ECON7200
Economics of Financial Markets
Lecture 2
Dr Shino Takayama
UQ School of Economics
1
Preview
• Before we can go on with the study of money, banking, and financial
markets, we must understand exactly what the phrase “interest
rates” means.
• An interest rate is the cost of borrowing or the price paid for the
rental of funds.
• Today, we see that a concept known as the yield to maturity is the
most accurate measure of interest rate.
Learning Objectives
• Calculate the present value of future cash flows and the yield to
maturity on the four types of credit market instruments.
• Recognize the distinctions among yield to maturity, current yield, rate
of return, and rate of capital gain.
• Interpret the distinction between real and nominal interest rates.
• Study about the Fisher Equation.
$100 $100
Year 0 1
PV 100
2
$100 $100
n
100/(1+i) 100/(1+i)2 100/(1+i)n
•Cannot directly compare payments scheduled in different points in the timeline
Simple Present Value
Simple present value
• Let denote today’s (present) value.
• Let denote future cash flow (payment) in year .
• Let denote the interest rate.
• Then Equation 1 states:
Intuition of Equation 1
• Intuitively, what Equation 1 tells us is that if you are promised $1 for
certain ten years from now, this dollar would not be as valuable to
you as $1 is today,
• because if you had the $1 today, you could invest it and end up with
more than $1 in ten years.
LearnX – Credit Market Instruments
• Simple Loan
• Fixed Payment Loan
• Coupon Bond
• Discount Bond
LearnX – Yield to Maturity
• Yield to maturity: the interest rate that equates the present value of
cash flow payments received from a debt instrument with its value
today
Example: Coupon Bond
• A coupon bond with $1,000 face value might pay you a coupon
payment of $100 per year for ten years, and at the maturity date
repay you the face value amount of $1,000.
• The coupon rate is then $100/$1,000, which is 0.10, or 10%.
Coupon bond
• Let denote the price of the bond.
• Let denote the yearly coupon payment.
• Let denote the face value of the bond.
• Let denote the number of years to maturity.
• We will use Equation 3:
=

1 +
+

1 + ଶ
+

1 + ଷ
+ ⋯ +

1 + ௡
+

1 + ௡
Table 1: From Equation 3
800 =
100
1 +
+
100
1 + ଶ
+
100
1 + ଷ
+ ⋯ +
100
1 + ଵ଴
+
1000
1 + ଵ଴
→ = 0.138052 …
S =
(ା)

(ା)

ା



(ା)
S =
(ା)

(ା)

ା ష

ା
(1+i)S = 1 +
(ା)

(ା)

ା ష
i S = 1
ା
S =


(ା)
We want to show
Thus
Finally
Another quick formula
Consol
• Consol or perpetuity: a bond with no maturity date that does
not repay principal but pays fixed coupon payments forever
consol theofmaturity toyield
paymentinterest yearly
consol theof price
/




c
c
c
i
C
P
iCP
cc PCi /: thisasequation above rewritecan 
For coupon bonds, this equation gives the current yield, easy to
calculate approximation to the yield to maturity
Consol – Alfred Nobel
“All of my remaining realisable
assets are to be disbursed as
follows: the capital, converted to
safe securities by my executors, is
to constitute a fund, the interest on
which is to be distributed annually
as prizes to those who, during the
preceding year, have conferred the
greatest benefit to humankind.”
- Alfred Nobel’s will -
Source: nobelprize.org
• Physics
• Chemistry
• Medicine
• Literature
• Peace
• Economics (added in 1968)
The yield to maturity & the current bond price
• The yield to maturity equals the increase in prices over the year
divided by the initial price.
• The yield to maturity is negatively related to the current bond price.
1.
P
F P Fi
P
  
Example: Discount Bond
• Let us consider a discount bond such as a one-year Treasury bill,
which pays off a face value of $1,000 in one year’s time.
• If the current purchase price of this bill is $900, then equating this
price to the present value of the $1,000 received in one year, using
Equation 1, gives:
$1000$900 .
1 i


