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ANNUITIES DUE

The perpetuity and annuity formulas assume that the first payment occurs at the end of the period. They tell you the value of a stream of cash payments starting one period hence.

However, streams of cash payments often start immediately. For example, Kangaroo Autos in Example 8 might have required three annual payments of $4,000 starting immediately.A level stream of payments starting immediately is known as an annuity due.

If Kangaroo`s loan were paid as an annuity due, you could think of the three payments as equivalent to an immediate payment of $4,000 plus an ordinary annuity of $4,000 for the remaining 2 years. This is made clear in Figure 1.12, which compares the cash-flow stream of the Kangaroo Autos loan treating the three payments as an annuity (panel a) and as an annuity due (panel b).

In general, the present value of an annuity due of t payments of $1 a year is the same as $1 plus the present value of an ordinary annuity providing the remaining t Ј 1 payments. The present value of an annuity due of $1 for t years is therefore

PV annuity due = 1 + PV ordinary annuity of t Ј 1 payments = 1 + [ 1 Ј 1 ] r r (1 + r)t Ј1

By comparing the two panels of Figure 1.12, you can see that each of the three cash flows in the annuity due comes one period earlier than the corresponding cash flow of the ordinary annuity. Therefore, the present value of an annuity due is (1 + r) times the present value of an annuity.2 Figure 1.12 shows that the effect of bringing the Kangaroo loan payments forward by 1 year was to increase their value from $9,947.41 (as an annuity) to $10,942.15 (as an annuity due). Notice that $10,942.15 = $9,947.41 1.10.

Home Mortgages

Sometimes you may need to find the series of cash payments that would provide a given value today. For example, home purchasers typically borrow the bulk of the house price from a lender. The most common loan arrangement is a 30-year loan that is repaid in equal monthly installments. Suppose that a house costs $125,000, and that the buyer puts down 20 percent of the purchase price, or $25,000, in cash, borrowing the remaining $100,000 from a mortgage lender such as the local savings bank. What is the appropriate monthly mortgage payment?

The borrower repays the loan by making monthly payments over the next 30 years (360 months). The savings bank needs to set these monthly payments so that they have a present value of $100,000. Thus

Present value = mortgage payment 360-month annuity factor = $100,000

Mortgage payment = $100,000

360-month annuity factor

Suppose that the interest rate is 1 percent a month.

Then Mortgage payment = $100,000 [ 1 Ј 1 ] . .01 .01(1.01)360 = $100,000 97.218 = $1,028.61

This type of loan, in which the monthly payment is fixed over the life of the mortgage, is called an amortizing loan. ¬Amortizing ­ means that part of the monthly payment is used to pay interest on the loan and part is used to reduce the amount of the loan. For example, the interest that accrues after 1 month on this loan will be 1 percent of $100,000, or $1,000. So $1,000 of your first monthly payment is used to pay interest on the loan and the balance of $28.61 is used to reduce the amount of the loan to $99,971.39. The $28.61 is called the amortization on the loan in that month.

Next month, there will be an interest charge of 1 percent of $99,971.39 = $999.71. So $999.71 of your second monthly payment is absorbed by the interest charge and the remaining $28.90 of your monthly payment ($1,028.61 Ј $999.71 = $28.90) is used to reduce the amount of your loan. Amortization in the second month is higher than in the first month because the amount of the loan has declined, and therefore less of the payment is taken up in interest. This procedure continues each month until the last month, when the amortization is just enough to reduce the outstanding amount on the loan to zero, and the loan is paid off.

Because the loan is progressively paid off, the fraction of the monthly payment devoted to interest steadily falls, while the fraction used to reduce the loan (the amortization) steadily increases. Thus the reduction in the size of the loan is much more rapid in the later years of the mortgage. Figure 1.13 illustrates how in the early years almost all of the mortgage payment is for interest. Even after 15 years, the bulk of the monthly payment is interest.

2 Your financial calculator is equipped to handle annuities due. You simply need to put the calculator in ¬begin ­ mode, and the stream of cash flows will be interpreted as starting immediately. The begin key is labeled BGN or BEG/END. Each time you press the key, the calculator will toggle between ordinary annuity versus annuity due mode.

ANNUITY DUE Level stream of cash flows starting immediately.

How Much Luxury and Excitement Can $96 Billion Buy?

Bill Gates is reputedly the world`s richest person, with wealth estimated in mid-1999 at $96 billion. We haven`t yet met Mr. Gates, and so cannot fill you in on his plans for allocating the $96 billion between charitable good works and the cost of a life of luxury and excitement (L&E). So to keep things simple, we will just ask the following entirely hypothetical question: How much could Mr. Gates spend yearly on 40 more years of L&E if he were to devote the entire $96 billion to those purposes? Assume that his money is invested at 9 percent interest. The 40-year, 9 percent annuity factor is 10.757. Thus

Present value = annual spending annuity factor $96,000,000,000 = annual spending 10.757 Annual spending = $8,924,000,000

Warning to Mr. Gates: We haven`t considered inflation. The cost of buying L&E will increase, so $8.9 billion won`t buy as much L&E in 40 years as it will today. More on that later.