Question

Explain the difference between present value and future value.

How can effective rate of interest be different that the nominal or stated rate of interest?

How can you tell in a word problem if the problem is calling for a future value amount or a present value amount?

4. Explain the difference between an ordinary annuity and an annuity due. Begin by explaining what an annuity is.

5.Explain how you use the ordinary annuity table to calculate an annuity "due.”

6. Comment on some situations where you would use the present value of ordinary annuity table.

Answer 1

What Is an Annuity?

an annuity is any asset that generates regular payments for a set time period. This type of investment is often used by those preparing for retirement or for a period of planned unemployment. Depending on the types of investments used, annuities may generate either fixed or variable returns.

Present Value of an Annuity

The present value of an annuity is simply the current value of all the income generated by that investment in the future – or, in more practical terms, the amount of money that would need to be invested today to generate consistent income down the road. Using the interest rate, desired payment amount and number of payments, the present value calculation discounts the value of future payments to determine the contribution necessary to achieve and maintain fixed payments for a set time period.

For example, the present-value formula would be used to determine how much to invest now if you want to guarantee monthly payments of $1,000 for the next 10 years.

Future Value of an Annuity

The future value of an annuity represents the amount of money that will be accrued by making consistent investments over a set period, assuming compound interest. Rather than planning for a guaranteed amount of income in the future by calculating how much must be invested now, this formula estimates the growth of savings, given a fixed rate of investment for a given amount of time.

The future-value calculation would be used to estimate the balance of an investment account, including interest growth, after making monthly $1,000 contributions for 10 years.

Effective Interest Rate

One other type of interest rate that investors and borrowers should know is called the effective rate, which takes the concept of compounding into account.

For example, if a bond pays 6% on an annual basis and compounds semiannually, then an investor who places $1,000 in this bond will receive $30 of interest after the first 6 months ($1,000 x .03), and $30.90 of interest after the next six months ($1,030 x .03). The investor received a total of $60.90 for the year, which means that while the nominal rate was 6%, the effective rate was 6.09%.

Mathematically speaking, the difference between the nominal and effective rates increases with the number of compounding periods within a specific period. Note that the rules pertaining to how the annual equivalent rate (AER) on a financial product is calculated and advertised are less stringent than for the annual percentage rate (APR).

Nominal Interest Rate

The nominal interest rate is conceptually the simplest type of interest rate. It is quite simply the stated interest rate of a given bond or loan. This type of interest rate is referred to as the coupon rate for fixed-income investments, as it is the interest rate guaranteed by the issuer that was traditionally stamped on the coupons that were redeemed by the bondholders.

The nominal interest rate is, in essence, the actual monetary price that borrowers pay to lenders to use their money. If the nominal rate on a loan is 5%, borrowers can expect to pay $5 of interest for every $100 loaned to them.

Annuity Due

Annuity due is an annuity whose payment is to be made immediately at the beginning of each period. A common example of an annuity due payment is rent, as the payment is often required upon the start of a new month as opposed to being collected after the benefit of rent has been received for an entire month.

1. Distinction between an Ordinary Annuity and an Annuity-Due

Each payment of an ordinary annuity belongs to the payment period preceding its date, while the payment of an annuity-due refers to a payment period following its date.

The meaning of the above statement may not be immediately obvious until we look at it graphically...

A more simplistic way of expressing the distinction is to say that payments made under an ordinary annuity occur at the end of the period while payments made under an annuity due occur at the beginning of the period.

A third possibility is to define an annuity due in terms of an ordinary annuity: an annuity-due is an ordinary annuity that has its term beginning and endingone period earlier than an ordinary annuity. This definition is useful because this is how we will compute an annuity due; i.e., in relation to an ordinary annuity (discussed further in "Calculating the Value of an Annuity Due" below).

Most annuities are ordinary annuities. Installment loans and coupon bearing bonds are examples of ordinary annuities. Rent payments, which are typically due on the day commencing with the rental period, are an example of an annuity-due.

Note that an ordinary annuity is sometimes referred to as an immediate annuity, which is unfortunate because it implies that the payments are made immediately (i.e., at the beginning of the period, which would be the case with an annuity-due). However, ordinary annuity is the more widely used term.