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## Time Value of Money

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**Time Value of Money**CHAPTER PLAY LIST SONGS: “Bang on the Drum All Day” by Todd Rundgren “All Work and No Play” by Van Morrison “Sixteen Tons” by Tennessee Ernie Ford**Learning Objectives**• LO 2-1 Explain what gives paper currency value and how the Federal Reserve Banks manage its distribution. • LO 2-2 Differentiate between simple and compound interest rates and calculate annual percentage yields and the value of paying yourself first. • LO 2-3 Calculate the future and present value of lump sums and annuities in order to know what to put aside to meet your personal financial goals.**Money Production**• Money is produced by our Government • Cash is produced by the Bureau of Engraving & Printing in D.C. and Fort Worth • Coins are produced by the US mint in Philadelphia and Denver • Money is then sent to the Federal Reserve banks.**Power of Compounding “The most powerful force in the**universe is compound interest” ~ Albert Einstein If Ben Franklin deposited $20 for you 250 years ago, what would its value be at 5%, 8% and 10% interest rates?**Compounding Interest**• Compounding: Process whereby the value of an investment increases exponentially over time due to earning interest on interest previously earned • Annual Percentage Rate (APR): Annual rate charged for borrowing or made by investing • Annual Percentage Yield (APY): Effective yearly rate of return taking into account the effect of compounding interest**Simple Interest vs. Compound Interest**Simple Interest on $1,000 @12% Deposit of $1,000 on Jan. 1… $1,000 + (1,000 x .12) = $1,120Resulting balance on Dec. 31 Compound Interest Quarterly on $1,000 @12% 12% ÷ 4 time periods = 3% per quarter Deposit of $1,000 on Jan. 1… $1,000.00 + (1,000.00 x .03) = $1,030.00 $1,030.00 + (1,030.00 x .03) = $1.060.90 $1,060.90 + (1,060.90 x .03) = $1,092.73 $1,092.73 + (1,092.73 x .03) = $1,125.51Resulting balance on Dec. 31**Annual Percentage Yield (APY)**Compound Interest Quarterly on $1,000 @12% 12% ÷ 4 time periods = 3% per quarter Deposit of $1,000 on Jan. 1… $1,000.00 + (1,000.00 x .03) = 1,000.00 + 30.00 = $1,030.00 $1,030.00 + (1,030.00 x .03) = 1,030.00 + 30.90 = $1,060.90 $1,060.90 + (1,060.90 x .03) = 1,060.90 + 31.83 = $1,092.73 $1,092.73 + (1,092.73 x .03) = 1,092.73 + 32.78 = $1,125.51 Resulting interest yield Dec. 31... 30.00 + 30.90 + 31.83 + 32.78 = $125.51 Annual Yield 125.51/1,000 = 12.55% APY**Annual Percentage Yield (APY)**APY = (1 + r/n)n – 1 r = stated annual interest rate n = number of times compounded/year**APY Examples**The 12% rate compounded annually yields (1 + .12/1)1 – 1 = 12.00% semiannually yields (1 + .12/2)2 – 1 = 12.36% quarterly yields (1 + .12/4)4 – 1 = 12.55% monthly yields (1 + .12/12)12 – 1 = 12.68% daily yields (1 + .12/365)365 – 1 = 12.75%**Lottery Winner???**• If you won the lottery would it be better to take the cash option now or choose to receive the amount in payments over your expected lifetime?**Time Value of Money**• A dollar now is worth more than a dollar in the future, even after adjusting for inflation, because a dollar now can earn interest or other appreciation until the time the dollar in the future would be received.**FYI**• Saving $2000 per year is only $166.67 per month or $38.46 per week or $5.48 per day. • A pack a day smoker spends $49 dollars a week, $208 dollars a month, and $2,548 a year • The average coffee drinker spends $1,100 per year • What could you give up today so you could have a million dollars in the future?