SVEN THOMMESEN FINANCE 2400/3200/3700 Spring 2018 [Updated 8/31/16] SOLUTION METHODS FOR SELECTED BASIC FINANCIAL RELATIONSHIPS VARIABLES USED IN THE FOLLOWING PAGES: N = the number of periods (months, years) I/YR = the applicable interest rate (% per year) = Present Value (value of a project at an early point in time) FV = Future Value (value of a project at a later point in time) PMT = Payment (the size of the recurring cash flow associated with the project) BEG = Begin Mode (used to set the calculator to expect an Annuity Due) END = End Mode (used to set the calculator to expect a Normal Annuity) P/YR = Payments Per Year (or in some cases, compounding periods per year) A = Present value of an annuity FVA = Future value of an annuity SFP = Sinking fund payment 1
FUTURE VALUE OF A SINGLE PAYMENT [LUMP SUM] EXPLANATION: The future value of a lump sum is the value the lump sum would grow to, if left to earn interest at a given rate over a specific time period, with no further contributions from the saver/investor. FV (1 i) n You save $2,500 and leave it for 12 years at 7% interest. How much do you have? FV = $2,500.00 * (1 + 0.07)^12 = $5,630.48 LOOK UP FVF(12 YEARS, 7%) = 2.252 FV = * FV-FACTOR = $2,500.00 * 2.252 = $5,630.00 BEG/END = END [regular annuity: interest added at end of each period] [interest added once a year] N = 12 I = 7 [interest rate in %] = -2,500.00 [deposit at start; note the sign!] PMT = 0 [no recurring payments here] Solve for FV -> $5,630.4790 [amount available at end] 2
PRESENT VALUE OF A SINGLE PAYMENT [LUMP SUM] EXPLANATION: The (discounted) present value is the answer to how much a future sum of money is worth in terms of today s dollars. Or, if you will, how much you would have to deposit in the bank today in order to have a specific sum available (with interest) at a future date. FV (1 i ) n You find a savings account which your parents started for you 15 years ago containing $7,600.00. The money has been earning 5% interest over that time. How much did your parents deposit back then? FV 7, 600.00 3, 655.73 n 15 (1 i) (1.05) SOLUTIONS USING THE TABLES: Look up the present value of $1 at 5% over 15 years: F(15 years, 5%) =.4810 Then: = FV * F = $7,600.00 *.4810 = $3,655.60 Alternatively, we can look up the future value factor: FVF(15 years, 5%) = 2.0789 Then: = FV / FVF = $7,600.00 / 2.0789 = $3,655.78 BEG/END = END [regular annuity: interest added at end of each period] [interest added once a year] N = 15 I = 5 [interest rate in %] FV = 7,600.00 [value of deposit at end] PMT = 0 [no recurring payments here] Solve for -> -3,655.73 [amount deposited at start] 3
PRESENT VALUE OF AN ORDINARY ANNUITY EXPLANATION: An ordinary annuity is a sequence of equal-size payments, spaced out equally over time, with the payments (cash flows) taking place at the END of each time period (month, year). The present value of such an annuity is the sum of the present values of the individual future payments. 1 1 (1 ) n i A PMT i A home mortgage requires the payment of $22,000.00 per year for 30 years. The interest rate charged is 6%. How much money was borrowed? 1 1 (1.06) 30 A 22, 000.00 302,826.29.06 Look up the factor for the present value of an annuity: AF(30 years, 6%) = 13.7648. Then: A = PMT * AF = $22,000.00 * 13.7648 = $302,826.60. BEG/END = END [regular annuity: payments made at end of each period] [yearly payments, and interest added once a year] N = 30 I = 6 [interest rate in %] FV = 0 [assume loan is to be fully paid off over the 30 years] PMT = -22,000.00 [the yearly payment made; observe the sign!] Solve for -> 302,826.29 [amount borrowed at start] 4
FUTURE VALUE OF AN ORDINARY ANNUITY EXPLANATION: An ordinary annuity is a sequence of equal-size payments, spaced out equally over time, with the payments (cash flows) taking place at the END of each time period (month, year). The future value of such an annuity is the sum of the future values of the individual cash flows. n (1 i) 1 FVA PMT i You invest $1,500 per year (at the end of each year) for 25 years. You earn a return on your investments of 9%. How much is in your investment account at the end of the 25 years? 25 (1.09) 1 FVA 1,500.00 127, 051.34.09 Look up the factor for the future value of an annuity: FVAF(25 years, 9%) = 84.701. Then: FVA = PMT * FVAF = $1,500.00 * 84.701 = $127,051.50. BEG/END = END [regular annuity: payments made at end of each period] [ yearly payments, and interest added once a year] N = 25 I = 9 [interest rate in %] = 0 [assume we start with an empty savings account] PMT = -1,500.00 [the yearly contribution made; observe the sign!] Solve for FV -> 127,051.34 [value of the account at end] 5
