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The Time Value of Money

The Time Value of Money. Chapter 5. LEARNING OBJECTIVES. 1.   Explain the mechanics of compounding when invested. 2.  Present value and future value. 3.  Ordinary annuity and its future value. 4.  An ordinary annuity and an annuity due. 5.  Non-annual future or present value of a sum .

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The Time Value of Money

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  1. The Time Value of Money Chapter 5

  2. LEARNING OBJECTIVES • 1.  Explain the mechanics of compounding when invested. • 2.  Present value and future value. • 3.  Ordinary annuity and its future value. • 4.  An ordinary annuity and an annuity due. • 5.  Non-annual future or present value of a sum . • 6. Determine the present value of a perpetuity.

  3. Power of time of value of money • History of Interest Rates $1000 ( 1 + .08)400 = ?

  4. Power of time value of money • Money Angles: by Andrew Tobias. 1. Chessboard with the King 2. Manhattan

  5. Terms • Compound Interest • Future value and Present Value • Annuities • Annuities Due • Amortized Loans • Compound Interest with Non-annual Periods • Present Value of an Uneven Stream· • Perpetuities

  6. COMPOUND INTEREST • FV1=PV (1+i) (5-1) • Where FV1=the future value of the investment at the end of one year • i=the annual interest (or discount) rate • PV=the present value, or original amount invested at the beginning of the first year

  7. Future value 1. Simple compounding 2. Complex compounding

  8. Future value

  9. FV1=PV (1+i) • =$100(1+0.06) • =$100(1.06) • =$106

  10. Compound twice a year

  11. Compound four times a year

  12. Compound 12 times a year

  13. Compound 360 times a year

  14. Continuous compounding

  15. Illustration of Compound Interest Calculations Year Beginning Value Interest Earned Ending Value 1 $100.00 $6.00 $106.00 2 106.00 6.36 112.36 3 112.36 6.74 119.10 4 119.10 7.15 126.25 5 126.25 7.57 133.82 6 133.82 8.03 141.85 7 141.85 8.51 150.36 8 150.36 9.02 159.38 9 159.38 9.57 168.95 10 168.95 10.13 179.08

  16. Future value and future value interest factor

  17. FVn=PV(FVIFi,n)

  18. Table 5-2 FVIFi,n or the Compound Sum of $1 N 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 1 1.010 1.020 1.030 1.040 1.050 1.060 1.070 1.080 1.090 1.100 2 1.020 1.040 1.061 1.082 1.102 1.124 1.145 1.166 1.188 1.210 3 1.030 1.061 1.093 1.125 1.158 1.191 1.225 1.260 1.295 1.331 4 1.041 1.082 1.126 1.170 1.216 1.262 1.311 1.360 1.412 1.464 5 1.051 1.104 1.159 1.217 1.276 1.338 1.403 1.469 1.539 1.611 6 1.062 1.126 1.194 1.265 1.340 1.419 1.501 1.587 1.677 1.772 7 1.072 1.149 1.230 1.316 1.407 1.504 1.606 1.714 1.828 1.949 8 1.083 1.172 1.267 1.369 1.477 1.594 1.718 1.815 1.993 2.144 9 1.094 1.195 1.305 1.423 1.551 1.689 1.838 1.999 2.172 2.358 10 1.105 1.219 1.344 1.480 1.629 1.791 1.967 2.159 2.367 2.594 11 1.116 1.243 1.384 1.539 1.710 1.898 2.105 2.332 2.580 2.853 12 1.127 1.268 1.426 1.601 1.796 2.012 2.252 2.518 2.813 3.138 13 1.138 1.294 1.469 1.665 1.886 2.133 2.410 2.720 3.066 3.452 14 1.149 1.319 1.513 1.732 1.980 2.261 2.579 2.937 3.342 3.797 15 1.161 1.346 1.558 1.801 2.079 2.397 2.759 3.172 3.642 4.177

  19. PV=$300, Vn=$774; i=9 % N= ？

  20. PV=$100; FVn=$179.10; n=10 years. I= ?

  21. PRESENT VALUE

  22. FV10=$500, n=10, i=6 % PV = ?

  23. (PVIF i, n) • present-value interest factor for I and n (PVIF i, n), (PVIF i, n) = 1/(1+i)

  24. Present value • FV10 =$1,500 • N= 10 years • discount rate= 8 % • PV=$1500(0.463) =$694.50

  25. Table 5-3 PVIFi,n or the Present Value of $1 N 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 1 0.990 0.980 0.971 0.962 0.952 0.943 0.935 0.926 0.917 0.909 2 0.980 0.961 0.943 0.925 0.907 0.890 0.873 0.857 0.842 0.826 3 0.971 0.942 0.915 0.889 0.864 0.840 0.816 0.794 0.772 0.751 4 0.961 0.924 0.888 0.855 0.823 0.792 0.763 0.735 0.708 0.683 5 0.951 0.906 0.863 0.822 0.784 0.747 0.713 0.681 0.650 0.621 6 0.942 0.888 0.837 0.790 0.746 0.705 0.666 0.630 0.596 0.564 7 0.933 0.871 0.813 0.760 0.711 0.655 0.623 0.583 0.547 0.513 8 0.914 0.837 0.766 0.703 0.645 0.592 0.544 0.500 0.460 0.424 9 0.905 0.820 0.744 0.676 0.614 0.558 0.508 0.463 0.422 0.386

  26. ANNUITIES • Annuity: equal annual cash flows. • Ordinary annuity: at the end of each period. • Annuity due: at the beginning of each eriod.

