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INTEREST

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i is the effective rate of interest per year. ... In the case of the Spragga Dap credit, the annual effective rate of interest is ... – PowerPoint PPT presentation

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Title: INTEREST


1
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..

2
  • 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

3
  • 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.

4
  • 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..

5
  • 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.

6
  • Example - Jessie borrows 1000 at 15 compound
    interest. How much does he owe after two years?
  • Solution 1000(1.15)21322.50.

7
  • 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.

8
  • 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.

9
  • 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.

10
  • 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.

11
  • 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.

12
  • 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

13
  • 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.

14
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?

15
  • 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

16
  • 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?

17
  • 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

18
  • 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).

19
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.

20
  • 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.

21
  • 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.

22
  • 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.

23
  • 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,

24
  • 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
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