International Financial Strategies Time Value of Money

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1 International Financial Strategies 1 Future Value and Compounding Future value = cash value of the investment at some point in the future Investing for single period: FV. Future Value PV. Present Value t. Time 2 1

2 Future Value and Compounding Investing for more than one period (you make interest on interest received) FV. Future Value PV. Present Value The process of accumulated interest is called compounding = earning interest on interest 3 Future Value and Compounding Periodic compounding interest compounded more often than once a year: FV. Future Value PV. Present Value t. Time n. Number of compounding periods per year 4 2

3 Future Value and Compounding Continuous compounding: o Is used in financial theory o We make the compounding period infinitesimally small othe formula is achieved by taking the limit on n to infinity 5 Present Value Current value of future cash flows discounted at the appropriate discount rate. Single-period and multiple periods: FV. Future Value PV. Present Value 6 3

4 Determining Discount Rate and Number of Periods There are only four parts to these equations. Given any three, we can always find the fourth one. FV. Future Value PV. Present Value (Discount Rate) t. Time (Number of Periods) log. Log Value (we may use natural logarithms) 7 Examples Assume we are offered an investment that costs us $100 and will double our money in eight years. To compare this to other investment, we would like to know what discount rate is implicit in these numbers. This discount rate is called rate of return. Suppose we are interested in purchasing an assets that costs $50,000. We currently have $25,000. If we can earn 12 percent on this money, how long will we have to wait? 8 4

5 Multiply Cash Flows Sometimes we need to calculate FV or PV of multiply cash flows Series of constant cash flows that occur at the end of each period for some fixed number of periods are called annuities. Annuities appear in financial arrangements like mortgages, consumer loans (car loans). APV. Present Value of Annuity A. Annuity 9 Examples You can afford to pay $500 per month to buy a new car. You call up your local bank and find out that IR is 1 percent per month for 48 months. How much can you borrow? You need to borrow $100,000 and you want to pay off the loan quickly by making 5 equal annual payments. If the interest rate is 18 percent, what will the payment be? 10 5

6 Multiply Cash Flows Of course, you can compute future value of annuity as well: AFV. Future Value of Annuity A. Annuity 11 Example You plan to contribute $2,000 every year to a retirement account paying 8 percent. If you retire in 30 years, how much will you have? 12 6

7 Multiply Cash Flows - Perpetuities Perpetuity is a special case of annuity. It is annuity with cash flows that continue forever (mathematically, it is a sum of infinitive geometric series): PPV. Present Value of Perpetuity P. Perpetuity 13 Example Preferred stock is an example of perpetuity. The buyer is promised a fixed cash dividend every period (usually every quarter) forever. This dividend must be paid before any dividend can be paid to regular stockholders. 14 7

8 Multiply Cash Flows Growing Annuities Growing annuities and perpetuities annuities and perpetuities usually have payments that grow over time. Let GR be the growth rate. We can the reformulate the formulas for annuities and perpetuities: GAPV. Present Value of Growing Annuity A. Annuity GR. Growth Rate of Annuity 15 Multiply Cash Flows Growing Annuities IF IR equals GR: GAPV. Present Value of Growing Annuity A. Annuity 16 8

9 Multiply Cash Flows Growing Annuities Future value of growing annuity: GAFV. Future Value of Growing Annuity A. Annuity GR. Growth Rate of Annuity 17 Multiply Cash Flows Growing Annuities IF IR equals GR: GAFV. Future Value of Growing Annuity A. Annuity 18 9

10 Multiply Cash Flows Growing Perpetuities IF GR is lower than IR we can compute present value of growing perpetuity: GPPV. Present Value of Growing Perpetuity P. Perpetuity GR. Growth Rate of Perpetuity 19 Exercises 20 10

11 - Exercises Suppose you have just celebrated your 19 th birthday. A rich uncle has set up a trust fund for you that will pay you $150,000 when you turn 30. If the relevant discount rate is 9 percent, how much is this fund worth today? 21 - Exercises You have been offered an investment that will pay you 10 percent per year. If you invest $15,000, how long until you have $30,

12 - Exercises You are looking into an investment that will pay you $12,000 per year for the next 10 years. If you require a 15 percent return, what is the most you would pay for this investment? 23 - Exercises You want to have $90,000 in your savings account 10 years from now, and you are prepared to make equal deposits into the account at the end of each year. If the account pays 5 percent interest, what amount must you deposit each year? 24 12

13 - Exercises The Insurance company is trying to sell you an investment that will pay you and your heirs $25,000 per year forever. If the required return on this investment is 8 percent, how much will you pay for the investment Exercises Today you have just receive your salary of $50,000 for all the work you did over the previous 12 months. You have decided that one year from now you will begin depositing 5 percent of you annual salary in an account that will earn 8 percent per year. Your salary will increase at 4 percent per year throughout your career. How much money will you have on the date of your retirement 40 year from today? 26 13

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