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 (1+i)t..
• 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.
• 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 1000[1+(.15)(16/365)]=1006.58.
• However…..
• Banker’s rule uses 360 days, which gives a
different result.
• Solution – 1000[1+(.15)(16/360)]=1006.67,
which is slightly higher.
• Canada uses exact simple interest.
Example - Jessie borrows $1000 at 15%
compound interest. How much does he owe
after two years?
• Solution = 1000(1.15)2=1322.50.
• 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.
• 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
• i=[1+ (i(m)/m)]m-1.
• Example – Find the accumulated value of
$1000 after three years at a rate of interest
of 24% per year convertible monthly.
• Solution- i=[1+(.24/12)]36-1=.26824.
• So the answer to the problem is
1000(1.26824)3=2039.88.
• 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 semiannually, which has the best deal.
• Solution- XXY has an effective annual
interest rate of [1+(.12/12)]12-1=.12683.
• In the case of the Spragga Dap credit, the
annual effective rate of interest is
• i=[1+(.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.
• 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(1+i).
• Example Suppose i was fixed at .12 and we
wanted to find i(m), we would use the
formula i=.12=[1+ (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.
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)/i……where v=(1/1+i)
• 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?
• Solution:
• First, we find i, which is (1.065)(1/6)-1, then
we proceed to set up the problem.
• 50000=X.a240
• An=[1-(1/1.01055)240]/.01055=87.1506 so…
• X=50000/87.1506=573.72
• Here’s 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?
• 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……….
• 5583.29=X.a60
• a60=[1-(1/1.013888)60]/.013888=40.5299
• Finally, 5583.3/40.5299=137.76
• 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(1+i).
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.
• 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
F=C.
• 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.
• The price of the bond can be obtained by
solving this formula:
• P=Fr.an+C(1+i)-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: F=C=500, r=.065, i=.08, n=10.
So the price of this bond is:
32.5a10+500(1.08)-10=449.67.
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,
• And if the investor is happy with a yield of
8% convertible semi-annually, what should
he pay for one of these bonds?
• F=C=1000, n=30, r=.05 and i=.04
• so the price is 50.a30+1000(1.04)-30=1172.92