Understanding and Benefiting from the Time Value of Money

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Contents• Introductory applied questions• Basic formulas and computations• Present Value and Future Value• Annuities• Practical applications of Time Value • Practice problems• Website links

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• Money is worth more as time progresses• An amount received today is worth more than that same

amount received in the future.

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• The future value of a dollar and the present value of a dollar are based on the interest rate earned and how many years and compounding times per year are involved.

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• The time value of money is all about knowing which opportunity creates the maximum value over a given period of time.

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Compounding• Simply refers to interest earning interest• Significantly influences how fast a dollar amount grows• May be computed annually or more frequently, such as

quarterly, monthly, daily, or continually

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If you invested $1,000 for one year at 10%, how much would you have at the end of a year?

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• $1,000 x 10% = $100 interest earned• Add $1000 principle to $100 interest• Balance at end of first year is $1,100

• Shortcut:$1,000 x 1.10 = $1,100 Thus,Present value (1 + interest rate) = Future Value

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Shortcut• Check out a FVIF table and get factor for 1 year and 10%

interest. Multiply this by the principle (PV) amount.

Thus the FV = PV x FVIF

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Future Value Interest Factors (1+I)n = FVIFPeriods 1% 3% 6% 8% 10%

1 1.010 1.030 1.060 1.080 1.100

2 1.020 1.061 1.124 1.166 1.210

3 1.030 1.093 1.191 1.260 1.331

4 1.041 1.126 1.262 1.360 1.464 10

If you left that $1,000 for two years, what would it be worth?

$1,100 x 1.10 = $ 1,210 1,100 x 1.102 = $ 1,210 or 1, 000 x 1.21 = $ 1,210

• Thus, Future Value = PV x (1+ Interest rate)number years

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Abbreviations

• PV = Current/earliest lump sum amount• FV = Ending/latest lump sum amount• I = Interest rate earned or paid• N = number of years/compounding periods• FVIF = future value interest factor = (1+I)N

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• FV = PV (1+I)n

Example: Invest $500 for 4 years at 8 percent rate of return, how much

will you have at the end of the fourth year?

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Solution

Using Formula & Table• FV = PV (1 + I )n

• FV = PV (FVIF i, n)• FV = $500 (1 + .08)4

= 500 (1.360)

FV = $ 680

With Calculator

$500 PV4 n8 I

FV = $680 14

If you are 20 years old and today invest $1000 dollars at 7%, what will your $1,000 be worth when you are 65 years old?

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Solution• What is the future value of $1000 today if invested for 45

years at 7% annual interest rate?

• PV = $1,000• i = 7 (the I/YR key)• N = 45• Solve for FV FV = $ 21,002

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If you didn’t get the correct answer, perhaps the next three slides will help you

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Trouble Shooting

• Only a few mistakes are common when computing time value of money problems on an HP 10B calculator.• Before starting a problem, make sure to check

for the following:• How many payments is your calculator set on. To

check, press shift (the yellow key), then C ALL (Input on the older HP calculator).• If a number other than what is needed appears, the

number of payments may be changed by pressing the correct number of payments, then shift then PMT

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Trouble Shooting (cont.)• Check to see whether your calculator is in the “begin” mode.

This means that payments are assumed to be made at the beginning of the time period rather than at the end of the time period, which is typically when payments are considered. If begin appears on your screen when it is not needed, press shift then BEG/END to change back to the end of the time period.

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Trouble Shooting (cont.)

• ALWAYS clear the machine by pressing SHIFT then C ALL. Never presume the bottom left “C” button clears everything. It doesn’t.• When using more than one compounding period

per year (as in monthly payments on a 5-year loan), press the number of years, then SHIFT then N. (eg., 5 SHIFT N after having the number of payments set on 12 per year - - 12 SHIFT PMT)• To change the number of decimal places shown

to say 6 places, press SHIFT, DISP (=) 6 20

Now, let’s try another problem…

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If you wait until you are 35 years old and invest $1000 dollars at 7%, what will your $1,000 be worth when you are 65 years old?

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Solution• $7,612

• 1000 PV• 7 I• 30 N• FV = 7,612.25

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Hint:• Any of these variables may be computed just as easily as FV,

such as a PV amount, an interest rate, or the number of years/payments, etc.

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Present Value Factors• The reciprocal of the Future Value Factors• PVIF = present value interest factor = 1/(1+I)N

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Present Value Interest Factors

Periods 1% 3% 6% 8% 10%

1 .990 .971 .943 .926 .909

2 .980 .943 .890 .857 .826

3 .971 .915 .840 .794 .751

4 .961 .888 .792 .735 .683 26

Present Value Example

You will need $8,000 in 5 years to make a down payment on a house. How much would you need to put aside today to achieve your goal if you are earning 6% on your money?

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Solution• $8,000 FV• 5 N• 6 I

• PV = $5,978 , i.e., is needed today to have the $8,000 in five years.

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Problem• If you invest $10,000 at 6% annual interest, how long will it

take your money to double?

