Chapter 4 – Introduction to Valuation: The Time Value of Money
A dollar today is worth more than a dollar tomorrow!!! How much more, that is the question.
Future Value
Suppose you have $100 and invest it for 1 year at 10%. How much will you have at the end of the year?
(100)(0.10) = 10 100 + 10 =110
How much will you have in 2 years?
(110)(0.10) = 11 110 + 11 =121
Why isn’t it $120? Where did the extra $1 come from?
Compounded interest
General Equations
FV = PV(1 + r)t FV = PV(FVIFr,t) You have $20,000 to invest. If you can earn 9% per year, how much will you have in ten years?
FV = 20,000(1 + 0.09)10 = $47,347.27 FV = 20,000(FVIF9%,10) = 20,000(2.3674) = $47,348.00
The GDP will grow at 2.5% for the next three years. The current GDP is $13,841.3 (billions). What will it be in three years?
13,841.3(1.025)3 = 14,905.57
Fin 311 Chapter 4 Lecture Notes
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You plan to purchase Apple stock for $185.00. You expect the share price to grow at 6% for the next 10 years. What will the price be in 10 years?
FV = 185(1.06)10 = $331.31 FV = 185(FVIF6%,10) = 185(1.7908) = $331.30
You plan to invest $1,500 in a financial asset with a rate of return of 12 percent. What will the value of the asset be in 35 years?
FV = 1,500(1.12)35 = $79,199.43
Present Value
Remember:
1 FV = PV(1 + r)t PV FV 1 r t
→ PV = FV(PVIFr,t)
NOTE: The PFIV is just the inverse of the FVIF Suppose you need $110 in 1 year and you can invest at 10%. How much must you invest now?
PV = 110
1 $100.00 1 1 0.10
PV = 110(PVIF10%,1) = 110(0.9091) = $100.00
Suppose you need $121 in 2 years and you can invest at 10%. How much must you invest now?
PV = 121
1 $100.00 2 1 0.10
PV = 121(PVIF10%,2) = 121(0.8264) = $99.99
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Fin 311 Chapter 4 Lecture Notes
Suppose you need $20,000 in 10 years. If you can earn 9% per year, how much do you have to invest today?
PV = 20,000
1 $8,448.22 10 1.09
PV = 20,000(PVIF9%,10) = 20,000(0.4224) = $8448.00
You will inherit $1.5 million in 40 years. Your bank is the trustee and will allow you to borrow against your inheritance. What is the maximum you can borrow? Assume r = 18%.
PV = 1,500,000
1 $1,998.99 1.1840
PV = 1,500,000(PVIF18%,40) = 1,500,000(0.0013) = $1,950.00
Interest Rate
Remember FV = PV(1 + r)t, we can solve for r.
FV FV t FV t t 1 r 1 r r 1 PV PV PV
You purchased one share of Apple stock in 1982 for $2.50. In 2007, (25 years later) it is valued at $185 per share. What is the average growth rate?
1
1
185 25 r 1 0.1879 18 .79 % 2.50
1
FV FVIF ,t r PV 185 74.0000 FVIF , 25 r 2.50
Look in the FVIF Table in the 25 period row until we find 74.0000. It is between 18 and 20 percent.
Fin 311 Chapter 4 Lecture Notes
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The GDP doubled between 1987 and 2007 (20 years). What is the average growth rate of the GDP?
r 2 20 1 0.0353 3.53%
1
2.0000 FVIFr , 20
Look in the FVIF Table in the 20 period row until we find 2.0000. It is between 3 and 4 percent.
Number of Periods
Remember FV = PV(1 + r)t, we can solve for t.
FV ln FV FV t PV 1 r ln t ln 1 r t PV ln 1 r PV
Suppose you want to buy lawn mower. You currently have $500 and the mower you want costs $1,500. If you can earn 8%, how long will you have to wait to buy the mower?
1,500 ln 500 ln 3 1.0986 14.27 t ln 1.08 ln 1.08 0.0770
FV FVIF r ,t PV 3 FVIF8%,t
Look in the FVIF Table in the 8 percent column until we find 3.0000. It is between 14 and 15 periods.
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Fin 311 Chapter 4 Lecture Notes
You want to buy a car. You know your grandparents will give you $10,000 when you graduation. You have researched the car market and decided you will buy a Honda Accord. It will cost $17,500 and you want to pay cash for it. You can invest the $10,000 gift at 12 percent. How long before you can buy the car?
17,500 ln 10,000 ln 1.75 0.5596 t 4.94 ln 1.12 ln 1.12 0.1133
FV FVIF r ,t PV 1.75 FVIF12%,t
Look in the FVIF Table in the 12 percent column until we find 1.75. It is between 4 and 5 periods.
Fin 311 Chapter 4 Lecture Notes
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Using the calculator Future Value
You have $20,000 to invest. If you can earn 9% per year, how much will you have in ten years?
