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If a payment of dollars is made in an account times a year at an interest , then the present value of the annuity after years is
where the amount is also the loan amount, and the periodic payment.
We have already developed the tools to solve most finance problems. Now we use these tools to solve some application problems.
One of the most common problems deals with finding the balance owed at a given time during the life of a loan. Suppose a person buys a house and amortizes the loan over 30 years, but decides to sell the house a few years later. At the time of the sale, he is obligated to pay off his lender, therefore, he needs to know the balance he owes. Since most long term loans are paid off prematurely, we are often confronted with this problem. Let us consider an example.
Mr. Jackson bought his house in 1975, and financed the loan for 30 years at an interest rate of 9.8%. His monthly payment was $1260. In 1995, Mr. Jackson decided to pay off the loan. Find the balance of the loan he still owes.
The reader should note that the original amount of the loan is not mentioned in the problem. That is because we don't need to know that to find the balance.
As for the bank or lender is concerned, Mr. Jackson is obligated to pay $1260 each month for 10 more years. But since Mr. Jackson wants to pay it all off now, we need to find the present value of the $1260 payments. Using the formula we get,
The next application we discuss deals with bond problems. Whenever a business, and for that matter the U. S. government, needs to raise money it does it by selling bonds. A bond is a certificate of promise that states the terms of the agreement. Usually the businesses sells bonds for the face amount of $1,000 each for a period of 10 years. The person who buys the bond, the bondholder , pays $1,000 to buy the bond. The bondholder is promised two things: First that he will get his $1,000 back in ten years, and second that he will receive a fixed amount of interest every six months. As the market interest rates change, the price of the bond starts to fluctuate. The bonds are bought and sold in the market at their fair market value . The interest rate a bond pays is fixed, but if the market interest rate goes up, the value of the bond drops since the money invested in the bond can earn more elsewhere. When the value of the bond drops, we say it is trading at a discount . On the other hand, if the market interest rate drops, the value of the bond goes up, and it is trading at a premium .
The Orange computer company needs to raise money to expand. It issues a 10-year $1,000 bond that pays $30 every six months. If the current market interest rate is 7%, what is the fair market value of the bond?
A bond certificate promises us two things – An amount of $1,000 to be paid in 10 years, and a semi-annual payment of $30 for ten years. Therefore, to find the fair market value of the bond, we need to find the present value of the lump sum of $1,000 we are to receive in 10 years, as well as, the present value of the $30 semi-annual payments for the 10 years.
The present value of the lump-sum $1,000 is
Note that since the interest is paid twice a year, the interest is compounded twice a year.
The present value of the $30 semi-annual payments is
Once again,
Therefore,
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