Not bad -- you retire a millionaire.

Q: With compounded returns, small differences in annual returns become very large over time.  Going back to the previous example, imagine that the 50-year, $10,000 investment is in a mutual fund.  Mutual fund companies charge fees, which lower your returns a bit, but some charge higher fees than others.  Compare two different stock mutual funds.  The first has fees of 1.4% per year (the industry average), which give you an annual return of 8.6%.  The second, with fees of 0.5%, has an annual return of 9.5%. --> Exactly how much more would you have with the second fund at the end of 50 years? A: Apply the future value formula, plugging in .086 for i in the first case and  .095 in the second case: First fund:     FV = $10,000*((1.086)^50)  = $618,716 Second fund: FV = $10,000*((1.095)^50) = $934,733 --> The second fund will be worth $316,056 more (or 51% more) at the end of 50 years.

A relatively simple application of future value is the Rule of 72 (or 70), which tells you how quickly an asset will double in value.  But first, let us see how Future Value leads us to two related concepts:  Present-Day Value, and Internal Rate of Return.

II.  PRESENT-DAY VALUE (PDV)

Present-day value, present value, and present discounted value all mean the same thing:  what some future payment or set of future payments is worth to you today, i.e., right here, right now.  The essence of this very important concept is that a dollar in the future is worth less than a dollar today, because if you had that dollar today, you could invest it and earn interest on it and end up in the future have more than just a dollar.  (This concept is sometimes called the time value of money.)

Breaking it down, word for word:
PRESENT DISCOUNTED VALUE
  |                    |                                  |
right now      "discounting" for            what it's worth
                    the way that interest
                    reduces the value of
                    future payments relative
                    to current payments

Or, in a nutshell:  PRESENT-DAY VALUE = what it's worth now

The present-day value (PDV) of any future payment will depend on three things:
* The size of the payment positively affects its PDV -- a million dollars tomorrow is still better than $100 today.
* The higher the current interest rate, the lower the PDV of any future payment.  The higher the interest rate, the greater the value of $1 today relative to $1 in the future.  If the interest rate is very high, you would forgo a lot of interest if you said, "Sure, I'll give you $100 today, and when you give me $100 four years from me now, that'll be just as good."  (It would be if the interest rate were zero, there were no inflation, and you were very patient, but even if there's no inflation and you have infinite patience, as long as i > 0%, then the PDV of $1 in the future is less than $1.)
* The longer you wait for the payment, the lower its PDV (the more it has to be "discounted").  As long as you have to wait at all for the payment, it's not worth as much as having the same amount of money right now.  (The character named Wimpy in the Popeye cartoons who says "I'd gladly pay you Tuesday for a hamburger today" is a man who understands the concept of PDV.)

The basic present-value formula should look very familiar to you, because it's just the Future Value (FV) formula rearranged.   Let FV be the amount of a future payment (i.e., its Future Value), and recall that i is the market interest rate and n is the number of years between now and the date of the payment, and the formula is

PDV = FV/(1+i)ⁿ The PDV of a stream of several future payments, to be made at different times, is equal to the sum of the separate PDV's of each of the payments.  That is, to compute the PDV of several future payments, you'll need to apply this formula for every single payment and then add all those PDV's up.
-- Is there ever a shortcut?  Yes, if the future payment amounts are all the same and are made at regular intervals.  (Coming up in the chapter 6 notes.)

III.  INTEREST RATES AND LONG-TERM RATES OF RETURN

The INTEREST RATE on a debt is calculated as the net annual payment on the debt, as a percentage of the total amount of the debt.
-- Ex.: Simple loan: You borrow $1000, and must pay back $1100 a year from now.
==> Net interest payment = $1100 - $1000 = $100
        Interest rate = $100/$1000 = 1/10 = 10%

Usually calculating the interest rate is not quite that simple, as the length of the loan or the asset's lifetime is not exactly one year.  But it's simple enough, as we can rearrange the basic Future Value equation to calculate the interest rate, as long as we know the values of Future Value (FV) and Present-Day Value (PDV).

-- The FV equation:  FV = PDV * [(1+i)^n]

---- (English translation:  A current sum of money, $PDV, earning interest at rate i, after n years will be worth $FV.)

Rearranging that equation to solve for i, [refer to your class notes to see all the steps], we end up with

i = { (FV/PDV)^(1/n) -1 } (*100%)

-- (The "*100%" is there because interest rates are normally expressed in percentage form.  For example, the numbers .08 and 8% are mathematically equivalent, but we would say the interest rate is 8%, not .08, when reporting it.)

