MGT 470 Ch 4 TVM (cs3ed) v1.0 Aug 15 1 Ch 4: Time Value of Money Time Has Value (The Time Value of Money – TVM):  Time affects the value of financial assets and transactions  When money is some how invested (and not just placed under a mattress or in a safe), the amount of money grows  Although money loses value over time due to inflation, the amount of money in an account that earns a positive ROR will be greater in the future than what it is today. Why is this true?  Thus the money in the account has different values at different points in time  This is what the term “Time Value of Money” refers to  The ROR should compensate for opportunity cost, inflation and risk  the increasing amount of money over time should more than make up for the value lost due to inflation and opportunity costs

2 1. To complete your business school education, you will need $10,000 a year for the next four years, starting next year (that is, you will need to pay for one year’s worth of schooling at the beginning of the year, one year from today). Your rich uncle offers to put you through school and he will deposit a lump sum into a saving account paying 7% p.a. sufficient to defray these expenses. The deposit will be made today. How large must the deposit be? (Assume your uncle is rich because he won the lottery and he doesn’t know much about finance stuff.) PMT = 10, PV = nominal = 7% 2. You are considering financing a new car which cost $51,300 with an amortized loan. The nominal rate is 2.9% p.a., the term of the loan is 6 years and you will make monthly payments. How much will each payment be? PMT = ? nominal = 2.9% PV = $51,300 m = 12 T = 6 n = m x T = 12 x 6 = 72 r periodic = nominal / m = 2.9/12 = P/Y=1, N=72, I/Y= , PV=51300; CPT,PMT: PMT = $ OR P/Y=12, N=72, I/Y=2.9, PV=51300’ CPT,PMT: PMT = $ Soln. Opt 2: P/Y=1, N=4, I/Y=7, PMT=10000; CPT,PV: $33, MGT 470 Ch 4 TVM (cs3ed) v1.0 Aug 15

3 3. How many years will it take for your savings account to accumulate $1m if it pays 4% interest p.a. compounded semiannually and you deposit $10k every 6-months at the end of the 6-month period? 0 PMT = (-)10,000 r = 4% FV = $1m n = ? n - 1n Today you deposited $1,300 in a bank that pays 5% p.a., compounded quarterly. How much money would you have in this account 20 months from now? nominal = 5% PV = $1,300 m = 4, T = 20/12 = n = m x T = FV = ? n = r periodic = r nominal /m = 5%/4 = 1.25% P/Y=1, N= , I/Y=1.25, PV=1300; CPT,FV: FV = $1, OR P/Y=4, N= , I/Y=5, PV=1300; CPT,FV: FV = $1, a) Find number of periods r periodic = nominal / m = 4%/2 = 2% P/Y=1, I/Y= 2, PMT=-10000, FV= ; CPT,N: N = periods OR P/Y=2, I/Y= 4, PMT=-10000, FV= ; CPT,N: N = periods b) Find number of years: Years = N/m = 55.48/2 = years MGT 470 Ch 4 TVM (cs3ed) v1.0 Aug 15

4 5. What is the future value of an annuity due yielding % p.a. that pays quarterly payments of $1,000 for 9 mos? (Do the math; do not use the financial functions on your calculator. Draw a cash flow diagram) FV = PV(1 + r/m) n + PV(1 + r/m) n-1 + ………….. = 1000( /4) ( /4) ( /4) 1 = 1000(1.0241) (1.0241) (1.0241) 1 = 1000(1.0741) (1.0488)+ 1000(1.0241) = 1, , , = $3, PMT = 1, FV = ? nominal = 9.64% T = 9/12, m = 4, n = T x m = 9/12 x 4 = 3 4 MGT 470 Ch 4 TVM (cs3ed) v1.0 Aug 15

5 6. On 1 January 2008 you deposited $700 in a mutual fund that has been yielding a constant 6% p.a. for the last several years. You plan to deposit $700 every 3 months thereafter for the next 4 years (i.e. the next deposit will be made 1 April, the subsequent deposit on 1 July, etc.) How much money will have accumulated after 4 years? PMT = 700 FV = ? nominal = 6% m = 4, T = 4 n = m x T = On 1 January 2008 you deposited $700 in a mutual fund that has been yielding a constant 6% p.a. for the last several years. You plan to deposit $700 every 3 months thereafter for the next 4 years (i.e. the next deposit will be made 1 April, the subsequent deposit on 1 July, etc.) How much money will have accumulated after 4 years to include a payment to be made at the beginning of the first quarter of the fifth year. Note: There is no payment at t = 16 Set to “BGN” mode: 2nd, BGN, 2nd, SET, CE/C r periodic = r nominal /m = 6%/4 = 1.5% P/Y=1, N=16, I/Y=1.5, PMT=700; CPT,FV: FV = $12, OR P/Y=4, N=16, I/Y=6, PMT=700; CPT,FV: FV = $12, Option 1: (Treat as an Annuity Due) PMT = FV = ? nominal = 6% m = 4, T = 4 n = m x T = Set to “BGN” mode: 2nd, BGN, 2nd, SET, CE/C r periodic = r nominal /m = 6%/4 = 1.5% [P/Y=1, N=16, I/Y=1.5, PMT=700; CPT,FV: FV = $12,740.95] + $700 = $13, OR P/Y=4, N=16, I/Y=6, PMT=700; CPT,FV: FV = $12,740.95] + $700 = 13, MGT 470 Ch 4 TVM (cs3ed) v1.0 Aug 15

