# 1 Time Value of Money (TVM) - the Intuition A cash flow today is worth more than a cash flow in the future since: uIndividuals prefer present consumption.

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1 Time Value of Money (TVM) - the Intuition A cash flow today is worth more than a cash flow in the future since: uIndividuals prefer present consumption to future consumption. uMonetary inflation will cause tomorrows dollars to be worth less than todays. uAny uncertainty associated with future cash flows reduces the value of the cash flow.

2 The Time-Value-of-Money uThe Basic Time-Value-of-Money Relationship: FV t+T = PV t X (1 + r) T where u r is the interest rate per period u T is the duration of the investment, stated in the compounding time unit u PV t is the value at period t (beginning of the investment) u FV t+T is the value at period t+T (end of the u investment)

3 Future Value and Compounding Compounding: How much will \$1 invested today at 8% be worth in two years? (The Time Line) Year012 \$1.1664 Future Value: FV 2 = \$1 x 1.08 2 = \$1.1664 \$1.1664 \$1 \$1.08 \$1.08 x 1.08 \$1 x 1.08 Or:

4 Housekeeping functions: 1. Set to 8 decimal places: 2. Clear previous TVM data: 3. Set payment at Beginning/End of Period: 3. Set # of times interest is calculated (compounded) per year to 1: TVM in your HP 10B Calculator Yellow = DISP 8 INPUT CLEAR ALL 0 BEG/END Yellow PMT P/YR Yellow 1

5 First, clear previous data, and check that your calculator is set to 1 P/YR: The display should show: 1 P_Yr Input data (based on above FV example) FV in your HP 10B Calculator INPUT CLEAR ALL +/- Yellow 1PV 8I/YR 2N FV Key in PV (always -ve) Key in interest rate Key in number of periods Compute FV Display should show: 1.1664

6 Q.Deposit \$5,000 today in an account paying 12%. How much will you have in 6 years? How much is simple interest? How much is compound interest? A.Multiply the \$5000 by the future value interest factor: \$5000 x (1 + r) T = \$5000 x ( ) 6 = \$5000 x 1.9738227 = \$__________ At 12%, the simple interest is ___ x \$5000 = \$ ___ per year. After 6 years, this is 6 x \$____ = \$ ______; the compound interest is thus: \$4869.11 - \$3600 = \$1,269.11 An Example - Future Value for a Lump Sum

7 (The Time Line) Year012 \$1 \$0.9259 \$1 / 1.08 \$0.9259 / 1.08 Or: Present Value and Discounting Discounting: How much is \$1 that we will receive in two years worth today (r = 8%)? \$0.8573 Present Value: PV 0 = \$1 / 1.08 2 = \$0.8573

8 First, clear previous data, and check that your calculator is set to 1 P/YR: The display should show: 1 P_Yr Input data (based on above PV example) PV in your HP 10B Calculator INPUT CLEAR ALL Yellow 1FV 8I/YR 2N PV Key in FV Key in interest rate Key in number of periods Compute PV Display should show: -0.85733882

9 Q.Suppose you need \$20,000 in three years to pay your university tuition. If you can earn 8% annual interest on your money, how much do you need to invest today? A.We know the future value (\$20,000), the rate (8%), and the number of periods (3). We are looking for the present amount to be invested (present value). We first define the variables: FV 3 = \$20,000r = 8 percent T= 3 yearsPV 0 = ? Set this up as a TVM equation and solve for the present value: ________ = PV 0 x (_____) -- Solve for PV: PV 0 = \$_________________ = \$15,876.64 \$15,876.64 invested today at 8% annually, will grow to \$20,000 in three years. Example 1 - Present Value of a Lump Sum

10 Q.Suppose you are currently 21 years old, and can earn 10 percent on your money. How much must you invest today in order to accumulate \$1 million by the time you reach age 65? A.We first define the variables: FV 65 = _______ r = _______ T= __________ PV 21 = ? Set this up as a TVM equation and solve for the present value: _________ = PV 21 x (_________) 44 Solve for PV: PV 21 = _____________ = \$15,091.13 If you invest \$15,091.13 today at 10% annually, you will have \$1 million by the time you reach age 65 Example 2 - Present Value of a Lump Sum