Note:  The term 1+i , which appears in the FV and PDV formulas, is called the gross interest rate.  The idea here is that the 1 includes the repayment of principal.  Interest compounds over time because the amount on which interest is paid becomes larger each year -- each year the amount gets multiplied by 1+i.

IV. THE RULE OF 72:

An asset with a compounded annual return of X% will double in value in approximately 72/X years.

This rule is very handy because the number 72 is evenly divisible by a lot of numbers (1, 2, 3, 4, 6, 8, 9, 12, 18).  As long as you remember your times tables from grade school, you can produce decent estimates of the rate of doubling without having to use a calculator.

The Rule of 70 is a common alternative and actually gives a closer approximation than 72/X does for small values of X (1, 2, 3, 4, 5).  But since simplicity is really the goal with these rules, I suggest you use 70 for numbers that divide evenly into 70 (2, 5, 7, 10, 14) and 72 otherwise..

Note well that this rule is only an approximation, and it doesn't work for particularly large values of X (those larger than, say, 25).  For values of X larger than, say, 25, the time to double will be more than 72/X years.  (For example, an asset earning 100% per year will double in value in exactly 1 year, not in 72/100 years.)  But, most financial assets have long-term annual returns well under 25% anyway.

Exs.:
* Savings account, which pays 3% interest
--> value of your account will double in ~72/3 = 24 years (or, a little more precisely, about 69/3 = 23 years)
* A CD that pays an annual return of 7% a year
--> value of that CD will double in ~70/7 = 10 years
* Stocks in the late 1990s; annual return was about 24% a year
--> value of a stock portfolio doubled in ~72/24 = 3 years

Ex.:  Back to the generous-grandparent problem.  Now suppose your grandfather wants to set up an investment account for you earning 10% per year and having a value of $1 million fifty years from now, when you're 70.  About how much does he need to put in the account now?
A:  We'll apply the Rule of 70, since 10 divides evenly into 70 --> that account will double in value every 7 years (= 70/10).  So over the next 50 years, your initial investment will double in value 7 times (since 7*7= 49, very close to 50).  Working backwards, we just need to divide $1,000,000 in half 7 times (with some rounding, since the point of the exercise is to keep the arithmetic fairly simple):  1) $500,000.  2) $250,000.  3) $125,000.  4)  $62,000.  5) $31,000.  6) $15,500.  7) $7,800.
-- So investing about $7,800 today makes you a millionaire 50 years later.  (We'll solve for the exact amount when we move on to present discounted value.)

V. COMPOUND ANNUAL INTEREST RATES AND ANNUAL PERCENTAGE YIELDS

Compounded interest has been called the most powerful force in the world. That might be exaggerated, but the magic of compounded interest is something you should acquaint yourself with. The main way that "the rich get richer" is through compounded interest or compounded dividends. It's also a way that many middle-class people become rich.
-- Real-life ex.: A former psychology professor at SUNY-Oswego put all his pension fund contributions (TIAA-CREF) into stocks, with all dividends reinvested (works much like compounded interest), and retired a millionaire.

At many banks, where interest rates on CD's and other accounts are listed, there are often two rates listed for each one:  the regular, or nominal, interest rate (i) and the annual percentage yield (APY), which the textbook calls the compound annual interest rate.  In a better, simpler world, these two rates would always be the same, but because of the way interest is calculated, they often are not. The annual percentage yield (APY) is the total value of the interest payments on a financial asset as a percentage of its original purchase price. The APY would be the exact same thing as the interest rate if the bank paid out the interest only at the end of the year; instead, in most cases interest is compounded (paid out) at several times during the year, e.g. monthly. Some banks have continuous compoundingof interest, whereby your account balance earns some tiny amount of interest every second of every day. If a bank compounds interest m times per year, then, instead of paying out i% interest at the end of the year, it pays out (i/m)% interest m times per year.
-- Ex.: If the interest rate on your credit-card balance is 24% and that interest is compounded every month, then, instead of being charged 24% interest on your unpaid balance at the end of the year, you'll be charged 2% (= 24%/12) interest at the end of every month, and your total interest charges will be much higher.
-- Ex.: If the interest rate on savings account is 5% and the bank compounds interest fourtimes per year, then the bank pays out 1.25% interest on those accounts at the end of every third month.
-- The greater the frequency of interest payments or compounding, the larger the total interest payments over the year. For an account holder or creditor (someone to whom money is owed), the more frequent the compounding, the better, and continuous compounding is the best. For a debtor (someone who owes money), the less frequent the compounding, the better, and continuous compounding is the worst.

For an asset with an interest rate of i and payments that are compounded m times per year, the annual percentage yield (APY), or effective interest rate or compounded interest rate, is calculated as follows: