# Lecture Notes for Week 6

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```					Lecture Notes for Week 6

• Week 5 discussed monetary policy while trying to omit entirely reference to the
interest rate. This was deliberate; although monetary policy is concerned with the
interest rate, the monetary authority (the Bank of Canada) can only affect the interest
rate by changing the money supply. It is better for you to think of monetary policy as
central bank control over the rate of growth of the money supply, a side effect of
which is control over the interest rate.

• There is a wide range of different interest rates in the economy. There is the interest
rate you pay on a mortgage, the interest rate you pay on a consumer loan (such as to
buy a car), the interest rate the bank pays you on a savings account, the interest rate
the bank pays you on a term deposit, the interest rate a large corporation pays on loans
from a bank, the interest rate paid to you on a bond due to mature in a year, the interest
rate paid to you on a bond due to mature in ten years, the interest rate paid to you on a
government bond, the interest rate paid to you on a junk bond (issued by a high-risk
company), and so on and on. We will talk of a single interest rate which you can think
of as a representative interest rate. In general this will not be misleading because all
these different interest rates tend to rise and fall together.

• To discuss the interest rate we need to introduce some technical details associated with
the very important concept of the inverse relationship between the interest rate and the
price of bonds. The logic of this is as follows. A typical bond has each year a coupon
representing the annual interest payment on that bond. If the current interest rate is six
percent and the bond has a face value (the dollar value the holder of the bond will be
paid when the bond matures) of a thousand dollars, this coupon would be \$60 because
\$60 is six percent of \$1,000. If the bond were to mature (come due for payment of the
face value to the bondholder) in five years, there would be five such \$60 coupons, one
for each year to maturity.

Now suppose the interest rate for some reason jumps to eight percent. You will say to
yourself "I should sell this bond and buy the new bonds that are paying 8%." But
everyone else thinks the same thing, so everybody is selling, nobody is buying these
old bonds. People deal with this by offering their bond for a lower price - the forces of
supply and demand bid down the price of this bond.

The opposite would occur if the interest rate were to fall, say to 5%. People ready to
invest their savings in bonds would say to themselves "Why buy new bonds that are
only paying 5% when old bonds are paying 6%?" They will try to buy old bonds from
you. With everybody trying to buy these old bonds the price of these old bonds will be
bid up.

The bottom line here is that market forces cause the price of bonds to fall when the
interest rate rises and the price of bonds to rise when the interest rate falls. And it
doesn't matter which happens first - when the price of bonds rises the interest rate
automatically falls and when the price of bonds falls the interest rate automatically
rises. And this happens virtually instantaneously because financial markets are very
efficient. This inverse relationship between interest rates and the price of bonds is as
close as we come in this course to a real economic "law," as opposed to the rules of
thumb that we use to capture other economic concepts in this course. This inverse
relationship is a very important concept in this course because it appears in several of
the stories we tell about how and why the macroeconomy is reacting to a shock.

• By how much does the price of bonds change when the interest rate changes? The
formula for figuring this out is on p.184. This formula is too difficult for us to use to
calculate price and interest rate changes, so we will use an approximation which is
simpler. Here is the logic. [I think you will find the exposition below to be better than
that provided on p.169-170 of the text.]

Suppose you buy a bond with a coupon of \$50 and is due to mature in 10 years at a
face value of \$1,000. And suppose the price of this bond is \$920. Over the ten years
you hold this bond you will expect its price to crawl up from \$920 to its face value of
\$1,000. This implies that the price of the bond should increase by approximately \$8
per year. So each year the approximate payoff to holding this bond should be the
coupon of \$50 plus the anticipated price increase of \$8, for a total payoff of \$58. As a
fraction of \$920, what you have to lay out to get this annual payoff, this is 58/920 =
0.063, an annual interest rate of 6.3%.

The formula to use to get this result for a bond with N years to maturity is as follows:

annual i = [coupon + (face value - price)/N] /price

A major mistake made by students is to memorize this formula. There is a
straightforward logic to it; if you understand the logic you can easily reproduce the
formula and you are in a much better position to answer questions based on this
concept. To make our life even simpler, the vast majority of the questions you will
encounter in this course will involve examples in which for a coupon bond there is one
year to maturity (N=1).

• There is another type of bond with which you must be familiar. This is a discount
bond, also called a zero-coupon bond, the prime example of which is a Treasury bill
(T-bill). The main difference between this bond and the regular bond discussed above
is that as its name implies there is no coupon. All the return to the bond holder takes
the form of the difference between the price paid for the bond and the face value at
which the bond pays off upon its maturity. So the formula to use here is as follows:

annual i = [(face value - price)/N] /price

Another major feature of T-bills is that their time to maturity is a year or less (because
they are issued by the government treasury to cover temporary shortfalls in tax
receipts during the year). This means that N in the formula above will be some
fraction of a year. In all the questions you will encounter in this course the time to
maturity of a T-bill will be some easily-determined fraction of a year. For example, a
90-day T-bill is three months, or approximately one-quarter of a year, so for this case
N is 1/4. So, for example, suppose the annual interest rate (interest rates are always
expressed in annual rates, and are sometimes called yields or yield to maturity) is
4.5% and you have been asked what is the price p of a T-bill due to mature in two
months at its par (face) value of \$1,000?