11 How Long is the Wait? If we deposit \$5000 today in an account paying 10%, how long do we have to wait for it to grow to \$10,000? Solve for T: FV t+T = PV t x (1 + r) T \$10000 = \$5000 x (1.10) T (1.10) T = 2 T = ln(2) / ln(1.10) = 7.27 years

12 First, clear previous data, and check that your calculator is set to 1 P/YR: The display should show: 1 P_Yr Input data (based on above example) T in your HP 10B Calculator INPUT CLEAR ALL Yellow FV I/YR N Key in FV Key in interest rate Compute T Display should show: 7.27254090 10,000 Key in PV +/-PV 5,000 10

13 An Example - How Long is the Wait? Q.You have \$70,000 to invest. You decided that by the time this investment grows to \$700,000 you will retire. Assume that you can earn 14 percent annually. How long do you have to wait for your retirement? A.We first define the variables: FV ? = \$700,000r = 14 percent PV 0 = \$70,000T= ? Set this up as a TVM equation and solve for T: _______________________ Solve for T: T = ln(10)/ln(1.14) = 17.57 years If you invest \$70,000 today at 14% annually, you will reach your goal of \$700,000 in 17.57 years

14 uAssume the total cost of a University education will be \$50,000 when your child enters college in 18 years. uYou have \$5,000 to invest today. uWhat rate of interest must you earn on your investment to cover the cost of your childs education? uSolve for r : FV t+T = PV t x (1 + r) T \$50000= \$5000 x (1 + r) 18 (1 + r) 18 = 10 ____________________ r= 0.13646 = 13.646% per year What Rate Is Enough?

15 First, clear previous data, and check that your calculator is set to 1 P/YR: The display should show: 1 P_Yr Input data (based on above example) Interest Rate (r) in your HP 10B Calculator INPUT CLEAR ALL Yellow FV N I/YR Key in FV Key in T Compute r Display should show: 13.64636664 50,000 Key in PV +/-PV 5,000 18

16 An Example - Finding the Interest Rate (r): Q.In December 1937, the market price of an ABC company common stock was \$3.37. According to The Financial Post, the price of an ABC company common stock in December 1999 is \$7,500. What is the annually compounded rate of increase in the value of the stock? A.Set this up as a TVM problem. Future value = ________Present value = _________ T = _________r = _________ FV 1999 = PV 1937 x (1 + r) T so, Solve for r: r = _____________________ =.1324 = 13.24%

17 Net Present Value (NPV) Example for NPV: You can buy a property today for \$3 million, and sell it in 3 years for \$3.6 million. The annual interest rate is 8%. Qa.Assuming you can earn no rental income on the property, should you buy the property? Aa.The present value of the cash inflow from the sale is: PV 0 = \$3,600,000/(1.08) 3 = \$2,857,796.07 Since this is less than the purchase price of \$3 million - dont buy We say that the Net Present Value (NPV) of this investment is negative: NPV = -C 0 + PV 0 (Future CFs) = -3,000,000+2,857,796.07 = -142,203.93 < 0

18 Example for NPV (continued): Qb.Suppose you can earn \$200,000 annual rental income (paid at the end of each year) on the property, should you buy the property now? Ab.The present value of the cash inflow from the sale is: PV 0 = [200,000 /1.08] + [200,000 /1.08 2 ] + [3,800,000/1.08 3 ] = \$3,373,215.47 Since this is more than the purchase price of \$3 million - buy We say that the Net Present Value (NPV) of this investment is positive: NPV = -C 0 + PV 0 (Future CFs) = -3,000,000+ 3,373,215.47 = 373,215.47 > 0 The general formula for calculating NPV: NPV = -C 0 + C 1 /(1+r) + C 2 /(1+r) 2 +... + C T /(1+r) T

19 Simplifications uPerpetuity A stream of constant cash flows that lasts forever uGrowing perpetuity A stream of cash flows that grows at a constant rate forever uAnnuity A stream of constant cash flows that lasts for a fixed number of periods uGrowing annuity A stream of cash flows that grows at a constant rate for a fixed number of periods

20 Perpetuity uA Perpetuity is a constant stream of cash flows without end. uSimplification: PV t = C t+1 / r 0 1 2 3 …forever... |---------|--------|---------|---------(r = 10%) \$100\$100\$100...forever… PV 0 = \$100 / 0.1 = \$1000 The British consol bond is an example of a perpetuity.

