1. The time value of money:

2. Don't waste your time with explanations: people only hear what they want to hear. Paulo Coelho 3. Time value of money: Which would you prefer -- $10,000 today or $10,000 in 5 years? Obviously, $10,000 today. You already recognize that there is TIME VALUE TO MONEY!! 4. Time value of money: Why TIME? Why is TIME such an important element in your decision? TIME allows you the opportunity to postpone consumption and earn INTEREST. 5. A dollar in hand today is worth more than a dollar to be received in the future because, if you had it now, you could invest it, earn interest, and end up with more than dollar in future. Any amount of money promised in the future is uncertain and riskier than others. 6. The time value of money principle is concerned with two topics: (1) future value, and (2) present value. As shown in the illustration below, the two are mirror images of one another. (Year 0 stands for "at the present time" or "right now" since year 1 would be 1 year from now, etc.) In a future value problem, we know the amount of money that we have to invest today (i.e., the present value). What we don't know is how much money we will have in the future (i.e., the future value). In a present value problem, we know the amount of money that we want to have (or expect to have) in the future. What we don't know is how much money we need to invest today in order to attain that money in the future. 7. Time line: An important tool used in the time value of money analysis; it is a graphical representation used to show the timing of cash flows. 0 1 2 3 4 5 6 7 Moving money through time that is, finding the equivalent value to money at different points in time involves translating values from one period to another. 8. Translating a value to the present is referred to as discounting. Translating a value to the future is referred to as compounding. Translating money from one period involves interest, which is how the time value of money and risk enter into the process. 9. Future value (FV): The amount to which a cash flow or a series of cash flows will grow over a period of time when compounded at a given interest rate. It can be calculated as; FV = PV (1+i)n Where, FV = future value or ending amount PV = present value or beginning amount i = interest rate per period n = the number of periods (1+i)n = future value (interest) factor 10. Compounding: The arithmetic process of determining the final value of a cash flow or series of cash flows when compound interest is applied. It is the process of going from todays values or present values (PVs) to future values (FVs), over a period of time (over the time line). 11. Present value: The value today of a future cash flow or a series of cash flows. The present value of a cash flow due n years in the future is the amount which, if it were on hand today, would grow to equal the future amount. It can be calculated as; Where, PV = present value FV = future value i = interest rate or rate of return n = number of periods 1/(1+i)n = present value (interest) factor 12. Present value: From the formula for present value we know that; As the number of discount periods, n, becomes larger, the discount factor becomes smaller and the present becomes less, and As the interest rate per period, i, becomes larger, the discount factor becomes smaller and the present value becomes less. Therefore, the present value is influenced by both the interest rate (i.e. the discount rate) and the number of discount periods. 13. Discounting: The process of finding the present value of a cash flow or a series of cash flows; discounting is the reverse of compounding. Translating a value back in time referred to as discounting requires determining what a future amount or cash flow is worth today. Discounting is used in valuation because we often want to determine the value today of some future value or cash flow (e.g. what a bond is worth today if it promised interest and principal repayment in the future). 14. Interest: The principal is the amount borrowed. Interest is the compensation for the opportunity cost of funds and the uncertainty of repayment of the amount borrowed; that is, it represents both the price of time and the price of risk. The price of time is compensation for the opportunity cost of funds. The price of risk is compensation for bearing risk. 15. Simple interest: Interest paid (earned) on only the principal amount, or principal borrowed (lent). Formula: SI = (Po)(i)(n) Where, SI = simple interest Po = principal amount (lent or borrowed) i = interest rate n = number of periods 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? SI = (Po)(i)(n) SI = $1000(0.07)(2) SI = $140 16. Simple interest (FV): What is the Future Value (FV) of the deposit? FV = Po + SI FV = $1000 + $140 FV = $1140 Future Value is the value at some future time of a present amount of money, or a series of payments, evaluated at a given interest rate. 17. Simple Interest (PV): What is the Present Value (PV) of the previous problem? The Present Value is simply the $1,000 you originally deposited. That is the value today! Present Value is the current value of a future amount of money, or a series of payments, evaluated at a given interest rate. 18. Compound Interest: Interest paid (earned) on any previous interest earned, as well as on the principal borrowed (lent). Example: Assume that you deposit $1,000 at a compound interest rate of 7% for 2 years. 0 1 2 $1,000 FV2 19. Future value single deposit: FV1 = Po (1+i)1 FV1 = $1,000 (1+0.07) FV1 = $1,070 Compound interest: You earned $70 interest on your $1,000 deposit over the first year. This is the same amount of interest you would earn under simple interest. The difference between the future value with compounded interest and that with simple interest is the interest-on- interest. 