CP

Semiannual and Other Compounding Periods 83

has $1 million of investment funds, expects to earn 10 percent on the investments,

expects inflation to average 5 percent per year, and wants to withdraw a constant real

amount per year. What is the maximum amount that he or she can withdraw at the

end of each year?

We explain in the spreadsheet model for this chapter, Ch 02 Tool Kit.xlsthat the

problem can be solved in three ways. (1) Use the real rate of return for I in a financial

calculator. (2) Use a relatively complicated formula. Or (3) use a spreadsheet model,

with Excel’s Goal Seek feature used to find the maximum withdrawal that will leave a

zero balance in the account at the end of 20 years. The financial calculator approach is

the easiest to use, but the spreadsheet model provides the clearest picture of what is

happening. Also, the spreadsheet approach can be adapted to find other parameters of

the general model, such as the maximum number of years a given constant income can

be provided by the initial portfolio.

To implement the calculator approach, first calculate the expected real rate of re-

turn as follows, where rris the real rate and rnomis the nominal rate of return:

Now, with a financial calculator, input N 20, I 4.761905, PV 
1000000, and

FV 0, and then press PMT to get the answer, $78,630.64. Thus, a portfolio worth

$1 million will provide 20 annual payments with a current dollar value of $78,630.64

under the stated assumptions. The actual payments will be growing at 5 percent per

year to offset inflation. The (nominal) value of the portfolio will be growing at first

and then declining, and it will hit zero at the end of the 20th year. TheCh 02 Tool

Kit.xlsshows all this in both tabular and graphic form.^9

Differentiate between a “regular” and a growing annuity.

What three methods can be used to deal with growing annuities?

Semiannual and Other Compounding Periods

In almost all of our examples thus far, we have assumed that interest is compounded

once a year, or annually. This is called annual compounding.Suppose, however, that

you put $100 into a bank which states that it pays a 6 percent annual interest rate but

that interest is credited each six months. This is called semiannual compounding.

How much would you have accumulated at the end of one year, two years, or some

other period under semiannual compounding? Note that virtually all bonds pay inter-

est semiannually, most stocks pay dividends quarterly, and most mortgages, student

loans, and auto loans require monthly payments. Therefore, it is essential that you un-

derstand how to deal with nonannual compounding.

Types of Interest Rates

Compounding involves three types of interest rates: nominal rates, iNom; periodic

rates, iPER; and effective annual rates, EAR or EFF%.

[1.10/1.05]
1.04.761905%.

Real raterr[(1rnom)/(1Inflation)]
1.0

(^9) The formula used to find the payment is shown below. Other formulas can be developed to solve for n and
other terms, but they are even more complex.
PVIF of a Growing Annuity PVIFGA [1
[(1 g)/(1 i)]n]/[(i
g)/(1 g)]
12.72.
Payment  PMT PV/PVIFGA $1,000,000/12.72
$78,630.64.

Time Value of Money 81