1. Chapter 4 Time Value of Money القيمة الزمنية للنقود

2. 4-2 Learning Goals 1.Discuss the role of time value in finance, the use of computational aids, and the basic patterns of cash flow. مناقشة أهمية القيمة الزمنية في مجال التمويل، واستخدام الوسائل الحسابية والأنماط الرئيسية للتدفق النقدي. 2.Understand the concept of future value and present value, their calculation for single amounts, and the relationship between them. فهم القيمة المستقبلية والقيمة الحالية والية احتسابها لمبالغ فردية والعلاقة بينها. 3.Find the future value and the present value of both an ordinary annuity and an annuity due, and the present value of a perpetuity. احتساب القيمة المستقبلية والقيمة الحالية لكل من الدفعات السنوية في نهاية الفترة والدفعات في بداية الفترة والقيمة الحالية للتدفقات الدائمة (للأبد).

3. 4-3 Learning Goals (cont.) 4.Calculate both the future value and the present value of a mixed stream of cash flows. حساب كل من القيمة المستقبلية والحالية للتدفقات النقدية غير المنتظمة (دفعات غير متساوية). 5.Describe the procedures involved in (1) determining deposits needed to accumulate to a future sum, (2) loan amortization, (3) finding interest or growth rates, and (4) finding an unknown number of periods. وصف الإجراءات اللازمة في (1) تحديد الودائع اللازمة لتتراكم وتصل إلى مبلغ مستقبلي، (2) استهلاك القرض، (3) احتساب أسعار الفائدة أو معدل النمو (4) احتساب عدد مجهول من السنوات.

4. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-4 Question Would it be better for a company to invest $100,000 in a product that would return a total of $200,000 after one year, or one that would return $220,000 after two years? The Role of Time Value in Finance أهمية ودور القيمة الزمنية Most financial decisions involve costs & benefits that are spread out over time. معظم القرارات المالية تستخدم المنافع والتكاليف التي ظهرت او انتشرت عبر الزمن. Time value of money allows comparison of cash flows from different periods. القيمة الزمنية للنقود تسمح بمقارنة التدفقات النقدية من فترات مختلفة

5. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-5 Answer It depends on the interest rate! The Role of Time Value in Finance (cont.)

6. 4-6 Basic Concepts مفاهيم رئيسية Future Value: compounding or growth over time القيمة المستقبلية: تنمو خلال الزمن. Present Value: discounting to today’s valueخصم القيمة اليوم Single cash flows & series of cash flows can be considered تدفق نقدي لمرة واحدة & سلسلة من التدفقات النقدية من الممكن أخذها بعين الاعتبار. Time lines are used to illustrate these relationships الخطوط الزمنية تستخدم للإشارة ولتوضيح هذه العلاقات.

7. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-7 Computational Aids الأدوات الحسابية Use the Equations استخدام المعادلات Use the Financial Tables استخدام الجداول المالية Use Financial Calculators استخدام الآلات الحاسبة المالية. Use Electronic Spreadsheets استخدام الجداول المالية مثل برنامج الاكسل

8. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-8 Computational Aids (cont.)

9. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-9 Computational Aids (cont.)

10. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-10 Computational Aids (cont.)

11. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-11 Computational Aids (cont.)

12. 4-12 Basic Patterns of Cash Flow الأنماط أو الأشكال الرئيسية للتدفقات النقدية The cash inflows and outflows of a firm can be described by its general pattern. التدفقات النقدية الداخلة والخارجة لشركة ممكن توضيحها من خلال شكلها او نمطها العام The three basic patterns include a single amount, an annuity, or a mixed stream: الأنماط الرئيسية الثلاثة تشمل مبلغ واحد و تدفقات منتظمة وتدفقات غير منتظمة.

13. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-13 Simple Interest الفائدة البسيطة With simple interest, you don’t earn interest on interest. لا يكسب فائدة على الفائدة Year 1: 5% of $100=$5 + $100 = $105 Year 2: 5% of $100=$5 + $105 = $110 Year 3: 5% of $100=$5 + $110 = $115 Year 4: 5% of $100=$5 + $115 = $120 Year 5: 5% of $100=$5 + $120 = $125

14. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-14 Compound Interest الفائدة المركبة With compound interest, a depositor earns interest on interest! المودع يكسب فائدة على الفائدة Year 1: 5% of $100.00= $5.00 + $100.00= $105.00 Year 2: 5% of $105.00= $5.25 + $105.00= $110.25 Year 3: 5% of $110.25 = $5.51+ $110.25= $115.76 Year 4: 5% of $115.76= $5.79 + $115.76= $121.55 Year 5: 5% of $121.55= $6.08 + $121.55= $127.63

15. 4-15 Time Value Terms رموز او مصطلحات القيمة الزمنية PV 0 =present value or beginning amount القيمة الحالية او الرصيد الابتدائي i= interest rate معدل الفائدة FV n =future value at end of “n” periods القيمة المستقبلية في نهاية عدد من الفترات n=number of compounding periods عدد الفترات او السنوات المركبة A=an annuity (series of equal payments or receipts) القسط السنوي (عدد من الدفعات أو المقبوضات المتساوية)

16. 4-16 Four Basic Models FV n = PV 0 (1+i) n = PV x (FVIF i,n ) PV 0 = FV n [1/(1+i) n ] = FV x (PVIF i,n ) FVA n = A (1+i) n - 1= A x (FVIFA i,n ) i PVA 0 = A 1 - [1/(1+i) n ] = A x (PVIFA i,n ) i Present Value Interest Factor Present Value Interest Factor = PVIF

17. 4-17 Future Value of a Single Amount القيمة المستقبلية لمبلغ واحد Future Value techniques typically measure cash flows at the end of a project’s life. طرق القيمة المستقبلية هي تقيس تماما التدفقات النقدية في نهاية حياة المشروع. Future value is cash you will receive at a given future date. القيمة المستقبلية هي النقدية التي سوف تحصل عليها في تاريخ مستقبلي محدد. The future value technique uses compounding to find the future value of each cash flow at the end of an investment’s life and then sums these values to find the investment’s future value. طريقة القيمة المستقبلية تستخدم التجميع لايجاد القيمة المستقبلية لكل تدفق نقدي في نهاية فترة الاستثمار ومن ثم تجمع هذه القيم لايجاد القيمة المستقبلية للاستثمار.

18. 4-18 $100 x (1.08) 1 = $100 x FVIF 8%,1 $100 x 1.08 =$108 Future Value of a Single Amount: القيمة المستقبلية لدفعة واحدة فقط If Fred Moreno places $100 in a savings account paying 8% interest compounded annually, how much will he have in the account at the end of one year ? اذا فرد مورينو يضع مبلغ 100$ في حساب توفير عليه فائدة مركبة 8% سنويا، كم هو سيكون لديه في حسابه بعد نهاية سنة؟؟؟ FV n = PV 0 (1+i) n حسب القانون التالي

19. FV n = PV 0 (1+i) n 4-19 FV 5 = $800 X (1 + 0.06) 5 = $800 X (1.338) = $1,070.40 Future Value of a Single Amount: The Equation for Future Value استخدام المعادلة لحساب القيمة المستقبلية لدفعة واحدة فقط Jane Farber places $800 in a savings account paying 6% interest compounded annually. She wants to know how much money will be in the account at the end of five years.

20. 4-20 Present Value of a Single Amountالقيمة الحالية لدفعة واحدة فقط Present value is the current dollar value of a future amount of money. القيمة الحالية هي قيمة الدولار الحالية لمبلغ مالي مستقبلي. It is based on the idea that a dollar today is worth more than a dollar tomorrow. هي على الأساس الفكرة ان دولار اليوم قيمته تكون افضل من دولار غدا. It is the amount today that must be invested at a given rate to reach a future amount. انه المبلغ اليوم الذي يجب ان يستثمر بمعدل محدد للوصول الي مبلغ مستقبلي. Calculating present value is also known as discounting.حساب القيمة الحالية يسمى أيضا بالخصم. The discount rate is often also referred to as the opportunity cost, the discount rate, the required return, or the cost of capital معدل الخصم غالبا يشير اليه بتكلفة الفرصة، معدل الخصم، معدل العائد، تكلفة راس المال..