The payoff to you of buying such a T-bill is (1000 - p) which when expressed as a
fraction of what you have to lay out to buy the T-bill is (1000-p)/p. This will give you
the interest rate over the two month period. Because two months is one-sixth of a year,
six times this will give you the annual interest rate. So 6(1000-p)/p should equal 0.045.
Note that this logic produces exactly what the formula above gives you. From this you
can figure out that p should be equal to 6000/6.045 = \$992.56.

• Here are some things to know about bond markets.
1. Bonds always have a face value that is a round number like \$1000.
2. Coupons usually have reasonably round numbers such as \$65.50.
3. The price of a bond is very seldom equal to its face value. When a bond is first issued
whoever is selling the bonds figures out a coupon that is approximately such that it
matches the current interest rate. But by the time the bonds are printed up for sale, a
couple of days later, the interest rate has changed and so the price of the bond needs to
change.
4. After bonds have been sold by the issuer, the people who buy the bonds can sell them
on the bond market. There is always a market for these "old" bonds, and buying and
selling them takes place all the time. So the interest rate and the price of these bonds
are continually changing.
5. We think of bonds as a conservative investment with low risk, and indeed some
investors do buy bonds and hold them to maturity, knowing exactly what their long-
run return will be. But the bond market is also a venue for speculators who are trying
to make money in the short run. Huge amounts of money are made and lost in the
bond market by speculators who are gambling on interest rate changes. If you think
that interest rates will soon fall, you can make a lot of money by buying bonds now. If
the interest rate does fall, the price of your bonds will increase and you can sell them
at a profit and go on to speculate on something else. Of course, if the interest rate
rises, the price of your bonds will fall and you will lose money!
6. To make big money on anticipated interest rate changes you need to buy/sell long-
term bonds, not short-term bonds. Why is this? Suppose you own a bond with coupon
\$70 and its price is equal to its face value of \$1,000, so that the interest rate is 7%.
Now suppose that the interest rate jumps to 8%. The price of your bond should fall. If
your bond is to mature in one year, the price should fall by about \$10 to create an extra
1% return to match the rise in the interest rate from 7% to 8%. But if your bond is to
mature in five years the price needs to fall by about \$50 to produce an extra 1% in
each of the five years to maturity! The bottom line here is that long-term bonds
experience much bigger price changes when the interest rate changes.
7. For those of you who like formulas, the result above can be seen by looking at an
earlier formula:
annual i = [coupon + (face value - price)/N] /price

When N is a larger number (more years to maturity) the change in price needs to be
bigger to have the same impact on the interest rate.

•   The discussion in Section 10.3 of the text needs to be supplemented. Our earlier
discussion of monetary policy had viewed it entirely in terms of controlling the money
supply. This was deliberate, to force you to think of controlling the money supply as
the essence of monetary policy. But in fact the interest rate has come to be viewed as
what is meant by monetary policy, primarily because most central banks, including the
Bank of Canada, articulate their monetary policy in terms of setting a desired interest
rate. But do not be fooled by this. In order to affect the interest rate, the Bank of
Canada must change the money supply, so the essence of monetary policy remains
controlling the money supply.

• Why does the Bank of Canada conduct its monetary policy in terms of the interest
rate? One reason is that it is much easier for the public to understand the Bank’s
monetary stance if it is described in terms of an interest rate. A second reason is that
using a monetary aggregate as a guide to policy can be problematic, for reasons we
discussed earlier in the context of the rules-versus-discretion debate. It is not clear
what definition of the money supply should be chosen for policy purposes, the
connection between monetary aggregates and spending behaviour is not constant, and
measuring the magnitude of any monetary aggregate is not easy. All these problems
are avoided by focussing on the interest rate as the target of monetary policy. The
money supply is adjusted by whatever is necessary to achieve this interest rate. As we
will see in Chapter 11, this interest rate policy runs into trouble in an inflationary
environment. But in periods of low inflation, such as we have been experiencing for
several years, this is a sensible way to conduct monetary policy.

• Just how does the Bank of Canada conduct its monetary policy? Every six weeks or
so, the Bank announces a target range of fifty basis points (i.e., half of a percentage
point) for the “overnight” interest rate. This rate is the interest rate at which
commercial banks borrow from / lend to other banks for short periods of time, usually
overnight. At the end of each day all the commercial banks figure out who owes what
to whom. The banks in a net debt position have to borrow funds from those in a net
credit position, paying the overnight rate, which is determined by the supply and
demand forces in this market. The Bank rate, the rate at which the Bank of Canada
stands ready to loan funds to the commercial banks, is set at the upper end of this
range, preventing the overnight rate from going above this range. The Bank stands
ready to borrow funds from the commercial banks at the lower end of this range,
ensuring that the overnight rate does not go below this range. The target overnight rate
announced by the Bank is set at a round quarter-percent, so that, for example, it could
be announced as 4.25 percent, in which case the range would be 4 percent to 4.5
percent. The Bank typically changes this rate by only a quarter- or a half-percent at a
time. To ensure that this change comes about, the Bank conducts open market
operations designed to influence the interest rate, so that ultimately it is control of the
money supply that is the foundation of monetary policy.

