What you will learn: Worksheet-2 Present Value Math I How to compute present and future values of single and annuity cash flows How to handle cash flow delays and combinations of cash flow streams How to handle monthly compounding How to use a financial calculator Why you should learn this: To be able to solve common life problems: retirement savings, leasing, mortgage loans PV Math is the foundation of all of finance: corporate, banking, and investments This topic repeats numerous times throughout the semester. If you master it now, you will master several future topics too: bonds, stocks, NPV of projects, company valuation. Today s Story: A dollar today grows to more than a dollar seven years from now? A dollar promised seven years from now is worth less than a dollar today. The reason: compound interest/return. PV math is the foundation of all finance. The value of a security, stock or bond, or a project is equal to the present value of all the future cash flows the project or the security will return. In Modules 4 and 5 we learn how to discount cash flows. First, future value problems: how much is in your account seven years from now? Second, present value problems: How much must be in the account now to be able to withdraw this, that and the other in the future. This, that, and the other can be single cash flows, annuities, or both. Then we combine future and present values. How much do you need to save per year if you want to withdraw in the future a known amount per year? Some interest compounds intra-year, semiannually or monthly. Exercise #1: Future Value. You deposit $200 in a bank. The compound interest rate is 8%. After one year you have: After two years you have: In general, if you deposit CF 0 today, at rate r, n years later you have: $CF 0 x (1+r) n = CF 0 x FVIF(r,n) = 200 x (1+0.08) n Table A1 FVIF is the fut val of $1 On a financial calculator: PV = 200 PMT=0 i=8 n=2 Press FV A. What is $600 after n=5 yrs at 7%? Formula: Table: Calculator: PV = PMT= i= n= FV= B. What is $600 after n=5 yrs at 7.5%? Formula: Calculator: PV = PMT= i= n= FV=
Exercise #2: Present Value. You want to have $200 in a bank n years from now. The compound interest rate is 8%. How much do you have to deposit today? If n=1: If n=2: In general, if you want to have CF n n years later, at rate r, you have to deposit today: $CF n x 1/(1+r) n = CF n x PVIF(r,n) = 200 x 1/(1+0.08) n Table A2 PVIF is the PV of $1 On a financial calculator: FV = 200 PMT=0 i=8 n=2 Press PV A. What is the PV of $600 after n=5 yrs at 7%? Formula: Table: Calculator: FV = PMT= i= n= PV= B. Note that: FVIF = (1+r) n PVIF = 1/(1+ r) n PVIF = 1/FVIF Your financial calculator has five buttons. These will allow us to calculate one quantity given the other four. It will also allow us to handle combinations of annuities and single cash flows. The buttons are PV, PMT, i, n, and FV These buttons correspond to the following time line: -PV PMT PMT PMT PMT PMT PMT&FV ------- -------- -------- ------- ---- -- -------- To compute single cash flow PVs or FVs, we set the number of periods and the interest rate (as a whole number, 8.5 not 0.085), zero out the PMT, enter the cash flow as PV and compute FV, or enter the cash flow as FV and compute PV. To compute ordinary annuity PVs or FV, we enter the consecutive cash flow as PMT, zero out the PV or the FV, and compute the other. We can also solve for the number of periods or the interest rate. Exercise #3: FV/PV Fun. A. How long will it take to triple your money at 6%? B. What rate do you need to earn to double your money in 15 years?
