Money and Banking

1
Money and Banking

• Spring 2007
• Martin Andreas Wurm
• University of Wisconsin - Milwaukee

2
Defining Interest Rates

• Required Reading Mishkin, Chapter 4

3
Defining Interest Rates

• 2. 3. Yield to maturity
• There are several ways of calculating interest
  rates
• When economist speak of interest rates, they
  normally use a concept known as yield to
  maturity.
• Yield of maturity is defined as the interest rate
  which equates the present discounted value of an
  instrument with its value today (i.e. its current
  price)
• Note The yield of maturity is specific to any
  debt instrument

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Defining Interest Rates

• 2. 3. Yield to maturity
• Understanding this concept is not necessarily
  easy right away.
• Before we talk about the economic meaning of the
  yield to maturity, lets, therefore, first go
  over an example of a simple loan in order to get
  more comfortable with what yield to maturity
  actually means

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Defining Interest Rates

• 2. 3. Yield to maturity
• 1. Simple loans
• Lets assume we have a future payment of 1.020
  which has a present value of 1.000.
• The yield to maturity is the interest rate for
  which
• Assuming (for simplicity) N1 and solving for i

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Defining Interest Rates

• 2. 3. Yield to maturity
• 1. Simple loans
• In our example, the yield to maturity is, hence,
  equal to
• Note that for any simple loan the simple interest
  rate equals the yield to maturity!

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Defining Interest Rates

• 2. 3. Yield to maturity
• 2. Other assets
• Computing the yield to maturity can be quite
  complicated in practice
• Think for example of a mortgage that requires you
  to pay back a fixed payment of 126 every month
  and that runs for 25 months.
• As we know from our previous discussion, the
  present discounted value of this mortgage is
  simply equal to

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Defining Interest Rates

• 2. 3. Yield to maturity
• 2. Other Assets
• Lets assume todays value of this stream of
  payments (i.e. the principal) to be equal to
  1.000, so that
• Solving this equation for i on a piece of paper
  is a painful procedure and financial analysts
  would solve this problem with the help of a
  financial calculator. In this case the yield to
  maturity is 12

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Defining Interest Rates

• 2. 3. Yield to maturity
• What is the purpose of doing this ???
• The yield to maturity essentially indicates how
  strongly you can discount an asset to still keep
  it profitable
• It is the maximum discount rate, at which the
  price of an asset is not above its present value

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Defining Interest Rates

• 2. 3. Yield to maturity
• To see what this means, consider the following
  example
• Assume you have some excess funds which you want
  to save in form of a coupon bond worth 1,000 .
• Lets say you know that if you put these funds
  into your savings account at the bank, the bank
  provides you an (exorbitantly high) interest rate
  of 9.
• Since you know that you can always get at least
  this interest rate it acts as your benchmark
  interest rate.

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Defining Interest Rates

• 2. 3. Yield to maturity
• Assume that the price of this coupon bond (its
  value P today) is equal to 1,000 and that it
  offers the following payment structure It pays a
  100 a year for ten years and a face value of
  1,000 at the end of that period.
• The present value of this coupon bond is as
  above equal to

12
Defining Interest Rates

• 2. 3. Yield to maturity
• Its yield of maturity, therefore, can be obtained
  from setting the price equal to the present value
  of this bond and solving for i
• Again, this computation is not easy, but it turns
  out, that the yield of maturity for this bond is
  exactly 10

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Defining Interest Rates

• 2. 3. Yield to maturity
• What does this tell you about the profitability
  of the asset?
• The yield to maturity is higher than your
  discount rate. Therefore, this coupon bond is
  more profitable than the savings account.
• To verify this compare the future values of both
  assets

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Defining Interest Rates

• 2. 3. Yield to maturity
• Now assume instead, that the price of this coupon
  bond today is 1,100 instead of 1,000. Is this
  bond still profitable?
• Be aware that the present discounted value of an
  asset is a negative function of the interest
  rate.
• In order for this equality to hold, the yield of
  maturity must, hence, be lower than before.

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Defining Interest Rates

• 2. 3. Yield to maturity
• It turns out that the yield of maturity for a
  coupon bond and a price of 1,100 is 8.48.
• Since this is lower than the discount rate of 9,
  this bond is no longer profitable compared to
  your outside option

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Defining Interest Rates

• 2. 3. Yield to maturity
• The below table indicates different yields to
  maturity given different bond prices today for
  the above payment structure

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Defining Interest Rates

• 2. 3. Yield to maturity
• Some interesting facts are implied by this table
• If a coupon bond is priced at its face value, the
  yield to maturity equals the coupon rate (the
  bond is sold at par)
• The price of a coupon bond (as well as of any
  other asset) and its yield to maturity are
  negatively related ceteris paribus.
• The yield to maturity of a coupon bond increases
  the lower the bond price falls below its face
  value.

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Defining Interest Rates

• 2. 3. Yield to maturity
• To summarize
• The yield to maturity indicates, which interest
  rate sets the present value (the price) of an
  asset equal to its present discounted value.
• Intuitively it can be understood as a way of
  determining the probability of an asset (compared
  to your outside options)
• The fundamental property of the yield to maturity
  is its negative relationship to the price of an
  asset (e.g. a bond) given everything else the
  same, which is a fundamental relation for many
  macroeconomic applications.

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Defining Interest Rates

• 2. 4. Other Measures of Interest Rates
• Since the yield to maturity can sometimes be hard
  to calculate, other measures of interest rates
  are occasionally reported.
• 1. The current yield is often used as an
  approximation for the yield to maturity of coupon
  bonds. It is given by
• The current yield is equal to the yearly coupon
  payment divided by the bond prices

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Defining Interest Rates

• 2. 4. Other Measures of Interest Rates
• 1. An example of current yield
• Assume, a bond sells at 6,000 and has a yearly
  coupon payment of 200, then the current yield is
  given by

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Defining Interest Rates

• 2. 4. Other Measures of Interest Rates
• 1. The current yield has the same central
  property as the yield to maturity
• As the price of a bond increases, the lower will
  be the current yield.
• Depending on a number of factors (mainly the
  maturity of a bond and the difference between
  bond price and face value), the current yield
  varies in magnitude, however.

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Defining Interest Rates

• 2. 4. Other Measures of Interest Rates
• 2. Yield on a discount basis
• The yield on a discount basis is a somewhat more
  antique method of approximating the yield to
  maturity
• It used to be a common measure before financial
  calculators became popular and actually leads to
  a quite inaccurate measure it underestimates
  actual interest rates (for details see Mishkin
  pp. 71, 72).