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二、Investment Tools: Quantitative Methods
1.A.: Time Value of Money
a: Calculate the future value (FV) and present value (PV) of a single sum of money.
Future Value:
FV = PV(1 + I/Y)N
Where PV = the amount of money invested today, I/Y = the rate of return, and N = the length of the holding period.
Example: Using a financial calculator, here's an example of how you would find the FV of a $300 investment (PV), given you earn a compound rate of return (I/Y) of 8% over a 10-year (N) period of time:
N = 10, I/Y = 8, PV = 300; CPT FV = $647.68 (ignore the sign).
Present Value:
PV = FV / (1 + I/Y)N
Example: Using a financial calculator, here's an example of how you'd find the PV of a $1,000 cash flow (FV) to be received in 5 (N) years, given a discount rate of 9% (I/Y).
N = 5, I/Y = 9, FV = 1,000; CPT PV = $649.93 (ignore the sign).
b: Calculate an unknown variable, given the other relevant variables, in single-sum problems.
Example 1: Solving for I/Y
In this example, you want to find the rate of return (I/Y) that you'll have to earn on a $500 investment (PV) in order for it to grow to $2,000 (FV) in 15 years (N). This very same problem could also be set up in terms of growth rates - e.g., what rate of growth (I/Y) is necessary for a company's sales to grow from $500 per year (PV) to $2,000 per year (FV) in 15 years (N).
N = 15, PV = -500, FV = 2,000; CPT I/Y = 9.68%
Example 2: Solving for N
In this example, you want to find out how many years (N) it will take for a $500 investment (PV) to grow to $1,000 (FV), given that we can earn 7% annually (I/Y) on your money.
I/Y = 7, PV = -500, FV = 1,000; CPT N = 10.24 years.
c: Calculate the FV and PV of an regular annuity and an annuity due.
Calculate the FV of an ordinary annuity:
Example: Find the FV of an ordinary annuity that will pay $150 per year at the end of each of the next 15 years, given the investment is expected to earn a 7% rate of return.
N = 15, I/Y = 7%, PMT = $150; CPT FV = $3,769.35 (ignore the sign).
Calculate the FV of an annuity due:
Example: Find the FV of an annuity due that will pay $100 per year for each of the next three years, given the cash flows can be invested at an annual rate of 10%.
Note: When solving for a FV of an annuity due, you MUST put your calculator in the beginning of year mode (BGN), otherwise you'll end up with the wrong answer.
N = 3, I/Y = 10%, PMT = $100; CPT FV = $364.10 (ignore the sign).
Calculate the PV of an ordinary annuity:
Example: Find the PV of an annuity that will pay $200 per year at the end of each of the next 13 years, given a 6% rate of return.
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