FV = $20,000(FVIF9%,10)
N I PV Pmt Cpt FV 10 9% $20,000 0 $47,347.27
The GDP will grow at 2.5% for the next three years. The current GDP is $13,841.3 (billions). What will it be in three years?
FV = $13,841.3(FVIF2.5%,3)
N I PV Pmt Cpt FV 3 2.5 $13,841.3 0 $14,905.57
You plan to purchase Apple stock for $185.00. You expect the share price to grow at 6% for the next 10 years. What will the price be in 10 years?
FV = 185(FVIF6%,10)
N I PV Pmt Cpt FV 10 6% $185 0 $331.31
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Fin 311 Chapter 4 Lecture Notes
You plan to invest $1,500 in a financial asset with a rate of return of 12 percent. What will the value of the asset be in 35 years?
FV = $1,500(FVIF12%,35)
N I PV Pmt Cpt FV 35 12% $1,500 0 $79,199.43
Suppose you invest $9,000 today and get an interest rate of 9 percent compounded month. How much will you have in 3 years?
FV = $9,000(FVIF9%/12,(3)(12))
N PV Pmt Cpt FV 3(12) = 36 $9,000 0 $11,777.81
I 9/12 = 0.75%
Present Value
Suppose you need $110 in 1 year and you can invest at 10%. How much must you invest now?
PV = 110(PVIF10%,1)
N I Cpt PV Pmt FV 1 10% $100 0 $110
Fin 311 Chapter 4 Lecture Notes
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Suppose you need $121 in 2 years and you can invest at 10%. How much must you invest now?
PV = 121(PVIF10%,1)
N I Cpt PV Pmt FV 2 10% $100 0 $121
Suppose you need $20,000 in 10 years. If you can earn 9% per year, how much do you have to invest today?
PV = 20,000(PVIF9%,10)
N I Cpt PV Pmt FV 10 9% $8,448.22 0 $20,000
You will inherit $1.5 million in 40 years. Your bank is the trustee and will allow you to borrow against your inheritance. What is the maximum you can borrow? Assume r = 18%.
PV = 1,500,000(PVIF18%,40)
N I Cpt PV Pmt FV 40 18% $1,998.99 0 $1,500,000
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Fin 311 Chapter 4 Lecture Notes
Interest Rate
You purchased one share of Apple stock in 1982 for $2.50. In 2007, (25 years later) it is valued at $185 per share. What is the average growth rate?
2.50 = 185(PVIFr,25)
N Cpt I PV Pmt FV 25 18.787% -$2.50 0 $185
or
185 = 2.50(FVIFr,25)
The GDP doubled between 1987 and 2007 (20 years). What is the average growth rate of the GDP?
1 = 2(PVIFr,20)
N Cpt I PV Pmt FV
or
20
2 = 1(FVIFr,20)
3.5265% -1 0 2
Eight years ago, you invested $10,000 in an account that is now worth $22,000. What interest rate per year did you earn?
10,000 = 22,000(PVIFr,8)
N Cpt I PV Pmt FV 8 10.36% -10,000 0 22,000
or
22,000 = 10,000(FVIFr,8)
Fin 311 Chapter 4 Lecture Notes
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An avid art collector bought a famous painting for $8.5 million. Unfortunately, when he sold the painting 7 years later, he only received $6.4 million. What was his rate of return on this investment?
8.5 = 6.4(PVIFr,7) or
N Cpt I PV Pmt FV 7 -3.97% -8.5 0 6.4
6.4 = 8.5(FVIFr,7)
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Fin 311 Chapter 4 Lecture Notes
Number of Periods
Suppose you want to buy lawn mower. You currently have $500 and the mower you want costs $1,500. If you can earn 8%, how long will you have to wait to buy the mower?
500 = 1,500(PVIF8%,t)
Cpt N I PV Pmt FV 14.27 8% -$500 0 $1,500
or
1,500 = 500(FVIF8%,t)
You want to buy a car. You know your grandparents will give you $10,000 when you graduation. You have researched the car market and decided you will buy a Honda Accord. It will cost $17,500 and you want to pay cash for it. You can invest the $10,000 gift at 12 percent. How long before you can buy the car?
10,000 = 17,500(PVIF12%,t)
Cpt N I PV Pmt FV 4.938 12% -$10,000 0 $17,500
or
17,500 = 10,000(FVIF12%,t)
You just won the lottery and received $25,000. You want to invest the money until you have enough to buy a $110,000 car. If you can earn 11 percent per year, how long do you have to wait until you buy the car?
25,000 = 110,000(PVIF11%,t)
Cpt N I PV Pmt FV 14.197 11% -$25,000 0 $110,000
or
110,000 = 25,000(FVIF11%,t)
Fin 311 Chapter 4 Lecture Notes
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