1. ENGINEERING ECONOMICS Lecture # 2 Time value of money Interest Present and Future Value Cash Flow Cash Flow Diagrams

2. Definitions Project : an investment opportunity generating cash flows over time Cash Flow: the movement of money (in or out) of a project Interest: The rent for loaned money Cash Flow Diagram: Describes type, magnitude and timing of cash flows over some horizon Time value of money: The change in the amount of money over given time period is called the time value of money. It is the most important concept in the engineering economy

3. The Time Value of Money Money NOW is worth more than money LATER!

4. $10,000 today Obviously, $10,000 today. TIME VALUE TO MONEY You already recognize that there is TIME VALUE TO MONEY!! $10,000 today $10,000 in 5 years Which would you prefer -- $10,000 today or $10,000 in 5 years?

5. Interest Cost of Money –Rental amount charged by lender for use of money –In any transaction, someone “earns” and someone pays Interest is the difference between an ending amount and beginning amount Interest is paid when money is borrowed (loan) and repayment involves an additional amount Interest is earned when money is saved, invested and rented for an additional return

6. oInterest rate, or the rate of capital growth, is the rate of gain received from an investment oWhen interest paid over specific time period is expressed as %age of principal (original) amount, it is called as interest rate oInterest Rate = (Interest per unit time/original amount) *100 oTime unit of interest rate is interest period oInterest Rate = Rate of return (ROR) = Rate of investment (ROI) Interest Rate

7. Types of Interest Compound InterestCompound Interest Interest paid (earned) on any previous interest earned, as well as on the principal amount u Simple Interest Interest paid (earned) on only the original amount, or principal amount

8. Simple Interest Formula Formula FormulaSI = P 0 (i)(n) SI:Simple Interest P 0 :Deposit today (t=0) i:Interest Rate per Period n:Number of Time Periods

9. $140SI = P 0 (i)(n) = $1,000(.07)(2) = $140 Simple Interest Example Assume that you deposit $1,000 in an account earning 7% simple interest for 2 years. What is the accumulated interest at the end of the 2nd year?

10. Simple Interest Interest earned/paid is directly proportional to capital involved. I = P * i * n Ex. $ 1000 loan for 2 years at 10 % per year – no compounding I = P * i * n = 1000 *.10 * 2 = $200 Payback = F = P + I = 1000 + 200 = $1200

11. A sum of money today is called a present value We designate it mathematically with a subscript, as occurring in time period 0 For example: P 0 = 1,000 refers to $1,000 today

12. A sum of money at a future time is termed a future value We designate it mathematically with a subscript showing that it occurs in time period n. For example: S n = 2,000 refers to $2,000 after n periods from now.

13. As already noted, the number of time periods in a time value problem is designated by n n may be a number of years n may be a number of months n may be a number of quarters n may be a number of any defined time periods

14. The interest rate or growth rate in a time value problem is designated by i i must be expressed as the interest rate per period. For example if n is a number of years, i must be the interest rate per year. If n is a number of months, i must be the interest rate per month.

15. The first of the general type of time value problems is called future value and present value problems. The formula for these problems is: S n = P 0 (1+i) n

16. An example problem: If you invest $1,000 today at an interest rate of 10 percent, how much will it grow to be after 5 years? S n = P 0 (1+i) n S n = 1,000(1.10) 5 S n = $1,610.51

17. Another example problem: How long will it take for $10,000 to grow to $20,000 at an interest rate of 15% per year? S n = P 0 (1+i) n 20,000 = 10,000(1.15) n n = 4.96 years (or, about 5 years)

18. One more example problem: If you invest $11,000 in a mutual fund today, and it grows to be $50,000 after 8 years, what compounded, annualized rate of return did you earn? S n = P 0 (1+i) n 50,000 = 11,000(1+i) 8 i = 20.84 percent per year

19. The next two general types of time value problems involve annuities An annuity is an amount of money that occurs (received or paid) in equal amounts at equally spaced time intervals. These occur so frequently in business that special calculation methods are generally used.

20. For example: If you make payments of $2,000 per year into a retirement fund, it is an annuity. If you receive pension checks of $1,500 per month, it is an annuity. If an investment provides you with a return of $20,000 per year, it is an annuity.

21. A common mathematical symbol for an annuity amount is PMT

22. For the future value of an annuity: FV = PMT[(1+i) n - 1]/i

23. For the present value of an annuity: PV = PMT[(1+i) n -1]/[i(1+i) n ]

24. An example problem: If you save $50 per month at 12 percent per annum, how much will you have at the end of 20 years? Note that since time periods are months, i = 12%/12 months = 1% per period, for 240 periods. FV = PMT[(1+i) n - 1]/i FV = 50[(1.01) 240 - 1]/.01 FV = $49,463

25. Another example problem: If you want to save $500,000 for retirement after 30 years, and you earn 10 percent per annum, how much must you save each year? FV = PMT[(1+i) n - 1]/i 500,000 = PMT[(1.1) 30 - 1]/.1 PMT = $3,040 per year

26. An example problem: If you borrow $100,000 today at 9 percent interest per annum, and repay it in equal annual payments over 10 years, how much are the payments? PV = PMT[(1+i) n -1]/[i(1+i) n ] 100,000 = PMT[(1+.09) 10 -1]/[.09(1.09) 10 ] PMT = $15,582 per year

27. A last type of time value problem involves what are called, perpetuities A perpetuity is simply an annuity that continues forever (perpetually). The formula for finding the present value of a perpetuity is: PV = PMT/i

28. A variation to perpetuity problems is the case of growing perpetuities If an annuity continues forever, and grows in amount each period at a rate g, then PV = PMT 1 /(i - g)

29. An example problem: If you invest in a stock that will pay a dividend of $10 next year and grow at 5 percent per year, and you require a 14 percent rate of return, how much is the stock worth to you today? PV = PMT 1 /(i - g) PV = 10/(.14-.05) PV = $111.11

30. Thank You