21. Present Value Interest Factor Present Value Interest Factor = PVIF 4-21 Four Basic Models FV n = PV 0 (1+i) n = PV x (FVIF i,n ) PV 0 = FV n [1/(1+i) n ] = FV x (PVIF i,n ) FVA n = A (1+i) n - 1= A x (FVIFA i,n ) i PVA 0 = A 1 - [1/(1+i) n ] = A x (PVIFA i,n ) i

22. 4-22 $300 x [1/(1.06) 1 ] =$300 x PVIF 6%,1 $300 x 0.9434 = $283.02 Present Value of a Single Amount: القيمة الحالية لدفعة واحدة فقط Using PVIF Tables باستخدام جداول القيمة الحالية لتدفقات نقدية متساوية Paul Shorter has an opportunity to receive $300 one year from now. If he can earn 6% on his investments, what is the most he should pay now for this opportunity? PV 0 = FV n [1/(1+i) n ]

23. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-23 PV = $1,700/(1 + 0.08) 8 = $1,700/1.851 = $918.42 Present Value of a Single Amount: Pam Valenti wishes to find the present value of $1,700 that will be received 8 years from now. Pam’s opportunity cost is 8%. PV 0 = FV n [1/(1+i) n ]

24. 4-24 Annuities الأقساط \ الدفعات المتساوية Annuities are equally-spaced cash flows of equal size. التدفقات المنتظمة هي تدفقات نقدية متساوية الحجم والمقدار Annuities can be either inflows or outflows. هي ممكن أن تكون اما تدفقات داخلة أو خارجة. An ordinary (deferred) annuity has cash flows that occur at the end of each period. الأقساط السنوية هي تدفقات نقدية تحدث في نهاية كل فترة زمنية. An annuity due has cash flows that occur at the beginning of each period. الدفعة المستحقة لها تدفقات نقدية والتي تحدث في بداية كل فترة.

25. 4-25 Types of Annuities أنواع الدفعات Note that the amount of both annuities total $5,000. Fran Abrams is choosing which of two annuities to receive. Both are 5-year $1,000 annuities; annuity A is an ordinary annuity, and annuity B is an annuity due. Fran has listed the cash flows for both annuities as shown in Table 4.1 on the following slide.

26. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-26 Future Value of an Ordinary Annuity

27. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-27 Future Value of an Ordinary Annuity (cont.) Fran Abrams wishes to determine how much money she will have at the end of 5 years if he chooses annuity A, the ordinary annuity and it earns 7% annually. Annuity a is depicted graphically below: ابرامز تتأمل بان تحدد كم من النقود سوف تستلم في نهاية 5 سنوات اذا هي اختارت دفعات متساوية، الدفعات المتساوية تكون الفائدة عليها 7%.

28. 4-28 Future Value of an Ordinary Annuity: Using the FVIFA Tables باستخدام جداول القيمة الحالية لتدفقات نقدية متساوية FVA n = A (1+i) n - 1 = i FVA = $1,000 (FVIFA,7%,5) = $1,000 (5.751) = $5,751

29. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-29 Present Value of an Ordinary Annuity Braden Company, a small producer of plastic toys, wants to determine the most it should pay to purchase a particular annuity. The annuity consists of cash flows of $700 at the end of each year for 5 years. The required return is 8%.

30. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-30 Present Value of an Ordinary Annuity: - The Long Method الطريقة المطولة PV 0 = FV n [1/(1+i) n ] استخدم هذا القانون

31. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-31 Present Value of an Ordinary Annuity: Using PVIFA Tables باستخدام جداول القيمة الحالية لتدفقات نقدية متساوية PVA =$700 (PVIFA,8%,5) =$700 (3.993) =$2,795.10 أو من خلال القانون المعطي سابقاً PVA 0 = A 1 - [1/(1+i) n ] i

32. 4-32 PV = Annuity/Interest Rate PV = $1,000/.08 = $12,500 Present Value of a Perpetuity القيمة الحالية لدفعات أبدية (دائمة أو مستمرة للأبد) A perpetuity is a special kind of annuity. هي نوع خاص من الدفعات With a perpetuity, the periodic annuity or cash flow stream continues forever.الدفعة الدورية النقدية تستمر بالتدفق للأبد (دائمة). For example, how much would I have to deposit today in order to withdraw $1,000 each year forever if I can earn 8% on my deposit?

33. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-33 Future Value of a Mixed Stream القيمة المستقبلية لتدفقات نقدية غير منتظمة

34. 4-34 Future Value of a Mixed Stream (cont.) FV n = PV 0 (1+i) n

35. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-35 Present Value of a Mixed Stream Frey Company, a shoe manufacturer, has been offered an opportunity to receive the following mixed stream of cash flows over the next 5 years.

36. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-36 Present Value of a Mixed Stream If the firm must earn at least 9% on its investments, what is the most it should pay for this opportunity? This situation is depicted on the following time line.

37. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-37 Present Value of a Mixed Stream PV 0 = FV n [1/(1+i) n ]

38. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-38 Compounding Interest More Frequently Than Annually Compounding more frequently than once a year results in a higher effective interest rate because you are earning on interest on interest more frequently. As a result, the effective interest rate is greater than the nominal (annual) interest rate. Furthermore, the effective rate of interest will increase the more frequently interest is compounded.

39. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-39 Compounding Interest More Frequently Than Annually (cont.) Fred Moreno has found an institution that will pay him 8% annual interest, compounded quarterly. If he leaves the money in the account for 24 months (2 years), he will be paid 2% interest compounded over eight periods.

40. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-40 Compounding Interest More Frequently Than Annually (cont.)

41. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-41 Compounding Interest More Frequently Than Annually (cont.)

42. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-42 Compounding Interest More Frequently Than Annually (cont.) A General Equation for Compounding More Frequently than Annually

43. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-43 Compounding Interest More Frequently Than Annually (cont.) A General Equation for Compounding More Frequently than Annually – Recalculate the example for the Fred Moreno example assuming (1) semiannual compounding and (2) quarterly compounding.

44. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-44 FV n (continuous compounding) = PV x (e kxn ) where “e” has a value of 2.7183. Continuous Compounding With continuous compounding the number of compounding periods per year approaches infinity. Through the use of calculus, the equation thus becomes: Continuing with the previous example, find the Future value of the $100 deposit after 5 years if interest is compounded continuously.

45. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-45 FV n (continuous compounding) = PV x (e kxn ) where “e” has a value of 2.7183. FVn = 100 x (2.7183).08x2 = $117.35 Continuous Compounding (cont.) With continuous compounding the number of compounding periods per year approaches infinity. Through the use of calculus, the equation thus becomes:

46. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-46 EAR = (1 + i/m) m - 1 Nominal & Effective Annual Rates of Interest The nominal interest rate is the stated or contractual rate of interest charged by a lender or promised by a borrower. The effective interest rate is the rate actually paid or earned. In general, the effective rate > nominal rate whenever compounding occurs more than once per year

47. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-47 Nominal & Effective Annual Rates of Interest (cont.) Fred Moreno wishes to find the effective annual rate associated with an 8% nominal annual rate (I =.08) when interest is compounded (1) annually (m=1); (2) semiannually (m=2); and (3) quarterly (m=4).

48. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-48 Special Applications of Time Value: Deposits Needed to Accumulate to a Future Sum

49. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-49 PMT = $30,000/5.637 = $5,321.98 Special Applications of Time Value: Deposits Needed to Accumulate to a Future Sum (cont.) Suppose you want to buy a house 5 years from now and you estimate that the down payment needed will be $30,000. How much would you need to deposit at the end of each year for the next 5 years to accumulate $30,000 if you can earn 6% on your deposits?

50. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-50 Special Applications of Time Value: Loan Amortization

51. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-51 Ray Noble wishes to find the rate of interest or growth reflected in the stream of cash flows he received from a real estate investment over the period from 2002 through 2006 as shown in the table on the following slide. Special Applications of Time Value: Interest or Growth Rates At times, it may be desirable to determine the compound interest rate or growth rate implied by a series of cash flows.

52. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-52 PVIF i,5yrs = PV/FV = ($1,250/$1,520) = 0.822 PVIF i,5yrs = approximately 5% Special Applications of Time Value: Interest or Growth Rates (cont.)

53. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-53 Ann Bates wishes to determine the number of years it will take for her initial $1,000 deposit, earning 8% annual interest, to grow to equal $2,500. Simply stated, at an 8% annual rate of interest, how many years, n, will it take for Ann’s $1,000 (PV n ) to grow to $2,500 (FV n )? Special Applications of Time Value: Finding an Unknown Number of Periods At times, it may be desirable to determine the number of time periods needed to generate a given amount of cash flow from an initial amount.

54. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 4-54 PVIF 8%,n = PV/FV = ($1,000/$2,500) =.400 PVIF 8%,n = approximately 12 years Special Applications of Time Value: Finding an Unknown Number of Periods (cont.)