Solving for i gives:
We have obtained .F Pi
P

Is the YtM always positive?
• Sweden in July 2009
• Denmark in July 2012
• Eurozone in June 2014
• Switzerland in Dec 2014
• Japan in Jan 2016
Some Quick Tips
It is often useful to be able to make approximate calculations quickly.
We will describe and apply two tricks that can help with this.
• The rule of 72: For a small interest rate i the doubling time (in years)
is approximately For example:
• ଻ଶ
௜
. And ଻ଶ
• ଻ଶ
௜
. And ଶସ
• ଻ଶ
௜
. And ଽ
• (or )
Last Week – The Manhattan Story
• According to American legend, Native
Americans sold Manhattan to Dutch traders
for $24 worth of beads in 1624.
• In a record, the value of the items used in the
purchase was 60 guilders, which is roughly
$1000 now.
• It is also often said that if they had put the
money in a bank and let the interest
compound, they could now buy all of
Manhattan’s real estate, which is now valued
at around $3 trillion. Is this right?
Source: Alfred Fredericks|Wikicommons
Last Week – The Manhattan Story
• Applying both rules from above, after 2022 – 1624 = 398 years:
• At a 6% interest rate, money doubles every 72/6 = 12 years
•  398/12 = 33.167: after 398 years, money has doubled roughly 33 times
• Then, calculate ଷଷ ଷ ଽ trillion
• Definitely yes!
• What about if the interest rate is 5%? Money doubles every 72/5 =
14.4 years
•  398/14.4 = 27.64: after 398 years, money has doubled roughly 27 times
• Calculate ଶ଻ ଻ ଺
• Not even close 
LearnX – The Distinction Between Interest
Rates and Returns
• Rate of Return (R): The payments to the owner, plus the change in
value expressed as a fraction of the purchase price
• : return from holding the bond from time to time
• ௧: price of bond at time
• ௧ାଵ: price of bond at time
• : coupon payment
• Definitions:
• ஼
௉೟
: current yield ௖
• ௉೟శభି௉೟
௉೟
: rate of capital gain
Effect of interest rates’ change
• Consider the situation where interest rates rise from 10% to 20%
from the first year to the second year (When you receive the
coupon for the first time, the interest rate is already 20%).
• We study One-Year Returns on Different-Maturity 10% Coupon-
Rate and Bonds.
Our example
Time Span
Purchase Coupon Bond (Face Value
$1000 & Coupon Rate %10)
Year 1
Interest Rate Rises to 20%
Year 1
Interest Rate Remains at 10%
Year 0
Interest Rate 10%
We consider the price of coupon bond
at Year 1. The maturity can be counted
from here for 2, 5, … years.
Example: if interest rates rise to 20%
• When year to maturity is 2 years from when you bought, the price
next year (i.e. 1 year remaining) is
•
• When year to maturity is 5 years from when you bought, the price
next year (i.e. 4 years remaining) is
௧ାଵ ଶ ଷ ସ ସ
Example: if interest rates remain at 10%
• When year to maturity is 2 years from when you bought, the price
next year (i.e. 1 year remaining) is
•
• When year to maturity is 5 years from when you bought, the price
next year (i.e. 4 years remaining) is
௧ାଵ ଶ ଷ ସ ସ
When initial current yield is 10%
Year to Maturity Price at Year 2 Rate of Capital Gain Rate of Return
2 $917 (917 – 1000)/1000 = – 0.083 10 – 8.3 = + 1.7%
5 $741 (741 – 1000)/1000 = – 0.259 10 – 25.9 = – 15.9%
10 $597 (597 – 1000)/1000 = – 0.403 10 – 40.3 = – 30.3%
20 $516 (516 – 1000)/1000 = – 0.484 10 – 48.4 = – 38.4%
30 $503 (503 – 1000)/1000 = – 0.497 10 – 49.7 = – 39.7%
10% of $1000 10% of $1000 $1000 (Face Value)
Intuition for Interest-Rate Risk
Our example
LearnX – The Distinction Between Interest
Rates and Returns
• A rise in interest rates is associated with a fall in bond prices.
• The more distant a bond’s maturity, the greater the size of the percentage
price change associated with an interest-rate change.
• 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.
LearnX – Maturity and the Volatility of Bond
Returns: 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.
• The risk level associated with an asset’s return that results from
interest-rate change is called interest-rate risk.
LearnX – The Distinction Between 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
LearnX – Fisher Equation
= nominal interest rate
= real interest rate
= expected inflation rate
When the real interest rate is low,
there are greater incentives to borrow and fewer incentives to lend.
The real inter
e
r
r
e
i i
i
i


 
est rate is a better indicator of the incentives to
borrow and lend.
Example: Fisher Equation (Nominal and Real Interest Rates)
• Suppose that you have made a one-year simple loan with a 5%
interest rate (i =5%) and you expect the price level to rise by 3% over
the course of the year ( = 3%).
• In this case, the interest rate you have earned in terms of real goods
and services is 2%.
• Now what if the interest rate rises to 8%, but you expect the inflation
rate to be 10% over the course of the year?
• Although you will have 8% more dollars at the end of the year, you
will be paying 10% more for goods.
• This is also exactly what the Fisher definition tells us, because:
e
8% 10% 2%.ri   
Incentives to borrow and lend
• As a lender, you are clearly less eager to make a loan in this case,
because in terms of real goods and services you have actually earned
a negative interest rate of 2%.
• As the borrower, you fare quite well because at the end of the year,
the amounts you will have to pay back will be worth 2% less in terms
of goods and services—you as the borrower will be ahead by 2% in
real terms.
• When the real interest rate is low, there are greater incentives to
borrow and fewer incentives to lend.
Figure 1 Real and Nominal Interest Rates (Three-Month Treasury Bill),
1953–2014
Sources: Nominal rates from Federal Reserve Bank of St. Louis FRED database: http://research.stlouisfed.org/fred2/. The real rate is
constructed using the procedure outlined in Frederic S. Mishkin, “The Real Interest Rate: An Empirical Investigation,” Carnegie-
Rochester 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.
Japan and the Interest Rates
Japanese nominal government bond yields
Source: Federal Reserve Bank of San Francisco
Japanese CPI inflation excluding fresh food
Australia and the Interest Rates
Summary of Concepts for this Week
• Present value
• A simple loan
• Fixed payment loan
• Coupon bond
• Yield to maturity
• Interest rates risk
• The distinction between interest rates and returns
• The distinction between real and nominal interest rates
• Fisher equation