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Chapter 5

Chapter 5. The Time Value of Money. 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

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Chapter 5

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  1. Chapter 5

  2. The Time Value of Money

  3. 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

  4. 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

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

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

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

  8. 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

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

  10. 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

  11. 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

  12. 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

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

  14. 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.

  15. 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

  16. 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

  17. 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.

  18. 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

  19. 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?

  20. 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

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

  22. Amortized Loans • Loans paid off in equal installments over time • Typically Home Mortgages

  23. 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

  24. 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.

  25. Amortization Schedule Amortize the payments on a 5-year loan for $10,000 at 6% interest. N Payment Interest Prin. Pay New 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 -----------

  26. 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

  27. 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

  28. 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

  29. 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

  30. Future Value of $1 Table N6%8%10%12% 1 1.06 1.0800 1.1000 1.1200 2 1.1236 1.1664 1.2100 1.2544 3 1.1910 1.2597 1.3310 1.4049 4 1.2625 1.3605 1.4641 1.5735 5 1.3382 1.4693 1.6105 1.7623 6 1.4185 1.5869 1.7716 1.9738 7 1.5036 1.7138 1.9487 2.2107 8 1.5938 1.8509 2.1436 2.4760 9 1.6895 1.9990 2.3579 2.7731 10 1.7908 2.1589 2.5937 3.1058

  31. 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

  32. Future Value of Annuity N6%8%10%12% 1 1.000 1.0000 1.000 1.000 2 2.060 2.0800 2.100 2.1200 3 3.1836 3.2464 3.310 3.3744 4 4.3746 4.5061 4.6410 4.7793 5 5.6371 5.8666 6.1051 6.3528 6 6.9753 7.3359 7.7156 8.1152 7 8.3938 8.9228 9.4872 10.8090 8 9.8975 10.6366 11.4359 12.2997 9 11.4913 12.4876 13.5795 14.7757 10 13.1808 14.4866 15.9374 17.5487

  33. Present Value of an Annuity N6%8%10%12% 1 .9434 .9259 .9091 .8929 2 1.8334 1.7833 1.7355 1.6901 3 2.6730 2.5771 2.4869 2.4018 4 3.4651 3.3121 3.1699 3.0373 5 4.2124 3.9927 3.7908 3.6048 6 4.9173 4.6229 4.3553 4.1114 7 5.5824 5.2064 4.8684 4.5638 8 6.2098 5.7466 5.3349 4.9676 9 6.8017 6.2469 5.7590 5.3282 10 7.3601 6.7101 6.1446 5.6502

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