20. Future value single deposit: FV1 = Po (1+i)1 = $1,000 (1+0.07) = $1,070 $1,000 FV1 = $1,070 FV2 FV2 = FV1 (1+i)1 = Po (1+i)(1+i) = $1,000(1.07)(1.07) = Po (1+i)2 = $1,000(1.07)2 = $1,144.90 You earned an extra $4.90 in year 2 with compound over the simple interest. This $4.90 is the interest earned on the 7% interest of $1,000, i.e. 1000 x 7% = 70 70 x 7% = 4.9 21. Types of annuities: Annuity: A series of equal payments or receipts of an equal amount at fixed intervals occurring for a specified number of periods. We see examples of annuities all around us in business. Interest payments on a fixed- income security (like a bond) are an annuity. Monthly payments on a loan are an annuity. Lease payments on a car or apartment are an annuity. The amount doesn't change from period to period. Ordinary (deferred) annuity: In an ordinary annuity payments or receipts occur at the end of each period. Annuity due: In an annuity due payments or receipts occur at the beginning of each period. 22. Ordinary annuity (FV): FVAn = PMT( FVIFAi,n ) FVAn = PMT[(1+i)n 1/i] Where; FVAn = future value of an annuity over n periods PMT = payments FVIFAi,n = annuity future value interest factor i = interest rate n = number of periods 23. Example: If PMT=$1,000, i=8%, n=3 years. Calculate future value of an ordinary annuity? FVAn = PMT( FVIFAi,n ) FVAn = PMT[(1+i)n 1/i] FVAn = $1,000 [(1.08)3 1/(0.08)] FVAn = $3,246. 24. Ordinary annuity (PV): PVAn = PMT( PVIFAi,n ) PVAn = PMT [1-[1/(1+i)n ]/i ] Where; PVAn = present value of an annuity of n periods PMT = payments PVIFAi,n = present value interest factor annuity i= interest rate n= number of years 25. Example: Periodic receipts of $1,000 at the end of each year, discount rate = 8%, n = 3 years. Calculate present value ordinary annuity? PVAn = PMT( PVIFAi,n ) PVAn = PMT [1-[1/(1+i)n ]/i ] PVAn = $1,000 [1-[1/(1.08)3 ]/(0.08)] PVAn = $2,577. 26. Annuity due (FV): FVAD = PMT (FVIFAi,n)(1+i) FVAD = PMT [(1+i)n 1/i](1+i) Where; FVAD = future value annuity due PMT = payment FVIFAi,n = future value interest factor of an annuity i= interest rate n= number of years 27. Example: PMT = $1,000, i= 5%, n= 5 years. Calculate future value of an annuity due? FVAD = PMT [(1+i)n 1/i](1+i) = $1,000 [1-[1/(0.05)5 ]/(0.05)](1.05) = $5,802.3 28. Annuity due (PV): PVAD = (1+i)(PMT)(PVIFAi,n) PVAD = (1+i) (PMT) [1-[1/(1+i)n ]/i] Where; PVAD = present value annuity due PMT = payments PVIFAi,n = present value interest factor of an annuity i= interest rate n= number of years 29. Example: PMT=$1,000, i=8%, n= 3 years. Calculate present value of annuity due? PVAD = (1+i)(PMT)(PVIFAi,n) PVAD = (1+i) (PMT) [1-[1/(1+i)n ]/i] = (1.08) (1,000) [1-[1/(1.08)3 ]/(0.08) = $2783.16 30. Perpetuity: An ordinary annuity whose payments or receipts continue forever. PVA00 = PMT/I where; PVA00 = perpetuity PMT = payment or receipts I = interest rate 31. Example: PMT= $1,000, I = 5%. Calculate perpetuity annuity? PVA00 = PMT/I = $1,000/0.05 = $2,000 32. Extra notes: Definition of 'Rate Of Return' The gain or loss on an investment over a specified period, expressed as a percentage increase over the initial investment cost. Gains on investments are considered to be any income received from the security plus realized capital gains. Definition of 'Real Rate Of Return' The annual percentage return realized on an investment, which is adjusted for changes in prices due to inflation or other external effects. This method expresses the nominal rate of return in real terms, which keeps the purchasing power of a given level of capital constant over time. 33. Inflation premium: Return on an investment over its normal rate of return. Investors seek this premium to compensate for the erosion in the value of their capital due to inflation. Definition of 'Nominal Interest Rate' The interest rate before taking inflation into account. The nominal interest rate is the rate quoted in loan and deposit agreements. The equation that links nominal and real interest rates is: (1 + nominal rate) = (1 + real interest rate) (1 + inflation rate). It can be approximated as nominal rate = real interest rate + inflation rate. 34. Definition of 'Market Risk' The possibility for an investor to experience losses due to factors that affect the overall performance of the financial markets. Market risk, also called "systematic risk," cannot be eliminated through diversification, though it can be hedged against. The risk that a major natural disaster will cause a decline in the market as a whole is an example of market risk. Other sources of market risk include recessions, political turmoil, changes in interest rates and terrorist attacks. Maturity Risk Fixed-income investment securities -- primarily bonds -- typically pay a fixed rate of interest and the face or principal amount when a bond matures. Available maturities range from 30 days to 30 years. Maturity risk is the potential for interest rates to change while your money is tied up in a bond until it matures. Buying a bond with a longer time to maturity increases the likelihood that interest rates could rise over that period. The maturity risk premium is the extra yield you will earn from buying a bond with a longer time to maturity. 35. Definition of 'Liquidity Risk' The risk stemming from the lack of marketability of an investment that cannot be bought or sold quickly enough to prevent or minimize a loss. Liquidity risk is typically reflected in unusually wide bid-ask spreads or large price movements (especially to the downside). The rule of thumb is that the smaller the size of the security or its issuer, the larger the liquidity risk. Definition of 'Default Risk' The event in which companies or individuals will be unable to make the required payments on their debt obligations. Lenders and investors are exposed to default risk in virtually all forms of credit extensions. To mitigate the impact of default risk, lenders often charge rates of return that correspond the debtor's level of default risk. The higher the risk, the higher the required return, and vice versa.