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Chapter 5

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The Time Value of Money

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Chapter Objectives Understand and calculate compound interest Understand the relationship between compounding and bringing money back to the present Annuity and future value Annuity Due Future value and present value of a sum with non- annual compounding Determine the present value of an uneven stream of payments Perpetuity Understand how the international setting complicates time value of money

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Compound Interest When interest paid on an investment is added to the principal, then during the next period, interest is earned on the new sum

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Simple Interest Interest is earned on principal $100 invested at 6% per year 1 st yearinterest is $6.00 2 nd yearinterest is $6.00 3 rd year interest is $6.00 Total interest earned:$18

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Compound Interest Interest is earned on previously earned interest $100 invested at 6% with annual compounding 1 st yearinterest is $6.00Principal is $106 2 nd yearinterest is $6.36Principal is $112.36 3 rd yearinterest is $6.74Principal is $119.11 Total interest earned: $19.10

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Future Value How much a sum will grow in a certain number of years when compounded at a specific rate.

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Future Value What will an investment be worth in a year? $100 invested at 7% FV = PV(1+i) $100 (1+.07) $100 (1.07) = $107

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Future Value Future Value can be increased by: – Increasing number of years of compounding – Increasing the interest or discount rate

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Future Value What is the future value of $1,000 invested at 12% for 3 years? (Assume annual compounding) Using the tables, look at 12% column, 3 time periods. What is the factor? $1,000 X 1.4049 = 1,404.90

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Present Value What is the value in today’s dollars of a sum of money to be received in the future ? or The current value of a future payment

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Present Value What is the present value of $1,000 to be received in 5 years if the discount rate is 10%? Using the present value of $1 table, 10% column, 5 time periods $1,000 X.621 = $621 $621 today equals $1,000 in 5 years

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Annuity Series of equal dollar payments for a specified number of years. Ordinary annuity payments occur at the end of each period

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Compound Annuity Depositing or investing an equal sum of money at the end of each year for a certain number of years and allowing it to grow.

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Compounding Annuity What will $1,000 deposited every year for eight years at 10% be worth? Use the future value of an annuity table, 10% column, eight time periods $1,000 X 11.436 = $11,436

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Future Value of an Annuity If we need $8,000 in 6 years (and the discount rate is 10%), how much should be deposited each year? Use the Future Value of an Annuity table, 10% column, six time periods. $8,000 / 7.716 = $1036.81 per year

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Present Value of an Annuity Pensions, insurance obligations, and interest received from bonds are all annuities. These items all have a present value. Calculate the present value of an annuity using the present value of annuity table.

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Present Value of an Annuity Calculate the present value of a $100 annuity received annually for 10 years when the discount rate is 6%. $100 X 7.360 = $736

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Present Value of an Annuity Would you rather receive $450 dollars today or $100 a year for the next five years? Discount rate is 6%. To compare these options, use present value. The present value of $450 today is $450. The present value of a $100 annuity for 5 years at 6% is XXX?

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Present Value table, five time periods, 6% column factor is 4.2124 $100 X 4.2124 = 421.24 Which option will you choose? $450 today or $100 a year for five years

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Annuities Due Ordinary annuities in which all payments have been shifted forward by one time period.

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Amortized Loans Loans paid off in equal installments over time – Typically Home Mortgages

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Payments and Annuities If you want to finance a new motorcycle with a purchase price of $25,000 at an interest rate of 8% over 5 years, what will your payments be? Use the present value of an annuity table, five time periods, 8% column – factor is 3.993 $25,000 / 3.993 = 6,260.96 Five annual payments of $6,260.96

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Amortization of a Loan Reducing the balance of a loan via annuity payments is called amortizing. A typical amortization schedule looks at payment, interest, principal payment and balance.

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Amortization Schedule Amortize the payments on a 5-year loan for $10,000 at 6% interest. NPaymentInterestPrin. PayNew Balance (PxRxT)(Payment -(Principal – Prin Pay) Interest) 1$2,373.96$600$1,773.96$8,226.04 2 $2,373.96$493.56$1,880.40$6,345.64 3 $2,373.96$380.74$1,993.22$4352.42 4 $2,373.96$261.15$2,112.81$2,239.61 5 $2,373.99$134.38$2239.61-----------

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Mortgage Payments How much principal is paid on the first payment of a $70,000 mortgage with 10% interest, on a 30 year loan (with monthly payments) Payment is $614 How much of this payment goes to principal and how much goes to interest? $70,000 x.10 x 1/12 = $583 Payment of $614, $583 is interest, $31 is applied toward principal

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Compounding Interest with Non-annual periods If using the tables, divide the percentage by the number of compounding periods in a year, and multiply the time periods by the number of compounding periods in a year. Example: 10% a year, with semi annual compounding for 5 years. 10% / 2 = 5% column on the tables N = 5 years, with semi annual compounding or 10 Use 10 for Number of periods, 5% each

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Non-annual Compounding What factors should be used to calculate 5 years at 12% compounded quarterly N = 5 x 4 = 20 % = 12% / 4 = 3% Use 3% column, 20 time periods

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Perpetuity An annuity that continues forever is called perpetuity The present value of a perpetuity is PV = PP/i PV = present value PP = Constant dollar amount of perpetuity i = Annuity discount rate

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Future Value of $1 Table N6%8%10%12% 11.061.08001.10001.1200 21.12361.16641.21001.2544 31.19101.25971.33101.4049 41.26251.36051.46411.5735 51.33821.46931.61051.7623 61.41851.58691.77161.9738 71.50361.71381.94872.2107 81.59381.85092.14362.4760 91.68951.99902.35792.7731 101.79082.15892.59373.1058

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Present Value of $1 N6%8%10%12% 1.9434.9259.9091.8929 2.8900.8573.8264.7972 3.8396.7938.7513.7118 4.7921.7350.6830.6355 5.7473.6806.6209.5674 6.7050.6302.5645.5066 7.6651.5835.5132.4523 8.6274.5403.4665.4039 9.5919.5002.4241.3606 10.5584.4632.3855.3220

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Future Value of Annuity N6%8%10%12% 11.0001.00001.0001.000 22.0602.08002.1002.1200 33.18363.24643.3103.3744 44.37464.50614.64104.7793 55.63715.86666.10516.3528 66.97537.33597.71568.1152 78.39388.92289.487210.8090 89.897510.636611.435912.2997 911.491312.487613.579514.7757 1013.180814.486615.937417.5487

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Present Value of an Annuity N6%8%10%12% 1.9434.9259.9091.8929 21.83341.78331.73551.6901 32.67302.57712.48692.4018 43.46513.31213.16993.0373 54.21243.99273.79083.6048 64.91734.62294.35534.1114 75.58245.20644.86844.5638 86.20985.74665.33494.9676 96.80176.24695.75905.3282 107.36016.71016.14465.6502

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Chapter 5 Introduction This chapter introduces the topic of financial mathematics also known as the time value of money. This is a foundation topic relevant.

Chapter 5 Introduction This chapter introduces the topic of financial mathematics also known as the time value of money. This is a foundation topic relevant.

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