**Secrets to Making Compounding Work**• Pay yourself first • Automatic transfer to savings and investment accounts • Transfer money on payday • Treat it like a bill • Start small • Increase the amount at least annually • Don’t touch the money**Time Value of Money**• Future Value: The projected value of an asset based on the interest rate and time in the account • Present Value: The current value of an asset to be received in the future based on the interest rate and time in the account • Lump Sum: Asingle, one-time payment • Annuity: Aseries of equal payments made at equal intervals over time (day, month, year)**Future Value (FV), Long-Hand Method**FV = PV (1 + i)n where FV = Future value PV = Present value i = Interest rate n = Number of periods**Problem**• At the time of your birth your aunt deposited $10,000 for you into an account earning 5% annually. How much money will you have on your 18th birthday? How much money will you have when you turn 30? How much money will you have by the time you retire at age 65?**Future Value (FV), Long-hand Method Example: $10,000 at**Birth, Earning 5% Annually, Age 18 FV = PV (1 + i)n FV = $10,000 (1 + 0.05)18 FV = $10,000 (1.05) 18 FV = $10,000 (2.41) FV = $24,066.19 on your 18th birthday**Future Value (FV), Long-hand Method Example:$10,000 at**Birth, Earning 5% Annually, Age 30 FV = PV (1 + i)n FV = $10,000 (1 + 0.05)30 FV = $10,000 (1.05)30 FV = $10,000 (4.32) FV = $43,219.42 on your 30th birthday**Future Value (FV), Long-hand Method Example: $10,000 at**Birth, Earning 5% Annually, Age 65 FV = PV (1 + i)n FV = $10,000 (1 + 0.05)65 FV = $10,000 (1.05)65 FV = $10,000 (23.84) FV = $238,399.01 on your 65th birthday**Future Value Interest Factor (FVIF) Table Method**Future Value Interest Factor (FVIF): Factor multiplied by today’s amount to determine value of said amount at a future date**Future Value (FV), Reference Table Method Example: $10,000**at Birth, Earning 5% Annually, Age 18 FV = PV (FVIFi,n) FV = $10,000 (FVIF5,18) FV = $10,000 (2.4066) FV = $24,066.00 on your 18th birthday**Future Value (FV), Reference Table Method Example: $10,000**at Birth, Earning 5% Annually, Age 30 FV = PV (FVIFi,n) FV = $10,000 (FVIF5,30) FV = $10,000 (4.3219) FV = $43,219.00on your30th birthday**Present Value of a Lump Sum**• Discounting: The reverse of compounding; finding the present value of a future amount by deducting the interest to be made**Present Value (PV), Long-Hand Method**PV = FV/(1 + i)nwhere PV = Present value FV = Future value i = Interest rate n = Number of time periods**Problem**• You want your newborn to have $10,000 at age 18. How much would have to be deposited, assuming a 6% interest rate compounded annually (simple interest)?**Present Value (PV), Long-hand Method Example: $10,000 in 18**Years, Earning 6% Annually PV = FV (1 + i)n PV = $10,000/(1 + 0.06)18 PV = $10,000/(1.06)18 PV = $10,000/(2.85) PV = $3,503.44 to be deposited**Present Value Interest Factor (PVIF) Table Method**Present Value Interest Factor (PVIF): Factor multiplied by a future amount so as to determine the value of said amount today**Present Value (PV), Reference Table Method Example: $10,000**in 18 Years, Earning 6% Annually PV = FV (PVIFi,n) PV = $10,000 (PVIF6,18) PV = $10,000 (.3503) PV = $3,503.00to be deposited**Future Value of an Annuity**• Ordinary Annuity: Stream of equal payments that occurs at the end of a period • Annuity Due: Stream of equal payments that occurs at the beginning of a period**Future Value of an Annuity (FVA), Long-Hand Method**FVA = PMT {[(1 + i)n – 1]/i} where FVA = Future value of an annuity PMT = Payment i = Interest rate n = Number of time periods**Problem**• Your parents want to contribute to your child’s education, but instead of a lump sum payment of $3,500, they plan to contribute $500 each year for 18 years. How much money will your child have in his or her educational fund after 18 years at 6% interest?