  27. Table 5-4 • Illustration of a Five-Year $500 Annuity Compounded at 6 percent • YEAR 0 1 2 3 4 5 • DOLLAR DEPOSITS AT END OF YEAR 500 500 500 500 500 • $500.00 • 530.00 • 562.00 • 595.50 • 631.00 • Future value of the annuity$2,818.50

  28. Ordinary annuity FVIFAk,n = [(1/k) ( (1+ k)n – 1)]

  29. Present value of an Annuity

  30. Table 5-6 Illustration of a Five-Year $500 Annuity Discounted to the Present at 6 percent YEAR 0 1 2 3 4 5 Dollars received at the 500 500 500 500 500 the end of year $471.50 445.00 420.00 396.00 373.50 PV annuity $2,106.00

  31. PVIFAK,n = (1/k) [( 1 – 1/(1+k)n]

  32. Table 5-7 PVIFi,n or the Present Value of an Annuity of $1 N 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 1 0.990 0.980 0.971 0.962 0.952 0.943 0.935 0.926 0.917 0.909 2 1.970 1.942 1.913 1.886 1.859 1.833 1.808 1.783 1.759 1.736 3 2.941 2.884 2.829 2.775 2.723 2.673 2.642 2.577 2.531 2.487 4 3.902 3.808 3.717 3.630 3.546 3.465 3.387 3.312 3.240 3.170 5 4.853 4.713 4.580 4.452 4.329 4.212 4.100 3.003 3.890 3.791 6 5.795 5.601 5.417 5.242 5.076 4.917 4.767 4.623 4.486 4.355 7 6.728 6.472 6.230 6.002 5.786 5.582 5.389 5.206 5.033 4.868 8 7.652 7.326 7.020 6.733 6.463 6.210 5.971 5.747 5.535 5.335 9 8.566 8.162 7.786 7.435 7.108 6.802 6.515 6.247 5.995 5.759 10 9.471 8.983 8.530 8.111 7.722 7.360 7.024 6.710 6.418 6.145

  33. n=10 years, I=5 percent, and current PMT=$1,000 PV= $1,000(7.722) = $7,722

  34. PMT Annuity: $5,000, n =5 years, i=8 percent, PMT:? $5,000 = PMT (3.993) $1,252.19=PMT

  35. AMORTIZED LOANS

  36. Loan Amortization Schedule Involving a $6,000 Loan at 15 Percent to Be Repaid in Four Years Year Annuity Interest Portion Repayment of Outstanding Of The Annuity1 The Principal Loan Balance Portion Of The After The An- Annuity2 nuity Payment 1 $2,101.58 $900.00 $1,201.58 $4,798.42 22,101.58 719.76 1,381.82 3,416.60 3 2,101.58 512.49 1,589.09 1,827.51 4 2,101.58 274.07 1,827.51

  37. ANNUITIES DUE • FVn (annuity due)=PMT(FVIFA I,n)(1+I) (5-10) FV5=$500(FVIFA5%,5)(1+0.06) =$500(5.637)(1.06) =$2,987.61  from $2,106 to $2,232.36, PV=$500(PVIFA6%,5)(1+0.06) =$500(4.212)(1.06) =$2,232.36

  38. End year Loan payment (1) Beginning principal (2) payments End of year principal(5) [(2)－(4)] Interest(3) [0.1 × (2)] Principal(4) [(1) － (3)] 1 $1892.74 $6000.00 $600.00 $1292.74 $4707.26 2 $1892.74 $4707.26 $470.73 $1422.01 $3285.25 3 $1892.74 $3285.25 $328.53 $1564.21 $1721.04 4 $1892.74 $1721.04 $172.10 $1720.64

  39. The Value of $100 Compounded at Various IntervalsFOR 1 YEAR AT i PERCENT I = 2% 5% 10% 15% Compounded annually $102.00 $105.00 $110.00 $115.00 Compounded semiannually 102.01 105.06 110.25 115.56 Compounded quarterly 102.02 105.09 110.38 115.87 Compounded monthly 102.02 105.12 110.47 116.08 Compounded weekly (52) 102.02 105.12 110.51 116.16 Compounded daily (365) 102.02 105.13 110.52 116.18

  40. PRESENT VALUE OF AN UNEVEN STREAM YEAR CASH FLOW YEAR CASH FLOW 1 $500 6 500 2 200 7 500 3 -400 8 500 4 500 9 500 5 500 10 500

  41. Present value of $500 received at the end of one year • = $500(0.943) = $471.50 • 2. Present value of $200 received at the end of tree years • = $200(0.890) = 178.00 • 3. Present value of a $400 outflow at the end of three years • = -400(0.840) = -336.00 • 4. (a) Value at the end of year 3 and a $500 annuity, years 4 through 10 • = $500 (5.582) = $2,791.00 • (b) Present value of $2,791.00 received at the end of year 3 • = 2,791(0.840) = 2,344.44 • 5. Total present value = $2,657.94

  42. Quiz 1 Warm up Quiz. Terms: : n = 5, m = 4, I =12 percent, and PV =$100 solve for fv

  43. Quiz 2 What is the present value of an investment involving $200 received at the end of years 1 through 4, a $300 cash outflow at the end of year 5 to 8, and $500 received at the end of years 9 through 10, given a 5 percent discount rate?

  44. Quiz 3 • A 25 year-old graduate has his $50,000 salary a year. How much will he get when he reaches to 60 (35 years later)year-old with a value rate of 8%(annual compounding). • The graduate will have his $80,000 salary at age of 30. How much will he get when he reaches to his age of 60(30 years later) with the value rate of 8%(semi-annual compounding).

  45. Quiz 4 • The graduate will have his $100,000 salary at age of 40. How much will he get when he reaches to his age of 60(20 years later) with the value rate of 12%(quarterly-annual compounding). • Compute the future value from 25-30/30-40/40-60 year old with the same rate and the compounding rate.

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