INTEREST

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INTEREST

• Simple Interest- this is where interest
  accumulates at a steady rate each period
  
• The formula for this is 1 it
• Compound Interest is where interest is earned on
  interest. This process is known as compounding.
  
• The formula for this is (1i)t..

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• Different components
• Principal is the original amount that was
  invested.
  
• i is the effective rate of interest per year.
• t is the time period in which the principal was
  invested.
  
• Accumulated Value is what your principal

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• has grown to, denoted A(t).
• Therefore .
• Interest Accumulated Value-Principal
• Compound Interest is the most important to
  remember due to the fact that it is used mostly
  in situations. It has exponential growth whereas
  simple interest has linear growth.

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• Example Someone borrows 1000 from the bank on
  January 1, 1996 at a 15 simple interest. How
  much does he owe on January 17, 1996?
  
• Solution Exact simple interest would give you
  10001(.15)(16/365)1006.58.
  
• However..

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• Bankers rule uses 360 days, which gives a
  different result.
  
• Solution 10001(.15)(16/360)1006.67, which
  is slightly higher.
  
• Canada uses exact simple interest.

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• Example - Jessie borrows 1000 at 15 compound
  interest. How much does he owe after two years?
  
• Solution 1000(1.15)21322.50.

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• Assuming a 3 rate of inflation 1 now will be
  worth 1.033 or 1.09 in three years.
  
• Example How much was 1000 worth 4 years ago
  assuming a 3 inflation rate?
  
• Solution It is worth 1000(1.03)-4, which is
  equal to 888.49.
  

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• Nominal rate of interest is a rate that is
  convertible other than once per year.
  
• i(m) is used to denote a nominal rate of interest
  convertible m times per year, which implies an
  effective rate of interest i(m) per mth a year,
  so the effective rate of interest is
• i1 (i(m)/m)m-1.

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• Example Find the accumulated value of 1000
  after three years at a rate of interest of 24
  per year convertible monthly.
  
• Solution- i1(.24/12)36-1.26824.
• So the answer to the problem is
  1000(1.26824)32039.88.

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• Also, this is just something to remember.
• Suppose XXY credit card is offering 12
  convertible monthly and Spragga Dap credit card
  is offering 12 convertible semi-annually, which
  has the best deal.
• Solution- XXY has an effective annual interest
  rate of 1(.12/12)12-1.12683.

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• In the case of the Spragga Dap credit, the annual
  effective rate of interest is
  
• i1(.12/2)2-1.1236, which is lower than the
  XXY credit card.
  
• So, the rule to remember is, given the same
  nominal rate, the effective annual rate of
  interest will be higher if it is compounded more.

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• Suppose we wanted to find a nominal rate of
  interest compounded continuously, which is the
  force of interest.
  
• There is a formula for this ln(1i).
• Example Suppose i was fixed at .12 and we wanted
  to find i(m), we would use the formula i.121
  (i(m)/m)m-1 and solve for i(m). We will see
  that

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• i(2).1166
• i(5).1146
• i(10).1140
• i(50).1135
• and if the nominal rate of interest is
  compounded continuously, then it would be
  
• ln(1.12).11333.

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ANNUITIES

• An annuity is a stream of payments.
• The present value of a stream of payments of 1
  is an.
  
• The formula for an is (1-vn)/iwhere v(1/1i)
• Suppose we were to take out a 50000 from the
  Spragga Dap bank. If the mortgage rate is 13
  convertible semi-annually, what would the monthly
  payment be to pay off this mortgage in 20 years?

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• Solution
• First, we find i, which is (1.065)(1/6)-1, then
  we proceed to set up the problem.
  
• 50000X.a240
• An1-(1/1.01055)240/.0105587.1506 so
• X50000/87.1506573.72

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• Heres a tricky one!
• Suppose Haskell Inc. supplies you with a loan of
  5000 that is supposed to be paid back in 60
  monthly installments. If i.18 and the first
  payment is not due until the end of the 9th
  month, how much should each one of the 60
  payments be?

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• Solution first we convert i into a monthly
  rate, which is 1.18(1/12)-1.
  
• Then we have to account for the fact that the
  5000 earned interest in the 1st 8 months. The
  new amount is 5000(1.013888)8 which is 5583.29
  so.
• 5583.29X.a60

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• a601-(1/1.013888)60/.01388840.5299
• Finally, 5583.3/40.5299137.76
• So we would need 60 payments of 137.76 to pay it
  off in 60 monthly installments.
  
• Note If we were supposed to take out a loan
  which was repaid starting immediately, we would
  use a double-dot which is an(1i).

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BONDS

• Investing in bonds is a good way to utilize your
  dollar. It is as simple as this. For a sum of
  money today, you will get interest annuity
  payments as well as another sum of money, known
  as redemption value, when the time period has
  elapsed.

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• There are a few key components to get familiar
  with when analyzing bonds.
  
• F is the face value or par value of the bond.
• r is the coupon rate per interest period.
  Normally, bonds are paid semi-annually.
  
• C is the redemption value of the bond. The phrase
  redeemable at par describes when FC.

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• i is the yield rate per interest period
• n is the number of interest periods until the
  redemption date.
  
• P is the purchase price of the bond to obtain the
  yield rate i.

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• The price of the bond can be obtained by solving
  this formula
  
• PFr.anC(1i)-n
• Example A bond of 500, redeemable at par in
  five years, pays interest at 13 per year
  convertible semi-annually. Find a price to yield
  an investor 8 effective per half a year.

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• Solution FC500, r.065, i.08, n10.
• So the price of this bond is
• 32.5a10500(1.08)-10449.67.
• Example Spragga Dap Corporation decides to
  issue 15-year bonds, redeemable at par, with face
  amount of 1000
  
  
  
  
  each. If interest payments are to
  be made at a rate of 10 convertible
  semi-annually,

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• And if the investor is happy with a yield of 8
  convertible semi-annually, what should he pay for
  one of these bonds?
  
• FC1000, n30, r.05 and i.04
• so the price is 50.a301000(1.04)-301172.92