1. Future Value 1- Ahmad plans to retire in fifteen years. Can he afford a $250,000 condominium when he retires if he invests $100,000 in a fifteen-year Mellon CD (certificate of deposit) which pays 7.5% interest, compounded annually? Solution: Yes, he could afford to purchase the condominium since he should have $295,887.74 when he retires. $295,887.74 = $100,000 (1.075)15.

2. 2- Can Ali afford the condominium if he purchases three consecutive five-year CD’s? The current five-year rate is 6%. Rates for the second and third five-year periods and expected to be 6.5% and 7.5%, respectively. Yes, he can still afford it: FV = 100,000 (1.06) 5 (1.065) 5 (1.075) 5 FV = 100,000 (1.3382)(1.37009)(1.43563) FV = 100,000 (2.6322) FV = 263,216.

3. 3. What is the future value of $26 billion invested by UAE for 325 years at an average rate of return of 7%? (In this context, did the UAE make a poor decision to sell Jabal-Ali Area to the Saudi Arabia?) FV = 26(1.07) 325 FV = 9.2194 x 1010 = $92.194 billion If the UAE had invested at a average annual rate of 7%, they would have over $92 billion after 325 years.

4. Time Value of Money Review Future Value of an Annuity The Future Value of an Annuity tells you how large a sum a stream of even payments will accumulate to in a given time given an investment rate and a compounding frequency. The formula is:

5. Time Value of Money Review Future Value of an Annuity Example: Assume that you want to save for a house. You plan on depositing $250 each month into an account which pays 8% per year with annual compounding. How much will you have after 10 years?

6. Sinking Funds * Money regularly set aside by a company to redeem its bonds, debentures or preferred stock from time to time as specified in the indenture or charter

7. **A sinking fund can be defined as an annuity invested in an order to meet a known commitment at some future date. Sinking funds are usually used for the following purposes: -Repayment of debts. -To provide funds to purchase a new asset when the existing asset is fully depreciated.

8. Time Value of Money Review Sinking Fund Factor The sinking fund factor tells you how much you put aside each month to have a fixed amount at the end of a given time period, assuming an interest rate and compounding frequency. The formula is:

9. Example of debt repayment using a sinking fund Let’s say that you want to save 5 years and at the end of that time you want to have $20,000. If you can invest at 10% with monthly deposits and compounding, how much must you deposit each month?

10. Time Value of Money Review Present Value of a Lump Sum The future value of a lump sum tells us how much we have to invest today to receive a fixed amount in the future. This essentially tells us what we should be willing to pay today for a fixed amount in the future. That is, the present value of a lump sum is the amount we should be willing to pay for the right to receive a certain cash flow in the future.

11. Time Value of Money Review Present Value of a Lump Sum The formula for this is just future value of a lump sum formula rearranged:

12. Present Value of a Lump Sum (single) The formula is simply a rearrangement of the future value of a lump sum: Ex. 3: What is the present value of $50,000 received in 10 years with a 10% discount rate and monthly compounding?

13. Present Value of a Lump Sum Ex. 4: What is the present value of $50,000 received in 10 years with a 10% discount rate and yearly compounding

14. Time Value of Money Review Present Value of a Series Present values are additive. This means that the present value of a stream of cash flows is simply the sum of the present values for each of the individual cash flows. Thus, for a series of T cash flows:

15. Time Value of Money Review Present Value of a Series Example 5: A bond will pay you $60 every six months for the next 2 years. At the end of the third year you will also receive principal of $100. If your discount rate is 8%, how much should you pay for this bond?

16. Time Value of Money Review Present Value of an Annuity If the future cash flows are all of equal amounts, you can use a shortcut equation, known as the present value of an annuity equation:

17. Present Value of an Annuity Example 6: What is the present value of winning a 20,000,000 lottery if you receive payments of $1,000,000 annually, your discount rate is 10% (annually compounded), and your first payment is in exactly one year?

18. Time Value of Money Review Present Value of an Annuity Note that you sometimes have to combine these formulas. For example, assume that you will receive $250 each month for 10 years, starting in 5 years. Your discount rate is 6% with monthly compounding. You can use the annuity formula to get the present value at time 5 years of the 10 years worth of payments:

19. Time Value of Money Review Present Value of an Annuity --To determine the present value today of that annuity, you have to discount back present value of the annuity: - Combining into one equation gives:

20. Capital Recovery Factor A capital recovery factor is the ratio of a constant annuity to the present value of receiving that annuity for a given length of time. Using an interest rate i, the capital recovery factor is:annuitypresent valueinterest rate where n is the number of annuities received. This is related to the annuity formula, which gives the present value in terms of the annuity, the interest rate, and the number of annuities.annuity formula If n = 1, the CRF reduces to 1+i. As n goes to infinity, the CRF goes to i.