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Time value of money -CFA level1 Exam Notes
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Compact Notes for CFA level 1 students By Professor James Forjan, PhD, CFA
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Time Value of Money BasicsThe time value of money (TVM) is a fundamental concept in finance that
recognizes the value of money today is worth more than the same
amount of money in the future. This is due to inflation, opportunity cost,
and the potential earning capacity of money.
Imagine you have $100 today. If you put it in a savings account with a 2%
annual interest rate, you would have $102 after one year. Therefore, $100
today is worth more than $102 a year from now.
This concept is essential when evaluating future cash flows, such as
calculating the present value (PV) or future value (FV) of money.
Present Value
The present value is the current worth of a future sum of money or stream
of cash flows, given a specific rate of return.
For example, let's say you expect to receive $1,000 in one year and the
interest rate is 5%. To calculate the present value, you would divide the
future value by (1 + interest rate).
PV = FV / (1 + interest rate) PV = $1,000 / (1 + 0.05) PV = $1,.05 PV
= $952.38
Therefore, the present value of $1,000 to be received in one year at a 5%
interest rate is $952.38.
Future Value
The future value is the value of a current sum of money or stream of cash
flows at a specified date in the future, given a specific rate of return.
For example, let's say you have $1,000 today and want to know how much
it will be worth in five years at a 3% interest rate. To calculate the future
value, you would multiply the present value by (1 + interest rate)^number
of periods.
FV = PV x (1 + interest rate)^number of periods FV = $1,000 x (1 +
0.03)^5 FV = $1,000 x 1.159274 FV = $1,159.27
Therefore, the future value of $1,000 today at a 3% interest rate for five
years is $1,159.27.
, Annuities
An annuity is a series of equal payments or receipts made at regular
intervals over a specified period. There are two types of annuities:
ordinary annuities and annuities due.
For example, let's say you want to save $1,000 per year for the next 10
years and the interest rate is 4%. To calculate the future value of the
annuity, you would use the formula:
FV = PMT x [(1 + interest rate)^number of periods - 1] / interest rate FV =
$1,000 x [(1 + 0.04)^10 - 1] / 0.04 FV = $1,000 x 14.80295 FV = $14,802.95
Therefore, the future value of a $1,000 annual payment for 10 years at a
4% interest rate is $14,802.95.
Notes on Calculating Future and Present Values
Calculating the future value (FV) of an investment is an important concept
in finance. It represents the total amount of money an investment will be
worth in the future, taking into account the compounding of interest.
To calculate the FV of a single sum of money, we can use the formula:
FV = PV x (1 + r)^n
where:
FV is the future value
PV is the present value (the initial amount of money invested)
r is the interest rate (as a decimal)
n is the number of periods (the amount of time the money is
invested for)
For example, let's say you invest $1,000 today at an interest rate of 5% per
year for 10 years. The FV of this investment would be:
FV = $1,000 x (1 + 0.05)^10 = $1,628.89
Calculating the present value (PV) of an investment is simply the reverse
of calculating the FV. It represents the current value of a future sum of
money, taking into account the time value of money and the interest rate.
To calculate the PV of a single sum of money, we can use the formula:
PV = FV / (1 + r)^n
, For example, let's say you expect to receive a payment of $2,000 in 5 years
at an interest rate of 6% per year. The PV of this payment would be:
PV = $2,000 / (1 + 0.06)^5 = $1,437.13
These formulas can also be used to calculate the FV and PV of multiple
payments, such as in an annuity. An annuity is a series of equal payments
made at regular intervals.
To calculate the FV of an annuity, we can use the formula:
FV = PMT x [(1 + r)^n - 1] / r
where:
FV is the future value
PMT is the periodic payment
r is the interest rate (as a decimal)
n is the total number of periods
For example, let's say you plan to make monthly payments of $200 into an
account that earns 3% interest per year for 10 years. The FV of this
annuity would be:
FV = $200 x [(1 + 0.03/12)^(12 x 10) - 1] / (0.03/12) = $31,952.13
To calculate the PV of an annuity, we can use the formula:
PV = PMT x [1 - (1 + r)^-n] / r
For example, let's say you expect to receive monthly payments of $500 for
the next 20 years at an interest rate of 4% per year. The PV of this annuity
would be:
PV = $500 x [1 - (1 + 0.04/12)^(-12 x 20)] / (0.04/12) = $87,555.27
Interest rates and required returns are critical concepts in finance, as they
represent the cost of borrowing and the compensation required for
investment risk, respectively.
Interest rates can be described in terms of nominal and real rates.
Nominal interest rates represent the stated rate of interest, while real
interest rates adjust for inflation and reflect the purchasing power of
money over time.
To calculate the real interest rate, we can use the formula:
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