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Solution• PV $10,000• FV 20,000• I 6• N 12 years (11.9 years)

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Rule of 72

• Any two numbers when multiplied together give you 72, will indicate the annual interest rate or the length of time until a sum doubles. For example, • 2 x 36 = 72 3 x 24 = 72• 4 x 18 = 72 6 x 12 = 72…..

• Thus, at 2% it takes 36 years for a sum to double. Likewise, at 36%, it takes 2 years for a sum to double.• Eg., in the previous example, 6 x 12 = 72

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Annuities

• Now, let’s consider making an investment or payment more than just one time.• Multiple payment situations are generically

referred to as annuities.• Payments may be made annually (once a

year) or more frequently• If payments are made more than once a year,

be sure the calculator knows the payment frequency by pressing the number of payments per year, then SHIFT then PMT

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Practice ProblemIf you are 25 years old and invest $1000 dollars each year at 7%,

what will your investments be worth when you are 65 years old?

If you waited until you were 35 to start making those payments, how much would you have at age 65?

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Solutions• Starting at age 25

$1000 PMT7 I40 N

FV $199,635

• Starting at age 35

$1000 PMT7 I30 N

FV $94,461

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Car Payments• You want to borrow $15,000 for a new car and make monthly

payments for 2 years. If the bank will lend you the money for 8%, how much will your monthly payments be?

• Hint: Solve for PMT rather than PV• Must set calculator on 12 payments per year (press 12 shift pmt)

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Solution to Car Payments• $15,000 PV• 8 I• 2 shift N (will assume 24 pmts)

PMT ?

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Car Loan Alternatives• What would your monthly payments be if you were charged

10% interest?• $692.17

• What would your monthly payments pay if you borrowed the money for 3 years at 8%?• $470.04

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Mortgages• If you were to borrow $100,000 for a house, consider the

following alternatives:• 15 years at 7.5 %• 30 years at 8%

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Mortgage Solutions• 15-year note

• 100,000 PV• 7.5 I• 15 shift N

(@12 pmts per year)• PMT 927.01

• 30-year note

• $100,000 PV• 8.0% I• 30 shift N• @12 pmts per year

• PMT $733.76

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How much more would you pay for the house over the life of the loan if you chose the 30-year note rather than the 15-year note?

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Solution: 15 vs. 30-year notes

15-year note

PMT $ 927.01

TOTAL PAID:(PMT x 12 x 15)

= $ 166,862

30-year note

PMT $733.76

TOTAL PAID:

= $ 264,154

Difference: $97,688

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Amortization• The depleting or repaying of borrowed funds• Term also used to mean the “using up of an asset” • May be used to determine the balance of a loan at any time

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Amortization Example

• On the 15 and 30-year mortgage examples, what would the ending balances be at the end of the 1st year of each loan?• HINT: Solve for the payment, then press shift

AMORT (FV) then =. You will then see the time period for which the amortization will be given (eg., period 1-12 for the first year when there are monthly payments). Then press = to get the amount of interest paid during the first year; press = again to get the amount of principal paid, then = again to get the ending balance after the first year)

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Amortization Answer

15- year loan1st year’s payments: ($11,124.15)Interest $7372.79Principle $3,751.36

Balance $ 96,248.64

30-year loan1st year’s payments: ($8,805.17)Interest $ 7,969.80Principle $ 835.37

Balance $ 99,164.63

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Ordinary Annuity vs. Annuity Due• When payments are made on a regular (say, annual) basis into

an investment, the timing of the payments is important.• If an amount is paid each year into a fund earning interest,

the present or future value of that investment fund is determined by whether the payment is made at the beginning or the end of each year.

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Ordinary Annuity vs. Annuity Due (cont.)• In a typical situation (ordinary annuity), the PV or FV is

computed assuming the payments are all made at the end of each period.

• 3 Questions:

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Ordinary vs. Annuity Due Examples• If payments of $1000 are made each year for 10

years at 12%, how much will be accumulated at the end of the 10th year?

• How much would be accumulated if the payments were made at the beginning of each year?

• How much would you pay today for an annuity that paid you $1000 a year for 10 years if interest rates were 12%? (assume payments at end of period) 47

Annuity SolutionsOrdinary annuity

$1000 pmt (1/year)10 n12 I

FV$17,548.74

PV 5,650.22

Annuity DueTurn on Begin (shift

BEG/END)$1000 PMT10 n12 I

FV $19,654.58

PV 6,328.25 48

Practice Problems

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• Oliver will need $15,000 for a down payment on a house, and has $10,000 in savings (in a certificate of deposit - CD) earning 5.5%. How long will it take Oliver to buy a house?

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• What equal annual payments are needed to repay $35,000 of student loans over eight years, if the interest rate being charged is 9%? (omit any retroactive interest charges)

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• Maura has just graduated from the university and will start earning $24,000 per year. In 10 years, what will she need to earn to maintain the same purchasing power if inflation is 3% per year?