**Future Value of an Annuity (PVA), Long-hand Method Example:**$500 Payments for 18 years, Earning 6% Annually FVA = $500{[(1 + 0.06)18 – 1]/0.06} FVA = $500{[(1.06)18 – 1]/0.06} FVA = $500[(2.85– 1)/0.06] FVA = $500(1.85/0.06) FVA = $500(30.91) FVA = $15,452.83 funded**Future Value Interest Factor of an Annuity (FVIFA)**Future Value Interest Factor of an Annuity (FVIFA): Factor multiplied by the annuity (payment) to determine the amount in the account at a future date**Future Value of an Annuity (FVA), Reference Table Method**Example:$500 Payments for 18 Years, Earning 6% Annually FVA = PMT x FVIFA i, n FVA = $500 x 30.906 FVA = $15,453.00funded**Calculating an Annuity Due**• Solve for an ordinary annuity by using the formula FVA = PMT {[(1 + i)n – 1]/i}, or the FVA table. • Multiply it by 1 plus the interest rate (1 + i) since the payments come at the beginning of the period. FVAd = FVA(1 + i) where FVAd = Future value of an annuity due FVA = Future value of an ordinary annuity i = Interest rate per period**Annuity Due from Previous Example**FVAd = FVA(1 + i) FVAd = $15,453.00(1 + 0.06) FVAd = $15,453.00(1.06) FVAd = $16,308.18funded**Present Value of an Annuity (PVA), Long-Hand Method**PVA = PMT ({1 – [1/(1 + i)n]}/i) where PVA = Present value of annuity PMT = Payment i = Interest rate per period n = Number of periods**Problem**• You want to be able to retire at age 65 with an income stream of $100,000 a year for the next 20 years. You think you will get a 5% return on your money. How much money will you need to save before you retire?**Present Value of an Annuity (PVA), Long-hand Method Example:**$100,000 Payment in 20 Years, Earning 5% PVA = $100,000 ({1 – [1/(1 + 0.05)20]}/0.05) PVA = $100,000 ({1 – [1/(1.05)20]}/0.05) PVA = $100,000 {[1 - (1/2.65)]/0.05} PVA = $100,000 [(1 - .38)/0.05] PVA = $100,000 (.62/0.05) PVA = $100,000 (12.46) PVA = $1,246,221.03 needed in retirement savings**Present Value Interest Factor of an Annuity (PVIFA)**Present Value Interest Factor of an Annuity (PVIFA): Factor multiplied by the annuity (payment) to determine the value of the annuity today**Present Value of an Annuity (PVA), Reference Table Method**Example: $100,000 Payments in 20 Years, Earning 6% PVA = PMT x PVIFA i, n PVA = $100,000 x PVIFA 6, 20 PVA = $100,000 x 12.462 PVA = $1,246,200.00needed in retirement savings**Calculating Loan Payments**PVA = PMT ({1 – [1/(1 + i)n]}/i) where PVA = Present value of an ordinary annuity PMT = Payment i = Interest rate per period n = Number of periods**Problem**If you are financing $15,000 at 6% interest for three years, making annual payments, you know the following: PVA = 15,000 i = .06 n = 3**Long-Hand MethodAnnual Loan Payment Calculation**• 15,000 = PMT ({1 – [1/(1 + .06)3]}/.06) • 15,000 = PMT {[1 – (1/1.063)]/.06} • 15,000 = PMT {[1 – (1/1.19)]/.06} • 15,000 = PMT [(1 – .84)/.06] • 15,000 = PMT (.16/.06) • 15,000 = PMT (2.67301) • Divide both sides by 2.67301 and your answer is: • $5,611.65 = PMT**Reference Table MethodAnnual Loan Payment Calculation**PVA = PMT (PVIFAi,n) 15,000 = PMT (2.673) 5,611.67 = PMT**Monthly Payments**To determine your monthly payments, divide your interest rate by 12. Your number of payments is now 36. For the above example, we would have the following: PVA = 15,000 i = .06/12 = 0.005 n = 36 PMT = ?**Long-Hand MethodMonthly Loan Payment Calculation**Using the formula, your equation would be as follows 15,000 = PMT ({1 – [1/(1 + .005)36]}/.005) 15,000 = PMT {[1 – (1/1.005)36]/.005} 15,000 = PMT {[1 – (1/1.19668)]/.005} 15,000 = PMT [(1 –.84)/.005] 15,000 = PMT (.16/.005) 15,000 = PMT (32.8710) 456.33 = PMT**Reference Table MethodMonthly Loan Payment Calculation**• Extrapolating, you would find PVIFA.5, 36 equals 32.871 • PVA = PMT (PVIFAi,n) • 15,000 = PMT (32.871) • 456.33 = PMT