8. You are considering leasing a car that cost $46,000. The lease will be for 5 years and requires monthly payments. Somewhere on the lease paperwork you notice a statement to the affect that you will be charged a nominal rate of % p.a. Assume the car will be worth $20,000 at the end of the lease. What kind of annuity is this?___________________ How much will your payments be? r periodic = r nominal /m = %/12 = % Set BGN, P/Y=1, N=60, I/Y=0.5238, PV=46000, FV= ; CPT PMT: PMT = $ OR Set BGN, P/Y=12, N=60, I/Y=6.2850, PV=46000, FV= ; CPT PMT: PMT = $ PMT = ? PV = $46,000 nominal = % 5 yrs m=12, T=5; n = m x T = 60 0 Turn-in = $20, (continued) Option 2: (Treat as an Annuity in Arrears) PMT = 700 FV = ? nominal = 6% m = 4, T = 4 n = m x T = r periodic = r nominal /m = 6%/4 = 1.5% P/Y=1, N=16, I/Y=1.5, PV=700, PMT=700; CPT,FV: FV = $13, OR P/Y=4, N=16, I/Y=6, PV=700, PMT=700; CPT,FV: FV = 13, MGT 470 Ch 4 TVM (cs3ed) v1.0 Aug 15

7 9. You are tasked with estimating the fair market value of a security that promises uneven future payments. The table below shows the monthly payment schedule (each cash flow occurs at the end of the month). You consider 6% p.a. to be the appropriate opportunity cost. What is the theoretical value of this security? 1 st month2 nd month3 rd month4 th month5 th month6 th month $350$390$480$660$820$940 r periodic = r nominal /m = 6%/12 = 0.5% CF, 2nd, CLR WORK (Clears Cash Flow Registers) 0, ENTER, ↓, 350, ENTER ↓, ↓, 390, ENTER ↓, ↓, 480, ENTER ↓, ↓, 660, ENTER ↓, ↓, 820, ENTER ↓, ↓, 940, ENTER NPV, 0.5, ENTER ↓, CPT = $3, $350 $390 $480 $660 $820 $940 MGT 470 Ch 4 TVM (cs3ed) v1.0 Aug 15

8 10. Today you open a new investment account for your company with a $30,000 deposit. The account has had an average yield of % p.a. over the last three years and compounds every month. You plan to deposit $30,000 into this account every quarter, at the beginning of the quarter. Your next deposit will be three months from now. How much will you have in this account 3 years from now? qtrs PMT = $30,000 FV = ? nominal = 7.56% compounded monthly m=4, T=3; n = m x T = 4 x 3 = 12 0 Set BGN, P/Y=4, C/Y=12, N=12, I/Y=7.56, PMT=30000; CPT,FV: FV = $407, OR Convert the monthly rate into a quarterly rate then solve for FV 2nd, ICONV, 7.56, ENTER, ↓, 12, ENTER, ↓, ↓, CPT: EFF% = % ↓, 4, ENTER. ↓, CPT: NOM% = % Set BGN, P/Y=4, N=12, I/Y=7.6077, PMT=30000; CPT,FV: FV = $407, MGT 470 Ch 4 TVM (cs3ed) v1.0 Aug 15

9 11. You are considering buying one of the two securities described below. What is the fair market value of each security and which one is priced closer to its fair market value? Security A: A 2-year investment period yielding 6.35% p.a.; monthly payments of $75; priced at $1950. Security B: A 2-year investment period yielding 6.25% p.a.; 3 payments of $62.50 at the end of each of each successive 6-month period and a final lump sum payment of $1,062.50; priced at $1300. Security A: PMT = 75 nominal = 6.35% PV = ? m = 12, T = 2; n = m x T = 24 r periodic = r nominal /m = 6.35%/12 = 0.529% P/Y=1, N=24, I/Y=0.529, PMT=75; CPT,PV: PV = $1, OR P/Y=12, N=24, I/Y=6.35, PMT=75; CPT,PV: PV = $1, Percent Above FMV: (1, ,686.21) / 1, = 15.64% PMT = PV = ? nominal = 6.25% PMT = Soln Opt 1 Security B: m = 2, T = 2; n = m x T = 4 Soln. Opt 1: r periodic = r nominal /m = 6.25%/2 = 3.125% CF0=0, CF1=62.5, CF2=62.5, CF3=62.5, CF4=1062.5, I/Y=3.125; NPV = $1, MGT 470 Ch 4 TVM (cs3ed) v1.0 Aug 15

