Time Value of Money Time Value of Money Time Value of Money Time Value of Money

Suppose your brother or sister owed you Suppose your brother or sister owed you $500. $500.

Would you rather have this money repaid Would you rather have this money repaid to you right away, in one payment, or to you right away, in one payment, or spread out over a year in four installment spread out over a year in four installment payments? payments?

Would it make a difference either way? Would it make a difference either way?

Time Value of Money Time Value of Money Time Value of Money Time Value of Money

You would probably be better off getting You would probably be better off getting your money right away, in one payment.your money right away, in one payment.

You could invest this money and earn You could invest this money and earn interest on it or you could use this money to interest on it or you could use this money to pay off an all or part of a loan. pay off an all or part of a loan.

The time value of money refers to the fact The time value of money refers to the fact that a dollar in hand today is worth more that a dollar in hand today is worth more than a dollar promised at some future time.than a dollar promised at some future time.

Time Value of Money Time Value of Money Time Value of Money Time Value of Money

… … The cost of any decision includes the cost of the The cost of any decision includes the cost of the best-forgone opportunity. best-forgone opportunity. If you pay $10.00 for a movie ticket, your cost of If you pay $10.00 for a movie ticket, your cost of attending the movie is not just the ticket price, but also attending the movie is not just the ticket price, but also the time and cost of what else you might have enjoyed the time and cost of what else you might have enjoyed doing instead of the movie. doing instead of the movie. Applying this concept to the $500 owed to you, you see Applying this concept to the $500 owed to you, you see that getting the money in installments will saddle you that getting the money in installments will saddle you with opportunity cost. By taking the money over time, with opportunity cost. By taking the money over time, you lose the interest on your investment or any other use you lose the interest on your investment or any other use for the initial $500, such as spending it on something you for the initial $500, such as spending it on something you would have enjoyed more. would have enjoyed more.

Time Value of Money Time Value of Money Time Value of Money Time Value of Money Recall the concept of opportunity costRecall the concept of opportunity cost

Future value (FV) refers to the amount of Future value (FV) refers to the amount of money to which an investment will grow over money to which an investment will grow over a finite period of time at a given interest rate. a finite period of time at a given interest rate.

Future value is the cash value of an Future value is the cash value of an investment at a particular time in the future.investment at a particular time in the future.

ProcessProcessProcessProcessFirst, consider future value.First, consider future value.

Investing For a Single Period:

Suppose you invest $100 in a savings account that pays Suppose you invest $100 in a savings account that pays 10 percent interest per year. In one year, you will have 10 percent interest per year. In one year, you will have $110. This $110 is equal to your original $110. This $110 is equal to your original principal principal of of $100 plus $10 in interest. We say that $110 is the $100 plus $10 in interest. We say that $110 is the future value of $100 invested for one year at 10 future value of $100 invested for one year at 10 percent.percent.If you invest for one period at an interest rate If you invest for one period at an interest rate rr, your , your investment will grow to investment will grow to (1 + r) (1 + r) per dollar invested. In per dollar invested. In our example, our example, rr is 10 percent, so your investment grows is 10 percent, so your investment grows to to 1 + .10 = 1.10 1 + .10 = 1.10 dollars per dollar invested. You dollars per dollar invested. You invested $100 in this case, so you ended up with invested $100 in this case, so you ended up with $100 $100 x x 1.10 = $110.1.10 = $110.

ProcessProcessProcessProcess

ProcessProcessProcessProcessInvesting For More Than One Period: Investing For More Than One Period: Consider your $100 investment that has now grown to Consider your $100 investment that has now grown to $110.$110.If you keep that money in the bank, you will earn If you keep that money in the bank, you will earn $110 $110 xx.10 = $11 .10 = $11 in interest after the second year, making a in interest after the second year, making a total of total of $110 + $11 = $121. $110 + $11 = $121. This $121 is the future This $121 is the future value of $100 in two years at 10 percent.value of $100 in two years at 10 percent.

Another way of looking at it is that one-year from now, Another way of looking at it is that one-year from now, you are effectively investing $110 at 10 percent for a you are effectively investing $110 at 10 percent for a year. This is a single-period problem, so you will end year. This is a single-period problem, so you will end up with $1.10 for every dollar invested, or up with $1.10 for every dollar invested, or $110 $110 xx1.1 = 1.1 = $121 $121 total. total.