Exercise #4: FV of two cash flows. A. You invest $10,000 @ 8%. You want to take out $6000 in 5 years. How much will you have in 10 years? Method 1: FV $10,000 to Y5, subtract $6,000, FV the rest to Y10 Method 2: FV $10,000 to Y10; FV $6,000 to Y10; subtract the latter from the first B. You invest $3000 in Y1 and $4000 in Y5 at 8%. How much will you have in Year 7? Method 1: FV $3000 to Y5, add $4000; FV the total to Y7 Method 2: FV $3000 to Y7, FV $4000 to Y7; add the two together Exercise #5: PV of two cash flows A. How much would you pay today for $4000 in Y3 and $2000 in Y5? r=8% Method 1: PV $2000 from Y5 to Y3, add to $4000; PV the total to Y0 Method 2: PV $2000 from Y5 to Y0; PV $4000 from Y3 to Y0; add the two
B. How much would you pay in Y1 for $4000 in Y3 and $2000 in Y5? i=8% Exercise #6: FV of [Ordinary] Annuity You are going to deposit $3,000 once year at the end of each year for the next 5 years. r=8% A. How much money is in your account 5 years from now? We can FV each cash flow separately and add the FVs up, i.e. we can: 3,000xFVIF (4,8%) + 3,000xFVIF (3,8%) + 3,000xFVIF (2,8%)+ 3,000xFVIF (1,8%)+ 3,000 = 3,000 x (1+.08) 4 + 3,000 x (1+.08) 3 + 3,000 x (1+.08) 2 + 3,000 x (1+.08) + 3,000 or we can use the [ordinary] annuity shortcut 3,000 x FVAF (5,8%) where FVAF (5,8%) = [( 1+r) n 1] / r Table A.4 PV = 0 PMT=3000 i= 8 n=5 Press FV B. How much money is in your account 6 years from now? [Hint. FV the total by 1 year] Exercise #7: PV of [Ordinary] Annuity You want to withdraw $3,000 once year at the end of each year for the next 5 years. r=8% A. How much money do you need to deposit now to fund the account fully? We can PV each cash flow separately and add the PVs up, i.e. we can: 3,000xPVIF (1,8%) + 3,000xPVIF (2,8%) + 3,000xPVIF (3,8%)+ 3,000xPVIF (4,8%)+ 3,000xPVIF (5,8%) = 3,000x1/(1+.08) + 3,000x1/(1+.08) 2 + 3,000x1/(1+.08) 3 +3,000x1/(1+.08) 4 +3,000x1/(1+.08) 5 or we can use the [ordinary] annuity shortcut 3,000 x PVAF (5,8%) where PVAF (n,r) = 1/r - 1/[r( 1+r) n ] Table A.3 FV = 0 PMT=3000 i= 8 n=5 Press PV
Exercise #8: PV of [Ordinary] Perpetuity You want to withdraw $3,000 once year at the end of each year forever. r=8% A. How much money do you need to deposit now to fund the account fully? The perpetuity shortcut has the same logic as the annuity shortcut, but it looks like this: CF 0 x 1/r = CF 0 / r = 3,000 x 1/0.08 = B. If the cash flow we will receive grows at a constant percent year per year, say g=4%, then CF 0 x 1/(r-g) = 3,000 x 1/(0.08-0.04) = We will use this formula in the chapter on stocks. Exercise #9: PV of a delayed [Ordinary] Annuity You want to withdraw $3,000 once year at the end of each year for 5 years, starting 3 years from now and ending 7 years from now. r=8%. How much money do you need to deposit now to fund the account fully? Method: PV the annuity to one year before the first cash flow using PVAF and then to Y0 using PVIF. 3,000 x PVAF (5,8%) x PVIF(,8%) On the calculator, this implies a two-step procedure: FV = PMT= i= n= Press PV then PV = PMT= i= n= Press FV
Exercise #10: PV of an Annuity Due You want to withdraw $3,000 once year at the beginning of each year for 5 years. r=8%. How much money do you need to deposit now to fund the account fully? [Hint: Think of the Annuity Due as an Ordinary Annuity that is delayed by minus one period. Thus the second step will involve FVing]. Draw Timeline: --------------- -------------------------------------------------------------------------------------- PV the annuity to one year before the first cash flow (i.e. Y -1) using PVAF and then to Y 0 using FVIF. 3,000 x PVAF (5,8%) x FVIF(1,8%) = 3,000 x PVAF (5,8%) x (1+0.08) = On the calculator, this also implies a two-step procedure: FV = PMT= i= n= Press PV then FV = PMT= i= n= Press PV or, instead of the second step, we can just multiply the result of the first step by 1+r Exercise #11: PV of a delayed perpetuity. You want to withdraw $3,000 once year at the end of each year forever, starting 3 years from now. r=8%. How much money do you need to deposit now to fund the account fully? Method: PV the annuity to one year before the first cash flow using 1/r and then to Y0 using PVIF. 3,000 x x. =