10 PMT = PV = ? nominal = 6.25% PMT = 1000 Soln Opt 2 Soln. Opt 2a: PVA(4, 3.125, 62.50) + PV(4, 3.125, 1000) [N=4, I/Y=3.125, PMT=62.5] + [N=4, I/Y=3.125, FV=1000] $ $ = $1, Soln. Opt 2b: P/Y=1, N=4, I/Y=3.125, PMT=62.5, FV=1000; CPT,PV: PV = $1, Percent Above FMV: (1,300 – 1,115.82) / 1, = 16.5% Security A is more fairly priced 10 MGT 470 Ch 4 TVM (cs3ed) v1.0 Aug 15

Retirement Plan You plan to retire on your 65th birthday. You want to withdraw $100, each year at the beginning of each year. You plan to live until your 88th birthday and you want your account balance to be $0 on that day (this is for planning purposes only). How much money you must deposit annually into a retirement account starting on your 25th birthday to fund the above described retirement plan. You estimate the average rate of return over the rest of your life will be 7% p.a. Set calculator to “Begin” mode P/Y=1, N=23, I/Y=7, PMT= ; CPT, PV: $1,206, PMT = $100k PV = ? nominal = 7% You make your deposit at the beginning of each year: Set calculator to “Begin” mode P/Y=1,N=40, I/Y=7, FV=1,206,124.05; Compute PMT: $5, PMT = ? FV = $1,206, nominal = 7% Step 2: Find the amount of the annual deposit. Step 1: Find the amount of money the account needs to have on your 65th birthday. MGT 470 Ch 4 TVM (cs3ed) v1.0 Aug 15

Martha Mills, manager of the Plaza Gold Emporium, wants to allow her customers to buy on credit, giving them 3 months in which to pay. However, Martha will have to borrow from her bank to establish the credit reserve. The bank will charge 7% p.a. interest compounded monthly. Martha wants to quote a simple rate to her customers that will exactly cover her financing cost. What simple (quoted, nominal) should Martha quote to her customers. Assume that all customers will take the full 3 months to pay. (Hint: the credit account compounds quarterly.) Approach: In order to break even, Martha wants credit sales to yield exactly equal to her cost of debt. Thus both arrangements must have the same EAR. For the bank loan, m = 12. For credit sales, m = 4. 1) Find EAR on the bank loan: Solution Opt 1: EAR = ( /12) 12 – 1 = 7.229% Solution Opt 2: 2nd, ICONV 7, ENTER ↓, ↓, 12, ENTER ↓, ↓, CPT : EFF% = 7.229% 2) Find the quoted (simple) rate to apply to credit sales Solution Opt 1: EFF% bank = (1 + r nominal /m)m = (1 + r nominal /4) = (1 + r nominal /4) = 1 + r nominal / = r nominal /4 nominal = 7.041% Solution Opt 2: 2nd, ICONV ↓, 7.229, ENTER ↓, 4, ENTER ↓, CPT: NOM = 7.041% MGT 470 Ch 4 TVM (cs3ed) v1.0 Aug 15

A friend of yours is working as an unpaid intern at a local brokerage firm and her boss is selling some securities which pay $50 at the end of each of the next 3 years and a final payment of $1050 at the end of the 4 th year. Your friend says she can get you some of these securities at a cost of $900 each. Your money is currently invested in a bank that pays 8% p.a. with quarterly compounding. You consider the securities as being just as safe and as liquid as your bank. What is the fair market value (the theoretical value) of these securities? PMT = PV = ? nominal = 8.24% PMT = 1050 Why use 8.24% for nominal? Find PV using uneven CF approach CF, 2nd, CLR WORK (clears CF registers) 0, ENTER ↓, 50, ENTER ↓, ↓, 50, ENTER ↓, ↓, 1050, ENTER NPV, 8.24, ENTER ↓, CPT: PV = $ Answer: This is the Opportunity Cost of Capital and it must be converted to the same compounding frequency as the securities Solution Opt. 1: EAR = (1 + r/m) m - 1 = ( /4) 4 – 1 = 8.24% Solution Opt. 2: 2nd, ICONV 8, ENTER ↓, ↓, 4, ENTER ↓, ↓, CPT: EFF% = 8.24% 2 nd Step: Find PV of the securities 1 st Step: Find the EAR on the bank account Why? MGT 470 Ch 4 TVM (cs3ed) v1.0 Aug 15