SINKING FUND PAYMENT [SAVING] EXPLANATION: A sinking fund is a savings or investment account into which you make periodic contributions in order to reach a specific target amount (FVA). Here we calculate the periodic payment necessary to reach a specific goal, given the time horizon and the expected interest rate. i SFP FVA n (1 i) 1 You need to save up $200,000 for your child s college education, which will commence 18 years from now. How much do you need to save each year (deposited at the end of the year) to reach this goal, assuming you can earn a return of 7% on your investments?.07 SFP 200, 000.0 5,882.52 18 (1.07) 1 The textbook does not have a table for this case. BEG/END = END [regular annuity: payments made at end of each period] [ yearly payments, and interest added once a year] N = 18 I = 7 [interest rate in %] = 0 [assume we start with an empty savings account] FV = 200,000.00 [our target or goal] Solve for PMT -> -5,882.52 [the required yearly contribution] 6
MORTGAGE PAYMENTS [LOANS]: YEARLY PAYMENTS EXPLANATION: This is the formula for computing the periodic payment required to fully pay off (amortize) a given loan amount. PMT i A 1 1 (1 ) n i You want to purchase a home being offered for sale at $300,000. After putting down a $20,000 down payment, you need to borrow the rest ($280,000.) The bank is offering a 30-year fixed rate mortgage with an interest rate of 4.95%. What will your yearly mortgage payment be? PMT.0495 280, 000.00 18,110.72 1 1 (1.0495) 30 The textbook does not have a table for this case. BEG/END = END [regular annuity: payments made at end of each period] [ yearly payments, and interest added once a year] N = 30 I = 4.95 [interest rate in %] = 280,000.00 [the amount to be borrowed] FV = 0 [no residual at the end; the loan is to be fully paid off] Solve for PMT -> -18,110.72 [the resulting yearly mortgage payment] 7
MORTGAGE PAYMENTS [LOANS]: MONTHLY PAYMENTS EXPLANATION: This is the formula for computing the MONTHLY payment required to fully pay off (amortize) a given loan amount. In our formula, this means that n becomes the number of months (30 x 12 = 360) and i becomes the MONTHLY interest rate. PMT i A 1 1 (1 ) n i You want to purchase a home being offered for sale at $300,000. After putting down a $20,000 down payment, you need to borrow the rest ($280,000.) The bank is offering a 30-year fixed rate mortgage with an interest rate of 4.95%. What will your monthly mortgage payment be? PMT.0495 280, 000.00 12 1, 494.56 1 1.0495 360 (1 ) 12 The textbook does not have a table for this case. BEG/END = END 2 [regular annuity: payments made at end of each period] [monthly payments, and interest added once a month] N = 30x12=360 [# periods (months)] I = 4.95 [YEARLY interest rate in %] = 280,000.00 [the amount to be borrowed] FV = 0 [no residual at the end; the loan is to be fully paid off] Solve for PMT -> -1,494.56 [the resulting monthly mortgage payment] 8
PRESENT VALUE OF A GROWING ANNUITY (2 pages) EXPLANATION: Say we have an annuity where the payments are not constant, but grow at a rate g. Example: a stock whose dividends grow from year to year. That means: PMT t+1 = (1+g) * PMT t The present value of a growing annuity lasting t periods is: 0 t C 1 g 1g r g 1 r r g 1 r t 1 C0 (1 g) 1 1 Where C0 = the most recent payment received, and C1 is the next payment to be received. (Use the applicable formula, depending on the information available.) NOTE that the above formula only applies when g < r. HINTS If PMT t+1 = PMT t * (1+g) then it is also true that PMT t+n = PMT t * (1+g) n In this case, it is also true that t+1 = t * (1+g) and that t+n = t * (1+g) n 9