21 Q1.ABC Life Insurance Co. is trying to sell you an investment policy that will pay you and your heirs \$1,000 per year (starting next year) forever. If the required annual return on this investment is 13 percent, how much will you pay for the policy? A1.The most a rational buyer would pay for the promised cash flows is C/r = \$1,000/0.13 = \$7,692.31 Q2.ABC Life Insurance Co. tells you that the above policy costs \$9,000. At what interest rate would this be a fair deal? A2.Again, the present value of a perpetuity equals C/r. Now solve the following equation: \$9,000 = C/r = \$1,000/r r = 0.1111 = 11.11% Examples - Present Value for a Perpetuity

22 Growing Perpetuity uA growing perpetuity is a stream of cash flows that grows at a constant rate forever. uSimplification: PV t = C t+1 / (r - g) 0 1 2 3 …forever... |---------|---------|---------|--------- (r = 10%) \$100\$102\$104.04 … (g = 2%) PV 0 = ________________ = \$1250

23 Q.Suppose that ABC Life Insurance Co. modifies the policy, such that it will pay you and your heirs \$1,000 next year, and then increase each payment by 1% forever. If the required annual return on this investment is 13 percent, how much will you pay for the policy? A.The most a rational buyer would pay for the promised cash flows is __________________ = \$8,333.33 Note:Everything else being equal, the value of the growing perpetuity is always higher than the value of the simple perpetuity, as long as g>0. An Example - Present Value for a Growing Perpetuity

24 Annuity uAn annuity is a stream of constant cash flows that lasts for a fixed number of periods. uSimplification:PV t = C t+1 (1/r) { 1 - [1 / ( 1 + r ) T ] } FV t+T = C t+1 (1/r) { [ ( 1 + r ) T ] - 1 } 0 1 2 3 years |----------|---------|---------|(r = 10%) \$100 \$100 \$100 PV 0 = 100 (1/0.1) { 1 - [ 1/(1.1 3 )] } = \$248.69 FV 3 = 100 (1/0.1) { [1.1 3 ] - 1 } = \$331

25 First, clear previous data, and check that your calculator is set to 1 P/YR: The display should show: 1 P_Yr Input data (based on above PV example) PV and FV of Annuity in your HP 10B Calculator INPUT CLEAR ALL Yellow PMT I/YR 3N PV Key in payment Key in interest rate Key in number of periods Compute PV Display should show: -248.68519910 100 FV Compute FV * Display should show: -331.00000000 PV 0 * Note: you can calculate FV directly, by following first 3 steps, and replacing PV with FV in the fourth step. 10

26 Q.A local bank advertises the following: Pay us \$100 at the end of the next 10 years. We will pay you (or your beneficiaries) \$100, starting at the eleventh year forever. Is this a good deal if if the effective annual interest rate is 8%? A.We need to compare the PV of what you pay with the present value of what you get: - The present value of your annuity payments: PV 0 = 100 (1/0.08){1 - [ 1/(1.08 10 )]} = \$671.01 - The present value of the banks perpetuity payments at the end of the tenth year (beginning of the eleventh year): PV 10 = C 11 /r = (100/0.08) = \$1,250 The present value of the banks perpetuity payments today: PV 0 = PV 10 /(1+r) 10 = (100/0.08)/(1.08) 10 = = \$578.99 Present Value of an Annuity - Example 1

27 Q.You take \$20,000 five-year loan from the bank, carrying a 0.6% monthly interest rate. Assuming that you pay the loan in equal monthly payments, what is your monthly payment on this loan? A.Since payments are made monthly, we have to count our time units in months. We have: T = 60 monthly time periods in five years, with a monthly interest rate of: r = 0.6%, and PV 0 = 20,000 With the above data we have: 20,000 = C (1/0.006){1 - [ 1/(1.006 60 )]} Solving for C, we get a monthly payment of: \$397.91. Note: you can easily solve for C in your calculator, by keying: Present Value of an Annuity - Example 2 PV 1) Key in the PV +/_ 20,000 I/YR 2) Key in interest rate 0.6 N 3) Key in # of payments 60 PMT 4) Compute PMT Display should show: 397.91389639

