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Economics 102: Macroeconomics16 chapters | 137 lessons | 14 flashcard sets

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Lesson Transcript

Instructor:
*James Walsh*

M.B.A. Veteran Business and Economics teacher at a number of community colleges and in the for profit sector.

A central concept in business and finance is the time value of money. We will use easy to follow examples and calculate the present and future value of both sums of money and annuities.

Donna was puzzled about something, so she went to talk to Becky about it. She told her friend that the problem is whether she would want a dollar today or a dollar one year from now. She doesn't see what the difference is, since it's still one dollar, no matter when you get it. Becky had to think about this for a while. When she sees Donna again, she tells her to take that dollar now and put it in a savings account. The bank will pay interest, so one year from now she'll have more than one dollar. To sum up the **time value of money**, money that you have right now will be worth more over time. So one dollar now will be worth more than a dollar in a year from now.

Donna went home and did some research and she discovered a formula for **future value**, or how much money put in the bank today will turn into at some point in the future with the interest. She needs to know three things:

- How much she has now
- What the interest rate is
- How many years she wants to put the money away for

Then she can use a formula to figure out how much she'll have at the end. The formula is:

FV = PV (1 + *r*)^*n*

In this formula, PV equals how much she has now, or the present value, *r* equals the interest rate she will earn on the money, *n* equals the number of periods she will put the money away for, and FV equals how much she will have at the end, or future value.

Let's imagine that Donna puts $100 in the bank for five years at five percent interest, and plug that into the equation.

FV = 100 (1 + .05)^5

FV = 100 * 1.2762

FV = $127.62

Pretty nice, huh?

Donna's parents think she's a pretty smart girl, especially after she shows her Dad these cool formulas. Dad knows he will need money in a few years to pay for Donna's college. He's wondering how much he can invest today in some CDs that would be worth $20,000 or so in 10 years when he'll need it. Donna shows him a formula for **present value**, or how much you need to save today to have a specific amount at some point in the future. Here's the formula:

PV = FV / (1 +*r*)^*n*

In this formula, PV equals how much he needs to have today, or present value; *r* equals the interest rate he'll earn; *n* equals the number of periods before he needs the money; and FV equals how much he will need in the future, or future value. So, if Dad needs the $20,000 in 10 years and can invest what he has for five percent, let's find out how much he needs to invest today.

PV = $20,000 / (1.05)^10

PV = $20,000 / 1.6289

PV = $12,278

Her dad is very happy to hear that.

An **annuity** is stream of equal payments. If Donna's parents give her an allowance of $20 every month on the first, that's an annuity. It isn't just one allowance payment, but a stream of them, since they happen every month, and it's always just the same amount. Donna went and told Becky all about the formulas and Becky told her parents about how they work. Becky's dad wants to save for Becky's college a different way. He wants to put $1,500 in the bank at the end of every year for 10 years. He wonders how much he'll have at the end. Becky looks up a formula for that. It's called the **future value of an annuity**, which is how much a stream of *A* dollars invested each year at *r* interest rate will be worth in *n* years. Here's what it looks like:

FV *A* = *A* * {(1 + *r*)^n -1} / *r*

In this formula FV *A* equals how much he will have at the end, or the future value of annuity. *A* equals $1,500, his yearly payment. *r* equals the interest rate he gets and *n* equals the number of periods he makes those deposits. So Becky puts the numbers in the formula. She comes up with this:

FV *A* = $1,500 * {(1 + *r*)^n -1} / *r*

FV *A* = $1,500 * {(1.6289-1)/ .05}

FV *A* = $1,500 * (0.6289/.05

FV *A* = $18,867

Becky's dad was happy to hear that, since that's pretty close to what he'll need.

Becky and her parents go to see Grandpa this weekend. He's getting ready to retire soon and he would like to know how much he needs today to be able to draw out $50,000 a year for the next 20 years. Becky furrows her brow and then looks at the formulas again. She finds one for Grandpa. It looks a bit tricky, but they'll figure it out. It's called the **present value of an annuity**, which tells you how much you'll need today to receive a stream of payments *A* each year for *n* years if the money is invested at *r* interest rate. Here it is:

In this formula PV *A* equals how much he needs today, or the present value of an annuity. *A* equals $50,000, the yearly payment he wants. *r* equals the interest rate he gets on his money and *n* equals the number of periods. The calculation is:

PV *A* = $50,000 * ({1 - (1 + .05)^-20} / .05)

PV *A* = $50,000 * (0.62311 / .05)

Becky gets out her calculator to solve this, since it has a big exponent. The answer is $623,110. Grandpa isn't happy to see that, but he wants the calculator and the formula so he can play around with some other numbers. He's thankful for the formulas.

The **future value** of a dollar is what a dollar today invested at *r* interest rate will be worth in *n* years. The formula is FV = PV (1 + *r*)^*n*. The **present value** of a dollar is what a dollar earned in the future is worth in today's money, where *r* is the interest rate the money earns, and *n* is the number of periods until it's received. The formula is PV = FV / (1 +*r*)^*n*. An **annuity** is a stream of equal payments. The **future value of an annuity** is how much a stream of *A* dollars invested each year at *r* interest rate will be worth in *n* years. The formula is FV *A* = *A* * {(1 + *r*)^n -1} / *r*. The **present value of an annuity** is how much you'll need today to receive a stream of payments *A* each year for *n* years if they money is invested at *r* interest rate. The formula is PV *A* = *A* * ({1 - (1 + *r*)^*n*} / *r*).

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Economics 102: Macroeconomics16 chapters | 137 lessons | 14 flashcard sets

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