Interest Rates

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Interest Rates
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Utility Approach to Interest Rates
Utility function
First period budget constraint
Second period budget constraint
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Utility Maximization
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Present Discounted Value
The present discounted value of a promised
payment Q one year from now must be discounted
by the interest rate r. Clearly, if you had the
money now you could put it in the bank and make
r interest on it.
Suppose now, we have a two year income stream
The general formula
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Lottery Example
What is the difference in present value terms
between a lump-sum lottery payment of 1,000,000
versus 20 installments of 50,000 when the
discount rate is 10?
The answer is a surprisingly low 425,678. In
general, we can find the present discounted value
of n-period annuity with an annual payment Q
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Mortgage Example
The formula for calculating the annual payments Q
of a loan of length n at interest rate i
I borrow 50,000 to buy a house. The banks
insists that I make equal payments of Q The
length of the loan is 25 years and the interest
rate is 10. Applying the formula above,
If I then cross multiply, 0.1(50,000)0.908Q
Q5,506 My total payments are then
255,506137,650 At an interest rate of 5, I
make 25 annual payments of 3,547 for a total of
88,675. These large differences in total payments
(and the implied monthly payments) is what makes
housing investment so sensitive to the interest
rate.
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Bond Prices and Yield to Maturity
The market value of a bond is determined by four
factors
(1) the face value paid at maturity, F. Without
loss of generality, we will assume that F100.
(2) the coupon paid each period to the
bondholder, C.
(3) the yield to maturity (YTM) of the security,
r.
(4) the number of years to maturity, n.
The bond price is the PDV of coupons and face
value discounted at the YTM
Conversely, if you know the bond price, you can
estimate the YTM
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Treasury Quotes and Prices
Todays Treasury quotes can be found here. (You
must register).
Calculation of yield to maturity
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The Yield Curve
The yield curve is the shape of the curve
depicting the rates of return on short-term and
long-term securities.
The yield curve was steep when the Fed began
tightening in 2005. It has been inverted for
about 6 months. Recent changes in slope.
Latest quote here.
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The Expectations Hypothesis
The leading theory for explaining the slope of
yield curve is what's known as the expectations
hypothesis. The idea is simply that an upward
sloping curve usually implies higher interest
rates in the future.
Example
The question before us then is whether to buy a
two-year bond now paying 7.5 or to only buy a
one-year bond in both periods given that we
expect interest rates to rise to next period. The
expectations hypothesis says that on average,
these investments will yield about the same.
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Implications for Yield To Maturity
According to the expectations theory, you must
expect to make the same money re-investing money
in two 1-year bonds as you do holding one 2-year
bond.
Equivalently
A general formulation for n-year bonds would be
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Yield Curve and Recessions
The Federal Reserve Bank of Cleveland reports
that negatively sloped segments of the yield
curve preceded all but 4 of the 17 business-cycle
downturns since 1910. The exceptions were 1926,
1945, 1948, and 1953. Many Wall Streeters also
think that the yield curve is the best predictor
of the stock market.
Jonathan Wright of the Federal Reserve has a new
model for predicting recessions based on the
yield curve and the level of the fed funds rate.
Calculate the latest recession probabilities.