14 PMT = PV = ? nominal = 8.24% PMT = 1000 Another way: Recognize the cash flows as such (this looks like a bond or a non-amortized loan) Solution Opt 1: PV= PVA(4, 8.24, 50) + PV(4, 8.24, 1000) = [N=4, I/Y=8.24, PMT=50] + [N=4, I/Y=8.24, FV=1000] = $ $ = $ Solution Opt 2: N=4, I/Y=8.24, PMT=50, FV=1000; CPT,PV: PV = $ Compare quoted price to fair market value $900 - $ = $6.74 Your friend is trying to make a profit of $6.74 MGT 470 Ch 4 TVM (cs3ed) v1.0 Aug 15

On 1 January you deposited $500 in a savings account that pays 3% p.a.. You plan to deposit $500 every three months thereafter (i.e. the next deposit will be on 1 April, the subsequent deposit on 1 July, etc.) How much money will have accumulated after 14 months? Wrong Ans: $2, PMT = $ FV = ? 5 m = 4 T = 14/12 = n = m x T = n = Cash FlowPVnr periodic FV 0$ %$ $ %$ $ %$ $ %$ $ %$ Solution Option 1: Compound each CF forward and sum them: Sum of FVs: $2, Solution Option 2: 1) Find the FV at t=4: PMT = $ FV = ? Set “END”, P/Y=4, N=4, I/Y=3, PV=500, PMT=500; CPT, FV: $2, ) Compound $2, forward periods PV = $ FV = ? n = PV = $2, P/Y=4, N= , I/Y=3, PV= ; CPT, FV: $2, MGT 470 Ch 4 TVM (cs3ed) v1.0 Aug 15

If you save $500 each year for 2 years and then $1,000 each for two years, how much must you save in the 5 th and 6 th years to have $10,000 at the end of 10 years if the interest rate is 5% p. a.? $500[(1.05) 9 +(1.05) 8 ] + $1,000[(1.05) 7 + (1.05) 6 ] + x[(1.05) 5 + (1.05) 4 ] = $10,000 $500(3.0288) + $1,000(2.7472) + x(2.4918) = $10,000 $1, $2, x(2.4918) = $10,000 x(2.4918) = $10,000 - $1, $2, x = $5, / x = $2, $500 $1,000 PMT = ? FV = $10, PMT = $100,000 PV = ? A football coach is leaving his current school. In doing so, he is giving up an annuity of $100,000 per year for 10 years that would begin when he turns 60. The coach is 45. His new school has offered to make up the loss of the annuity with a lump sum payment when he moves. How much should the new school pay if the interest rate is 7% p.a.? Step 1: Compute the value of the annuity at age 60 P/Y=1, SET BGN, N=10, I/Y=7, PMT=100000; CPT, PV: PV = $751, FV = $751, PV = ? 45 Step 2: Discount the value computed in Step 1 to age 45 P/Y=1, N=15, I/Y=7, FV= ; CPT, PV: PV = $272, MGT 470 Ch 4 TVM (cs3ed) v1.0 Aug 15

17 Formulas: Future Value: FV = PV(1 + r/m) n Present Value: PV = FV / (1 + r/m) n Find r: Find n: n = LN(FV / PV) / LN(1 + r/m) Find FV of an Ordinary Annuity: FV A = PMT [( (1 + r/m) n – 1) / (r/m)] Find FV of an Annuity Due: FV A,due = FV A (1 + r/m) Find PV of an Ordinary Annuity: PV A = PMT[((1 + r/m) n – 1) / ((r/m) (1 + r/m) n) ] Find PV of an Annuity Due: PV A,due = PV A (1 + r/m) Find PMT of an Annuity: Ordinary Annuity (FV is Given) PMT = FV A [(r/m)/((1 + r/m) n – 1)] Ordinary Annuity (PV is Given) PMT = PV A [(r/m)(1 + r/m) n / ( (1 + r/m) n – 1)] Annuity Due (FV is Given) PMT = FV A,due [(r/m) / ( (1 + r/m) n – 1)] / (1 + r/m) Annuity Due (PV is Given) PMT = PV A,due [(r/m)(1 + r/m) n / ( (1 + r/m) n – 1)] / (1 + r/m) Effective Annual Rate (EAR): EAR = ( 1 + r nominal / m ) m – 1 PV of a Perpetuity: PMT/(r/m) r = n FV / PV - 1