• In the text, particularly in Curiosity 10.2, page 173, reference is made to the discount
rate and to the federal funds rate. The discount rate is the U.S. equivalent of the
Canadian Bank rate, and the federal funds rate is the U.S. equivalent of the Canadian
overnight rate.

• An earlier story (from Chapter 9) explained how monetary policy worked in terms of
the quantity theory: An excess supply of money caused people to increase spending to
decrease their money holdings, etc. It is important that you recognize that this story is
merely a steppingstone to the story we tell in this chapter 10. Our new story is that
monetary policy works by influencing the interest rate, which in turn influences
aggregate demand for goods and services. From now on in this course we abandon the
quantity theory transmission mechanism to focus on the interest rate transmission
mechanism, as described in section 10.4 of the text. Make sure to remember this when

• Moving on to Chapter 11 of the text, we come to what I consider to be the single most
important concept in the entire course - the difference between real and nominal
interest rates. The nominal interest rate is the interest rate we observe. It is the one we
talk about in the news, the one that we pay on our loans, and the one paid on bond
holdings. It is the one that affects bond prices, as discussed earlier. The real interest
rate is connected to the nominal interest rate by the following simple relationship:

nominal irate = real irate + expected inflation

What this is in effect saying is that there is some fundamental real interest rate to
which is added a premium for expected inflation to account for the erosion of the
dollar's purchasing power caused by inflation.

Here is the best way to see where this relationship comes from. Suppose there is no
inflation and the interest rate is 3%. People borrowing money are happy to pay 3% to
be able to buy furniture now instead of waiting for a year to buy. Lenders are content
with a 3% return on their funds as a reward for loaning them out to others. Now
suppose that inflation jumps to 5% per year and everybody sees this.

Lenders will say to themselves "If I continue to loan out at 3%, during the period of
the loan the purchasing power of my money will erode by 5% so at the end of the year
when the loan is paid back I will have lost 2% purchasing power." The lender will
want to charge 8% to protect against the anticipated inflation."

Borrowers will say to themselves "If I buy furniture now I will save 5% because in a
year's time this furniture will cost 5% more because of inflation." So the borrower is
willing to pay the extra 5% to the lender. They settle for 8% interest which in real
terms corresponds to the original 3% - what the lender wants as a real return, and what
the borrower is willing to pay in real terms for the ability to have the furniture now
instead of waiting for a year.

• There are several important things you should know here.
1. The real interest rate is unobservable; we must "estimate" it by estimating the
expected rate of inflation.
2. News commentary almost always is concerned with the observable, nominal interest
rate. Whenever news commentary talks about the real interest rate it usually states
explicitly the words "real interest rate." But not always - you need to be on guard for
this.
3. Sometimes people refer to last year's real interest rate as measured by the nominal
interest rate minus last year's actual inflation. This is what is called an "ex post" real
interest rate, and is not what we mean by the real interest rate. In our meaning last
year's real interest rate is last year's nominal interest rate minus the inflation that
people expected last year. It involves an expected (not an actual) inflation.
4. The real interest rate is fairly modest (in the range of about 3 or 4 percent) and doesn't
change by a lot. The nominal interest rate can be very high (if expected inflation is
high) and can change by big amounts (if expected inflation changes by big amounts).
5. An implication of the preceding item is that if the interest rate is undesirably high it is
because of high expected inflation. If the central bank wants to lower this interest rate
its appropriate strategy should be to lower inflation which should eventually lower
expected inflation.
6. The nominal interest rate is the rate that determines the price of bonds.
7. The real interest rate is the rate that determines aggregate demand for goods and
services. Recall the person who was buying furniture in an earlier example. The rise of
the nominal interest rate from 3% to 8% did not cause this person to stop buying the
furniture. Paying 8% nominal interest corresponded to 3% real interest. There was no
change in the real interest rate and so no change in the consumer's desire to borrow to
buy the furniture. A similar story can be told to explain why investment demand is
affected by the real rather than nominal interest rate. By buying equipment now
instead of a year from now the business will save 5% because of the anticipated 5%
rise in the price of the equipment. The 5% higher cost of the equipment should be
covered by the 5% higher price (because of the inflation) of the product the business
will produce with the equipment.
8. This difference between real and nominal interest rates is the most important concept
in the entire course. This is because this concept enables you to interpret a wider
variety of news clips than does any other concept in the course. Make sure you are
familiar with this wide range of applications of this concept. The flowchart on p.191 is
only one application of this concept; make sure you understand all the examples in
Appendix 11.1, p.205-6.

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