You are offered a stock which just paid a dividend of $2.50 per share. The company has promised to increase dividends by 4% per year forever. You plan to hold this stock in your portfolio for 10 years. How much is this stock worth to you? You use a discount rate of 11%. First, note that the next dividend will be: C 1 = C 0 * (1+g) = $2.50 (1.04) = $2.60. Then: 0 t 10 C 1 1 g $2.60 1.04 1 1 $17.78 r g 1 r 0.11 0.04 1.11 SOLUTION USING TABLES: There are no tables commonly available for the present value of growing annuities. (You would need a separate table for each possible value of g!) You have to solve these problems using the basic formula. Financial calculators are not equipped to deal with growing annuities. You have to solve these problems using the basic formula. 10
PRESENT VALUE OF A GROWING PERPETUITY EXPLANATION: If we have a growing annuity where the payments are supposed to last forever (t goes to infinity), then what? As long as the growth rate g is smaller than the discount rate r, the last term in the formula goes to zero. Then: BASIC FORMULA FOR A GROWING PERPETUITY: 0 C0 (1 g) r g 1 r r g 1 r r g r g t C1 1g C1 1g C1 1 1 (Use one of the last two formulas, depending on which cash flow is known.) You are offered a stock which just paid a dividend of $2.50 per share. The company has promised to increase dividends by 4% per year forever. You plan to hold this stock in your portfolio forever. How much is this stock worth to you? You use a discount rate of 11%. First, note that the next dividend will be: C 1 = C 0 * (1+g) = $2.50 (1.04) = $2.60. Then: 0 C C (1 g) 2.50 1.04 1 0 r g r g 0.11 0.04 $37.14 11
PRESENT VALUE OF A FIXED PERPETUITY EXPLANATION: If we have an annuity with fixed payments where the payments are supposed to last forever (t goes to infinity), then what? BASIC FORMULA FOR A FIXED PERPETUITY: 0 C C C C r g 1 r r g 1 r r g r t 1 1g 1 1g 1 1 1 You are offered a stock which just paid a dividend of $2.50 per share. The company has promised to maintain this dividend forever. You plan to hold this stock in your portfolio forever. How much is this stock worth to you? You use a discount rate of 11%. 0 C 2.50 $22.73 r 0.11 12
PRESENT VALUE OF AN ANNUITY DUE EXPLANATION: An annuity due is a sequence of equal-size payments, spaced out equally over time, with the payments (cash flows) taking place at the BEGINNING of each time period (month, year). The present value of such an annuity is the sum of the present values of the individual future payments. 1 1 (1 ) n i A PMT (1 i i ) A long term home lease requires the payment of $22,000.00 per year for 30 years, at the beginning of each year. The applicable discount rate is 6%. What is the present value of this lease? 1 1 (1.06) 30 A 22, 000.00 (1.06) 320,995.86.06 Look up the factor for the present value of an ordinary annuity: AF(30 years, 6%) = 13.7648. Then: A = PMT * AF = $22,000.00 * 13.7648 = $302,826.60. Since this is an annuity due, multiply by the interest factor (1+i): $302, 826.60 * (1.06) = $320,996.20. BEG/END = BEGIN [annuity due: payments made at start of each period] [yearly payments, and interest added once a year] N = 30 I = 6 [annual interest rate in %] FV = 0 [no additional money due at end of lease] PMT = -22,000.00 [the yearly payment made; observe the sign!] Solve for -> 320,995.86 [Net present value of lease at inception] 13
FUTURE VALUE OF AN ANNUITY DUE EXPLANATION: An annuity due is a sequence of equal-size payments, spaced out equally over time, with the payments (cash flows) taking place at the BEGINNING of each time period (month, year). The future value of such an annuity is the sum of the future values of the individual future payments. n (1 i) 1 FVA PMT (1 i ) i You invest $1,500 per year (at the beginning of each year) for 25 years. You earn a return on your investments of 9%. How much is in your investment account at the end of the 25. year? 25 (1.09) 1 FVA 1,500.00 (1.09) 138, 485.97.09 Look up the factor for the future value of an ordinary annuity: FVAF(25 years, 9%) = 84.701. Then: FVA = PMT * FVAF = $1,500.00 * 84.701 = $127,051.50. Since this is an annuity due, multiply by the interest factor (1+i): $127,051.50 * (1.09) = $138,486.14. BEG/END = BEGIN [annuity due: payments made at beginning of each period] [yearly payments, and interest added once a year] N = 25 I = 9 [annual interest rate in %] = 0 [assume we start with an empty savings account] PMT = -1,500.00 [the yearly contribution made; observe the sign!] Solve for FV -> 138,485.97 [value of the account at end] 14