,
Current Yield, Yield on Discount basis
3. Illustrate inverse relationship of bond prices
and interest rates
4. Explain difference between nominal and real
interest rates
5. Explain difference between interest rates and
rates of return
The End
Questions?