28 Annuity Due –Self Study uAn annuity due is a stream of constant cash flows that is paid at the beginning of each period and lasts for a fixed number of periods (T). uSimplification:PV t = C t + C t+1 (1/r) { 1 - [1 / ( 1 + r ) T-1 ] } FV t+T = C t (1/r) { ( 1 + r ) T+1 - (1+r) } 0 1 2 3 years (T = 3) |----------|---------|---------| (r = 10%) \$100\$100 \$100 PV 0 = 100 + 100 (1/0.1) { 1 - [ 1/(1.1 2 )] } = \$273.55 FV 3 = 100 (1/0.1) { [1.1 4 ] - 1.1 } = \$364.10

29 First, clear previous data, and check that your calculator is set to 1 P/YR: The display should show: 1 P_Yr Input data (based on above example) PV and FV of Annuity Due in your HP 10B Calculator INPUT CLEAR ALL Yellow PMT I/YR 3N PV Key in payment Key in interest rate Key in number of PAYMENTS Compute PV Display should show: -273.55371901 100 FV Compute FV Display should show: -364.10000000 PV 0 0 Yellow Set payment to beginning of period When finished - dont forget to set your payment to End of period BEG/END 10

30 Growing Annuity uA Growing Annuity is a stream of cash flows that grows at a constant rate over a fixed number of periods. uSimplification for PV: PV t = C t+1 [ 1/(r-g) ] { 1 - [(1+g)/(1+r)] T } 0 1 2 3 |---------|----------|---------|(r = 10%) \$100\$102\$104.04(g = 2%) PV 0 = 100 [ 1/(0.10-0.02) ] { 1 - (1.02/1.10) 3 } = \$253

31 Q.Suppose that the bank rewords its advertisement to the following: Pay us \$100 next year, and another 9 annual payments such that each payment is 4% lower than the previous payment. We will pay you (or your beneficiaries) \$100, starting at the eleventh year forever. Is this a good deal if if the effective annual interest rate is 8%? A.Again, we need to compare the PV of what you pay with the present value of what you get: - The present value of your annuity payments (note: g = -4%): PV 0 = C 1 [1/(r-g)]{1- [(1+g)/(1+r)] T } = 100[1/(0.08-(-0.04))]{1-[(1+(-0.04))/(1.08)] 10 } = [100/0.12]{1-[0.96/1.08] 10 } = \$576.71 - The present value of the banks perpetuity payments today: \$578.99 (see example above) An Example - Present Value of a Growing Annuity

32 uA special case - when r < g, we still use the above formula uExample: 0 1 2 3 |------------------|-------------------|------------------|(r = 4%) \$100 \$100x1.07 \$100x1.07 2 (g = 7%) PV 0 = 100 [ 1/(0.04-0.07) ] { 1 - (1.07/1.04) 3 } = \$296.86 Growing Annuity - Special Cases – Self Study (-) (+)

33 Growing Annuity - Special Cases – Self Study uA special case - when r = g, we cannot use the above formula uExample-1: 0 1 2 3 |------------------|-------------------|------------------|(r = 5%) \$100 \$100x1.05 \$100x1.05 2 (g = 5%) uIn general, when cashflow starts at time t+1, use:

34 uExample-2: 0 1 2 3 |------------------|-------------------|------------------|(r = 5%) \$100 \$100 % 1.05 \$100 % 1.05 2 \$100 % 1.05 3 (g = 5%) uIn general, when cashflow starts at time t, use: Growing Annuity - Special Cases –Self Study

35 Growing Annuity uSimplification for FV: FV t+T = C t+1 [ 1/(r-g) ] { (1+r) T - (1+g) T } 0 1 2 3 |---------|----------|---------|(r = 10%) \$100\$102\$104.04(g = 2%) FV t+T = ____ [ 1/(___-___) ] {___ 3 - ____ 3 } = \$337.24

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