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4. The Relationship Between Present and Future Value Calculating Perpetuities Calculating Values for Different Durations of Compounding Periods Comparing Interest Rates Calculating Values for Fractional Time Periods Loans and Loan Amortization Additional Detail on Present and Future Values The Time Value of Money > Additional Detail on Present and Future Values Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/finance?campaign_content=book_192_section_59&campaign_term=Finance&utm_campaign=powerpoint&utm_medium=dir ect&utm_source=boundless

5. The future value (FV) measures the nominal future sum of money that a given sum of money is "worth" at a specified time in the future assuming a certain interest rate, or more generally, rate of return. The FV is calculated by multiplying the present value by the accumulation function. PV and FV vary jointly: when one increases, the other increases, assuming that the interest rate and number of periods remain constant. As the interest rate (discount rate) and number of periods increase, FV increases or PV decreases. The Relationship Between Present and Future Value Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/finance/textbooks/boundless-finance-textbook/the-time-value-of-money-5/additional-detail-on-present-and-future-values- 59/the-relationship-between-present-and-future-value-271- 1835?campaign_content=book_192_section_59&campaign_term=Finance&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess FV of a single payment View on Boundless.com The Time Value of Money > Additional Detail on Present and Future Values

6. Perpetuities are a special type of annuity; a perpetuity is an annuity that has no end, or a stream of cash payments that continues forever. To find the future value of a perpetuity requires having a future date, which effectively converts the perpetuity to an ordinary annuity until that point. Perpetuities with growing payments are called Growing Perpetuities; the growth rate is subtracted from the interest rate in the present value equation. Calculating Perpetuities Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/finance/textbooks/boundless-finance-textbook/the-time-value-of-money-5/additional-detail-on-present-and-future-values- 59/calculating-perpetuities-272- 1836?campaign_content=book_192_section_59&campaign_term=Finance&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess View on Boundless.com The Time Value of Money > Additional Detail on Present and Future Values

7. The units of the period (e.g. one year) must be the same as the units in the interest rate (e.g. 7% per year). When interest compounds more than once a year, the effective interest rate (EAR) is different from the nominal interest rate. The equation in skips the step of solving for EAR, and is directly usable to find the present or future value of a sum. Calculating Values for Different Durations of Compounding Periods Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/finance/textbooks/boundless-finance-textbook/the-time-value-of-money-5/additional-detail-on-present-and-future-values- 59/calculating-values-for-different-durations-of-compounding-periods-273- 8314?campaign_content=book_192_section_59&campaign_term=Finance&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess EAR with Continuous Compounding View on Boundless.com The Time Value of Money > Additional Detail on Present and Future Values

8. A nominal interest rate that compounds has a different effective rate (EAR), because interest is accrued on interest. The Fisher Equation approximates the amount of interest accrued after accounting for inflation. A company will theoretically only invest if the expected return is higher than their cost of capital, even if the return has a high nominal value. Comparing Interest Rates Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/finance/textbooks/boundless-finance-textbook/the-time-value-of-money-5/additional-detail-on-present-and-future-values- 59/comparing-interest-rates-274- 1469?campaign_content=book_192_section_59&campaign_term=Finance&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Fisher Equation View on Boundless.com The Time Value of Money > Additional Detail on Present and Future Values

9. The balance of an account only changes when interest is paid. To find the balance, round the fractional time period down to the period when interest was last accrued. To find the PV or FV, ignore when interest was last paid an use the fractional time period as the time period in the equation. The discount rate is really the cost of not having the money over time, so for PV/FV calculations, it doesn't matter if the interest hasn't been added to the account yet. Calculating Values for Fractional Time Periods Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/finance/textbooks/boundless-finance-textbook/the-time-value-of-money-5/additional-detail-on-present-and-future-values- 59/calculating-values-for-fractional-time-periods-275- 1470?campaign_content=book_192_section_59&campaign_term=Finance&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Compounding Interest View on Boundless.com The Time Value of Money > Additional Detail on Present and Future Values

10. Each amortization payment should be equal in size and pays off a portion of the principal as well as a portion of the interest. The percentage of interest versus principal in each payment is determined in an amortization schedule. If the repayment model for a loan is "fully amortized," then the very last payment pays off all remaining principal and interest on the loan. Loans and Loan Amortization Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/finance/textbooks/boundless-finance-textbook/the-time-value-of-money-5/additional-detail-on-present-and-future-values- 59/loans-and-loan-amortization-276- 1471?campaign_content=book_192_section_59&campaign_term=Finance&utm_campaign=powerpoint&utm_medium=direct&utm_source=boundl ess Amortization Schedule View on Boundless.com The Time Value of Money > Additional Detail on Present and Future Values

