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.
N = 13, I/Y = 6, PMT = 200; CPT PV = $1,770.54
Calculate the PV of an annuity due:
Example: Find the PV of a 3-year annuity due that will make a series of $100 beginning of year payments, given a 10% discount rate.
Note: There are two ways to approach this question. The first is to put your calculator in BGN mode and then input all the variables as you normally would. The second is to shorten the annuity by one year (N - 1) and find the PV of that shortened annuity as if it were an ordinary annuity, then add the first annuity payment (PMT0) to it to come up with the PV of this annuity due. In this second alternative, you will leave your calculator in the END mode.
1. BGN mode: N = 3, I/Y = 10, PMT = 100; CPT PV = $273.55
2. END mode: N = 2, I/Y = 10, PMT = 100; CPT PV = $173.55 + 100 = PV = $273.55
d: Calculate an unknown variable, given the other relevant variables, in annuity problems.
Example: Find the PMT required to fund a retirement program of $3,000 at the end of 15 years, given a rate of return of 7%.
N = 15, I/Y = 7%, FV = 3,000; CPT PMT = $119.38 (ignore the sign).
Example: Suppose that you will deposit $100 at the end of each year for 5 years into an investment account. At the end of 5 years, the account will be worth $600. What is the rate of return
N = 5, FV = 600, PMT = 100; CPT I/Y = 7 years.
Example: Solve for the PMT given a 13-year annuity with a discount rate of 6%, and a PV of $2,000.
N = 13, I/Y = 6, PV = 2,000; CPT PMT = $225.92.
Example: Suppose t
hat you have $1,000 in the bank today. If the interest rate is 8%, how many annual, end-of-year payments of $150 can you withdraw
I/Y = 8, PMT = 150, PV = -1,000; CPT N = 9.9 years.
Example: What rate of return will you earn on an annuity that costs $700 today and promises to pay you $100 per year for each of the next 10 years
N = 10, PV = 700, PMT = 100; CPT I/Y = 7.07%.
p
e: Calculate the PV of a perpetuity.
Example: Assume a certain preferred stock pays $4.50 per year in annual dividends (and they're expected to continue indefinitely). Given an 8% discount rate, what's the PV of this stock
PVperpetuity = PMT / I/Y
PVperpetuity = 4.50 / .08 = $56.25
This means that if the investor wants to earn an 8% rate of return, she should be willing to pay $56.25 for each share of this preferred stock.
f: Calculate an unknown variable, given the other relevant variables, in perpetuity problems.
Example: Continuing with our example from LOS 1.A.e, what rate of return would the investor make if she paid $75.00 per share for the stock
I/Y = PMT / PVperpetuity
4.50 / 75.00 = 6.0%
g: Calculate the FV and PV of a series of uneven cash flows.
