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Robert Winslow

Econ 330 HW4

Interest Rates

This assignment is under construction.

Problem 1:

Calculate the Present Value, to the nearest dollar, for each of the following cash flows. Assume a 10% interest rate for each. Show your work. (The relevant formulas are in chapter 4 of Mishkin)

  1. 1100 dollars next year, 1210 dollars two years from now, and 1331 dollars three years from now.
  2. 7400 dollars, 21 years from now.
  3. 2000 dollars every year, forever.
  4. 2000 dollars every year, for 7 years

SOLUTIONS

  1. $3000
\[PV = \frac{1100}{(1.10)^1} + \frac{1210}{(1.10)^2} + \frac{1331}{(1.10)^3}\] \[PV = 1,000 + 1,000 + 1,000 = 3,000\]

Present Value = $3,000

  1. $1000
\[PV = \frac{7400}{(1.10)^{21}} = \frac{7400}{7.40024994} \approx 1000\]

(Note: I chose the numbers to make things very convenient. (1.10)^21 ≈ 7.400)

  1. $20,000

Formula for perpetuity:
\(PV = \frac{C}{r} = \frac{2000}{0.10} = 20,000\)

You can find this formula in the book or derive it yourself, or you could use a spreadsheet and just add of the present value of future cash flows until things diverge.

  1. $9737

Here is the full expanded version of problem #4, writing the present value of the 7-year annuity as the sum of all seven individual discounted cash flows:

\[\begin{aligned} PV &= \frac{2000}{(1.10)^1} + \frac{2000}{(1.10)^2} + \frac{2000}{(1.10)^3} + \frac{2000}{(1.10)^4} + \frac{2000}{(1.10)^5} + \frac{2000}{(1.10)^6} + \frac{2000}{(1.10)^7} \\ &= 2000 \times \left( \frac{1}{1.10} + \frac{1}{1.10^2} + \frac{1}{1.10^3} + \frac{1}{1.10^4} + \frac{1}{1.10^5} + \frac{1}{1.10^6} + \frac{1}{1.10^7} \right) \\ &\approx 2000 \times 4.8684 \\ &\approx 9736.8 \end{aligned}\]

Problem 2:

Calculate the Yield to Maturity for each of the following assets. Show your work, and verify that the present value of future payments equals the initial price you paid.

  1. You pay 1000 dollars today and receive 1340 dollars six years from now.
  2. You pay 1000 dollars today. You receive 100 dollars every year for the next nine years. Then ten years from now, you receive 1100 dollars.

PROBLEM 2 Solutions:

  1. 5%.

YTM is value of i that solves

\[1000 = \frac{1340}{(1+i)^6}\]

Problem 3:

This goes a bit beyond what’s in the book. Suppose you have a fixed payment loan that pays C dollars each year for 30 years. If the interest rate is i, then the present value of the cash flow from this loan is :

\[PV = \frac{C}{(1+i)} + \frac{C}{(1+i)^2} + ... + \frac{C}{(1+i)^{30}}\]

With a bit of algebra, we can derive a simpler form:

\[PV=\frac{C}{i}\cdot\left(1-\frac{1}{(1+i)^{30}}\right)\]
  1. Algebraically derive the second formula from the first.
  2. How much money could you borrow using a fixed payment loan with 30 annual fixed payments of 10,000 dollars each, and an interest rate of 5%. Show your work.

Problem 4:

Suppose a Discount Bond promises 2000 dollars in one year’s time. For each of the following prices for this bond, calculate the interest rate (yield to maturity): 2000 dollars, 1900 dollars, 1800 dollars, 1700 dollars, 1600 dollars. Show your work.

Problem 5:

Suppose the state of South Dakota reduces state taxes without reducing state spending, and has a state-level budget deficit as a result.

In about 20-30 words, describe what this might do to the supply and demand for South Dakota Municipal bonds, and why.

Attach a sketch of this shock to your homework, ala the depictions of shocks in Mishkin ch5. (I would prefer that you insert it into your submission file, if possible.) What do you expect would happen to equilibrium Price, Interest Rate, and Quantity sold for this bond?