11. Free to share, print, make copies and changes. Get yours at www.boundless.com Appendix

12. Key terms amortization The distribution of the cost of an intangible asset, such as an intellectual property right, over the projected useful life of the asset. amortization schedule a table detailing each periodic payment over the life of the loan amortized loan a form of debt where the principal is paid down over the life of the debt according to some amortization schedule, typically through equal payments capitalization The process of finding the future value of a sum by evaluating the present value. compounding period The length of time between the points at which interest is paid. discounting The process of finding the present value using the discount rate. Future Value The value of an asset at a specific date. It measures the nominal future sum of money that a given sum of money is "worth" at a specified time in the future, assuming a certain interest rate, or more generally, rate of return, it is the present value multiplied by the accumulation function. growth rate The percentage by which the payments grow each period. inflation An increase in the general level of prices or in the cost of living. present value Also known as present discounted value, is the value on a given date of a payment or series of payments made at other times. If the payments are in the future, they are discounted to reflect the time value of money and other factors such as investment risk. If they are in the past, their value is correspondingly enhanced to reflect that those payments have been (or could have been) earning interest in the intervening time. Present value calculations are widely used in business and economics to provide a means to compare cash flows at different times on a meaningful "like to like" basis. present value a future amount of money that has been discounted to reflect its current value, as if it existed today time period assumption business profit or loses are measured on timely basis Free to share, print, make copies and changes. Get yours at www.boundless.com The Time Value of Money

13. time value of money the value of an asset accounting for a given amount of interest earned or inflation accrued over a given period Free to share, print, make copies and changes. Get yours at www.boundless.com The Time Value of Money

14. Compounding Interest The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikimedia. CC BY-SA http://upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Compound_Interest_with_Varying_Frequencies.svg/800px- Compound_Interest_with_Varying_Frequencies.svg.png View on Boundless.comCC BY-SAhttp://upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Compound_Interest_with_Varying_Frequencies.svg/800px- Compound_Interest_with_Varying_Frequencies.svg.pngView on Boundless.com The Time Value of Money

15. Amortization Schedule An example of an amortization schedule of a $100,000 loan over the first two years. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Amortization schedule." GNU FDL http://en.wikipedia.org/wiki/Amortization_schedule View on Boundless.comGNU FDLhttp://en.wikipedia.org/wiki/Amortization_scheduleView on Boundless.com The Time Value of Money

16. Fisher Equation The nominal interest rate is approximately the sum of the real interest rate and inflation. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Fisher equation." GNU FDL http://en.wikipedia.org/wiki/Fisher_equation View on Boundless.comGNU FDLhttp://en.wikipedia.org/wiki/Fisher_equationView on Boundless.com The Time Value of Money

17. FV Periodic Compounding Finding the FV (A(t)) given the PV (Ao), nominal interest rate (r), number of compounding periods per year (n), and number of years (t). Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Compound interest." GNU FDL http://en.wikipedia.org/wiki/Compound_interest View on Boundless.comGNU FDLhttp://en.wikipedia.org/wiki/Compound_interestView on Boundless.com The Time Value of Money

18. Solving for n This formula allows you to figure out how many periods are needed to achieve a certain future value, given a present value and an interest rate. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Compound interest." GNU FDL http://en.wikipedia.org/wiki/Compound_interest View on Boundless.comGNU FDLhttp://en.wikipedia.org/wiki/Compound_interestView on Boundless.com The Time Value of Money

19. Calculating the effective annual rate The effective annual rate for interest that compounds more than once per year. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Effective annual rate." GNU FDL http://en.wikipedia.org/wiki/Effective_annual_rate View on Boundless.comGNU FDLhttp://en.wikipedia.org/wiki/Effective_annual_rateView on Boundless.com The Time Value of Money

20. FV of a single payment The PV and FV are directly related. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Time value of money." GNU FDL http://en.wikipedia.org/wiki/Time_value_of_money View on Boundless.comGNU FDLhttp://en.wikipedia.org/wiki/Time_value_of_moneyView on Boundless.com The Time Value of Money

21. EAR with Continuous Compounding The effective rate when interest compounds continuously. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Effective annual rate." GNU FDL http://en.wikipedia.org/wiki/Effective_annual_rate View on Boundless.comGNU FDLhttp://en.wikipedia.org/wiki/Effective_annual_rateView on Boundless.com The Time Value of Money

22. Free to share, print, make copies and changes. Get yours at www.boundless.com The Time Value of Money Which of the following describes the relationship between present value and future value? A) When one increases, the other increases, assuming all variables are constant. B) When present value increases, the future value decreases, assuming all variables are constant. C) The higher the interest rate, the higher the present value and the lower the future value. D) The more time that passes, the higher the present value and the lower the future value.

23. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ The Time Value of Money Which of the following describes the relationship between present value and future value? A) When one increases, the other increases, assuming all variables are constant. B) When present value increases, the future value decreases, assuming all variables are constant. C) The higher the interest rate, the higher the present value and the lower the future value. D) The more time that passes, the higher the present value and the lower the future value.

24. Free to share, print, make copies and changes. Get yours at www.boundless.com The Time Value of Money You own a perpetuity that pays $1000 annually. It has a 5% annual interest rate and a 2% annual growth rate. What is the present value of the perpetuity? A) $20,000 B) $50,000 C) $33,333 D) $14,286

25. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ The Time Value of Money You own a perpetuity that pays $1000 annually. It has a 5% annual interest rate and a 2% annual growth rate. What is the present value of the perpetuity? A) $20,000 B) $50,000 C) $33,333 D) $14,286

26. Free to share, print, make copies and changes. Get yours at www.boundless.com The Time Value of Money A bond currently valued at $100,000 has a quarterly interest rate of 5%. The bond matures in 3 years. What is its future value? A) $1,157,625 B) $1,219,391 C) $1,050,945 D) $1,160,755

27. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ The Time Value of Money A bond currently valued at $100,000 has a quarterly interest rate of 5%. The bond matures in 3 years. What is its future value? A) $1,157,625 B) $1,219,391 C) $1,050,945 D) $1,160,755

28. Free to share, print, make copies and changes. Get yours at www.boundless.com The Time Value of Money Which of the following is a definition for the term "real interest rate"? A) The rate of return that capital could earn in an alternative investment of equivalent risk. B) The amount of interest actually accrued in a given period. C) All of these answers. D) The interest rate accrued after accounting for inflation.

29. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ The Time Value of Money Which of the following is a definition for the term "real interest rate"? A) The rate of return that capital could earn in an alternative investment of equivalent risk. B) The amount of interest actually accrued in a given period. C) All of these answers. D) The interest rate accrued after accounting for inflation.

30. Free to share, print, make copies and changes. Get yours at www.boundless.com The Time Value of Money Which of the following is a definition for cost of capital? A) The rate of return that capital could earn in an alternative investment of equivalent risk. B) The interest rate accrued after accounting for inflation. C) The amount of interest actually accrued in a given period. D) All of these answers.

31. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ The Time Value of Money Which of the following is a definition for cost of capital? A) The rate of return that capital could earn in an alternative investment of equivalent risk. B) The interest rate accrued after accounting for inflation. C) The amount of interest actually accrued in a given period. D) All of these answers.

32. Free to share, print, make copies and changes. Get yours at www.boundless.com The Time Value of Money You purchase a three year certificate of deposit (CD) for $100,000 on January 1st, 2000. This CD has an annual interest rate of 5%. The interest compounds continuously. What is the balance for the CD account on July 1, 2001? A) $107,593 B) $107,500 C) $105,000 D) $110,250

33. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ The Time Value of Money You purchase a three year certificate of deposit (CD) for $100,000 on January 1st, 2000. This CD has an annual interest rate of 5%. The interest compounds continuously. What is the balance for the CD account on July 1, 2001? A) $107,593 B) $107,500 C) $105,000 D) $110,250

34. Free to share, print, make copies and changes. Get yours at www.boundless.com The Time Value of Money Which of the following statements about the amortization of a loan is true? A) A loan is fully amortized once the amortization schedule is drafted. B) 50% of each payment is for interest while the rest is applied to the principal balance. C) All of these answers. D) Amortization is the process of paying off a debt over time through regular payments.

35. Free to share, print, make copies and changes. Get yours at www.boundless.comwww.boundless.com Boundless - LO. "Boundless." CC BY-SA 3.0 http://www.boundless.com/CC BY-SA 3.0http://www.boundless.com/ The Time Value of Money Which of the following statements about the amortization of a loan is true? A) A loan is fully amortized once the amortization schedule is drafted. B) 50% of each payment is for interest while the rest is applied to the principal balance. C) All of these answers. D) Amortization is the process of paying off a debt over time through regular payments.

