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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 (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
…
• 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 semi-annually, 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
• 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.
• 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