The process of leaving the initial investment The process of leaving the initial investment plus any accumulated interest in a bank for plus any accumulated interest in a bank for more than one period is called more than one period is called compounding.compounding.

Compounding the interest means earning Compounding the interest means earning interest on interest so we call the result: interest on interest so we call the result: compound interest.compound interest.

With With simple interestsimple interest, the interest is not , the interest is not reinvested, so interest is earned each period is reinvested, so interest is earned each period is on the original principal only. on the original principal only.

ProcessProcessProcessProcess

Suppose you locate a two-year investment that Suppose you locate a two-year investment that

pays 14 percent per year. If you invest $325, pays 14 percent per year. If you invest $325,

how much will you have at the end of two years? how much will you have at the end of two years?

√ √ At the end of the first year, you will have At the end of the first year, you will have $325 $325

xx (1 + .14) = $370.50. (1 + .14) = $370.50.

√ √ If you reinvested this entire amount, and If you reinvested this entire amount, and

thereby compound the interest, you will have thereby compound the interest, you will have

$370.50 $370.50 x x 1.14 = $422.37 1.14 = $422.37 at the end of the second at the end of the second

year.year.

Interest on Interest... Interest on Interest...

The total interest you earn is The total interest you earn is $422.37 — 325 = $97.37.$422.37 — 325 = $97.37.

Your $325 original principal earns Your $325 original principal earns $325 $325 x x .14 = .14 = $45.50 $45.50 in interest each year, for a two-year total of in interest each year, for a two-year total of $91 in $91 in simple interestsimple interest. . The remaining The remaining $97.37 -- 91 = $6.37 $97.37 -- 91 = $6.37 results from results from compounding.compounding. How much will you have in the third year? How much will you have in the third year?

Year One Year One $370.50$370.50

Year Two Year Two $422.37$422.37

$422.37 x 1.14 + $481.50$422.37 x 1.14 + $481.50

Another $45.50 in simple interest and $13.63 in Another $45.50 in simple interest and $13.63 in compounded interest = $ 481.50compounded interest = $ 481.50

Year Three Year Three $481.50$481.50

Suppose you go in for an interview for Suppose you go in for an interview for a part-time job. a part-time job.

The boss offers to pay you $50 a day for The boss offers to pay you $50 a day for a 5-day, 10-week position a 5-day, 10-week position

OR OR you can earn only one cent on the first you can earn only one cent on the first day but have your daily wage doubled day but have your daily wage doubled every additional day you work. every additional day you work.

Which option would you take?Which option would you take?

Option OneOption One

$2500 $2500

$50 * 5 days * 10 weeks$50 * 5 days * 10 weeks

Option TwoOption Two

$5,629,499,534,213.12 $5,629,499,534,213.12

FV= PV(1 + i)FV= PV(1 + i) N N

FV = Future Value FV = Future Value PV = Present Value PV = Present Value i = the interest rate per period i = the interest rate per period n= the number of compounding periods n= the number of compounding periods

What is the What is the future value of future value of $34 in 5 years if $34 in 5 years if the interest rate the interest rate is 5%? is 5%?

FV= PV( 1 + i )FV= PV( 1 + i ) n n FV= $ 34 ( 1+ .05 )FV= $ 34 ( 1+ .05 )55 FV= $ 34 (1.2762815)FV= $ 34 (1.2762815)FV= $43.39 FV= $43.39

I will give you $1000 in 5 years. I will give you $1000 in 5 years. How much money should you give me now to How much money should you give me now to make it fair to me? You think a good interest make it fair to me? You think a good interest rate would be 6%.rate would be 6%.

You can go backwards too!You can go backwards too!

FV= PV ( 1 + i ) FV= PV ( 1 + i ) NN $1000 = PV ( 1 + .06) $1000 = PV ( 1 + .06) 55 $1000 = PV (1.338) $1000 = PV (1.338) $1000 / 1.338 = PV $ 747.38 = PV$1000 / 1.338 = PV $ 747.38 = PV

So you give me $747.38 today and in 5 So you give me $747.38 today and in 5 years I'll give you $1000. years I'll give you $1000. Sound fair??Sound fair?? You will get 6% interest on your money.You will get 6% interest on your money.