36. Attribution Wikipedia. "Cost of capital." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Cost_of_capitalCC BY-SA 3.0http://en.wikipedia.org/wiki/Cost_of_capital Wikipedia. "Fisher equation." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Fisher_equationCC BY-SA 3.0http://en.wikipedia.org/wiki/Fisher_equation Wiktionary. "inflation." CC BY-SA 3.0 http://en.wiktionary.org/wiki/inflationCC BY-SA 3.0http://en.wiktionary.org/wiki/inflation Wikipedia. "Compound interest." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Compound_interestCC BY-SA 3.0http://en.wikipedia.org/wiki/Compound_interest Wikipedia. "Compound interest." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Compound_interestCC BY-SA 3.0http://en.wikipedia.org/wiki/Compound_interest Wikipedia. "Accounting assumptions." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Accounting_assumptionsCC BY-SA 3.0http://en.wikipedia.org/wiki/Accounting_assumptions Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//finance/definition/compounding-periodCC BY-SA 3.0http://www.boundless.com//finance/definition/compounding-period Wikipedia. "time value of money." CC BY-SA 3.0 http://en.wikipedia.org/wiki/time%20value%20of%20moneyCC BY-SA 3.0http://en.wikipedia.org/wiki/time%20value%20of%20money Wikipedia. "Loan." CC BY-SA 3.0 http://en.wikipedia.org/wiki/LoanCC BY-SA 3.0http://en.wikipedia.org/wiki/Loan http://loanhr.com/loan-amortization/meaning-of-loan-amortization-and-its-requirement/. CC BY http://loanhr.com/loan- amortization/meaning-of-loan-amortization-and-its-requirement/CC BYhttp://loanhr.com/loan- amortization/meaning-of-loan-amortization-and-its-requirement/ Wikipedia. "Amortization schedule." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Amortization_scheduleCC BY-SA 3.0http://en.wikipedia.org/wiki/Amortization_schedule Wikipedia. "Amortizing loan." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Amortizing_loanCC BY-SA 3.0http://en.wikipedia.org/wiki/Amortizing_loan Wiktionary. "amortization." CC BY-SA 3.0 http://en.wiktionary.org/wiki/amortizationCC BY-SA 3.0http://en.wiktionary.org/wiki/amortization Wikipedia. "Amortization schedule." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Amortization_scheduleCC BY-SA 3.0http://en.wikipedia.org/wiki/Amortization_schedule Wikipedia. "Amortization (business)." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Amortization_(business)CC BY-SA 3.0http://en.wikipedia.org/wiki/Amortization_(business) Wikipedia. "Present value." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Present_valueCC BY-SA 3.0http://en.wikipedia.org/wiki/Present_value Wikipedia. "Present value." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Present_valueCC BY-SA 3.0http://en.wikipedia.org/wiki/Present_value Free to share, print, make copies and changes. Get yours at www.boundless.com The Time Value of Money

37. Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//finance/definition/discounting--2CC BY-SA 3.0http://www.boundless.com//finance/definition/discounting--2 Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//finance/definition/capitalizationCC BY-SA 3.0http://www.boundless.com//finance/definition/capitalization Wikipedia. "Future value." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Future_valueCC BY-SA 3.0http://en.wikipedia.org/wiki/Future_value Wikipedia. "Terminal value (finance)." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Terminal_value_(finance)CC BY-SA 3.0http://en.wikipedia.org/wiki/Terminal_value_(finance) Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//finance/definition/growth-rateCC BY-SA 3.0http://www.boundless.com//finance/definition/growth-rate Wikipedia. "Time value of money." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Time_value_of_moneyCC BY-SA 3.0http://en.wikipedia.org/wiki/Time_value_of_money Wikipedia. "Perpetuities." CC BY-SA 3.0 http://en.wikipedia.org/wiki/PerpetuitiesCC BY-SA 3.0http://en.wikipedia.org/wiki/Perpetuities Wikipedia. "Compound interest." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Compound_interestCC BY-SA 3.0http://en.wikipedia.org/wiki/Compound_interest Wikipedia. "Effective annual rate." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Effective_annual_rateCC BY-SA 3.0http://en.wikipedia.org/wiki/Effective_annual_rate Wikipedia. "Effective annual rate." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Effective_annual_rateCC BY-SA 3.0http://en.wikipedia.org/wiki/Effective_annual_rate Wikipedia. "Future Value." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Future%20ValueCC BY-SA 3.0http://en.wikipedia.org/wiki/Future%20Value Wikipedia. "Present value." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Present_valueCC BY-SA 3.0http://en.wikipedia.org/wiki/Present_value Free to share, print, make copies and changes. Get yours at www.boundless.com The Time Value of Money