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• Would you lend anyone money for 30 years expecting to receive no interest?

• If so, what would you pay today for a 30-year zero coupon $100,000 bond? Hint, pick an interest rate that you need to receive over the years, say 8%.

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• If a credit company charges 12% per year on monthly accounts of its customer loans, what is the effective annual interest rate?

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• At a 15% cost of money, which is more valuable, a) an annuity of $2000 per year for 10 years, b) a lump sum of $15,000 now, or c) a lump sum of $35,000 at the end of 10 years?

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• Louisa and Michael just got married and received $11,000 cash in wedding gifts. If they place this money in a mutual fund earning 11%, how much will they have on their first anniversary? Their 10th anniversary? Their 25th anniversary? Their 50th anniversary?

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• It is your 18th birthday and you have just won a lottery worth $250,000. However, you may not receive any funds until you are 21. If your windfall earns 8% between now and your 21st birthday, how much will you receive on you 21st birthday?

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• If you are 21 and start depositing $2,000 each year into an IRA which earns 9% interest, how much will you have accumulated at age 65?

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• What return are you receiving on a Series EE bond that you bought in 1999 and that will mature in 2016?

• Hint: a Series EE sells for half of par and pays no annual interest

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• Daniel Paul, age 1, just inherited $50,000 from his great grandmother. If this amount is invested for him at 13%, how long will it take the little tot to become a millionaire?

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• Charles Casco has just retired at age 65. He has $700,000 accumulated in his pension plan and would like to withdraw it in equal annual dollar amounts so that he has withdrawn it all by age 85. How much should he withdraw each year until age 85 assuming 7% interest?

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.• You plan to make the following deposits to save money to buy

a car in five years. At 9% interest, find the value of your deposits at the end of the 5th years:• You deposit $100 at the end of the 1st year• You deposit $200 at the end of the next 2 years• You deposit $500 at the end of the 4th year• You deposit $1000 at the end of the 5th year

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Capital Budgeting Problem• A new robot for your factory will cost you $45,000 but will

save you variable/manual labor costs of $14,000 per year for 10 years. What is the present value of the investment if you must pay 8.5% for money to buy the robot?

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• Suppose Olivia deposits $10,500 into a 403B plan at the end of each of the next 25 years, what will her investment be worth in 25 years if her deposits earn 9% each year?

• What would this investment be worth at the end of the 25th year if she makes the deposits at the beginning of the year instead?

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• What is the most that you should pay for a 30-year zero coupon bond if you are earning 12% on the money you would use to buy the bond?

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• If you buy an annuity today for $15,000, it will pay you $500 per month for 10 years, what rate of return are you earning on your money?

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Applicable Links

• On-line calculators• Mutual Funds and Misc. Investments• Stock Market • Money rates • Mortgage and Real Estate• Economic Data • Insurance Quotes• Tax Services

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On-line Calculators

• www.techmtg.com/calc/mortgage.htm• www.moneyadvisor.com/calc/• www.financenter.com• www.investorguide.com• http://pathfinder.com/• http://tqd.advanced.org/• www.calcbuilder.com• www.401kafe.com/tools/calc.html• http://quote.bloomberg.com/• www.banking.yahoo.com/calculators.html

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Mutual Funds and Miscellaneous Investments• www.brill.com• www.closed-endfunds.com• www.fabian.com• www.findafund.com• www.fundspot.com• www.indexfundsonline.com• www.investmentdiscovery.com• www.morningstar.com• www.fundz.com• www.bondsonline.com• www.publicdebt.treas.gov

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Stock Market

• www.411stocks.com• www.financialweb.com• www.investorguide.com• www.money.net• www.quote.com• www.investingstocks.com• www.quote.yahoo.com• www.prospectusonline.com• www.quotecentral.com

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Money Rates

• www.bankrate.com• www.banx.com• www.onmoney.com• http://finance.yahoo.com• www.bog.frb.fed.us/releases/H15• http://www.bankrate.com/brm/default.asp• http://bankcd.com/• http://www.money-rates.com/• http://www.rate.net/

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Mortgage and Real Estate

• www.microsurf.com• www.realestate.com• www.tnrealtor.com• www.netquote.com• www.loan.yahoo.com• http://www.mortgage101.com• http://www.homepath.com• http://www.homemortgageguide.com• http://www.mortgageexpert.org

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Economic Data• www.bea.doc.gov• www.frbchi.org• www.fedstats.gov/key• www.huduser.org• http://www.mortgagemart.com/armindex.html• www.mtsu.edu/~dgraddy/future/ecdata.htm

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Insurance Quotes

• www.insweb.com• www.quotesmith.com• www.quicken.com• www.insurance.yahoo.com• www.lowestpremium.com• www.quickquote.com• www.netquote.com• www.insbuyer.com

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Tax Services• www.moneycentral.com• www.irs.gov• www.taxresources.com• www.EL.com/elinks/taxes• http://taxes.yahoo.com/• http://www.hrblock.com• http://www.ssa.gov

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