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.

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