How to Calculate Present Values
15 Feb 2025
The future value of $100 invested at 7% per year for 1 year is:
$100 × (1 + 0.07) = $107
The present value of $107 received in 1 year, assuming a 7% interest rate, is:
$107 ÷ (1 + 0.07) = $100
In general,
future value = present value × ( 1 + r )n
present value = future value × ( 1 / ( 1 + r )n )
where r is the interest rate and discount rate, respectively, and n is the number of periods. The expression 1/(1+r)n is also called the discount factor.
The discounted cash flow is a method used to value a stream of cash flows extending over time:
PV = CF1 / (1 + r) + CF2 / (1 + r)2 + ... + CFn / (1 + r)n
where PV is the present value of the cash flows, CFi is the cash flow in period i, r is the discount rate, and n is the number of periods.
To determine whether an investment is worthwhile, we take the sum of the present values of all future cash flows, including the initial cash outlay. This sum is called the net present value (NPV), and any investment with a positive NPV is considered worthwhile.
A perpetuity is a stream of cash flows that continues indefinitely. The present value of a perpetuity is given by:
PV = CF / r
where CF is the cash flow and r is the discount rate.
A delayed perpetuity is a perpetuity that starts at a future date. To calculate the present value of a delayed perpetuity, calculate the "present" value of the perpetuity as if it started today, and then discount it further to the present:
PV = ( CF / r ) × ( 1 / ( 1 + r )n )
where n is the number of periods until the perpetuity starts. Notice the discount factor in the second term.
An annuity is a stream of cash flows that lasts for a fixed number of periods. Since an annuity is a perpetuity that stops after n periods, the present value of an annuity is given by the difference between the present value of a perpetuity and the present value of a delayed perpetuity:
PV = CF × ( ( 1 / r ) - ( 1 / ( r × ( 1 + r )n ) ) )
where CF is the cash flow and r is the discount rate.
A growing perpetuity is a perpetuity where the cash flow grows at a constant rate. The present value of a growing perpetuity is given by:
PV = CF / ( r - g )
where CF is the cash flow, r is the discount rate, and g is the growth rate.
A growing annuity is an annuity where the cash flow grows at a constant rate. The present value of a growing annuity is given by:
PV = CF × ( 1 / ( r - g )) × (1 - (1 + g)n / (1 + r)n )
where CF is the cash flow, r is the discount rate, g is the growth rate, and n is the number of periods.
Interest rates are typically quoted as annual percentage rates (APR). For example, a 5% annual interest rate means that an interest of 5% is paid on the principal amount per year. However, compounding does not necessarily occur annually. A 5% annual interest rate compounded semi-annually means that 2.5% interest is paid every 6 months. A 5% annual interest rate compounded quarterly means that 1.25% interest is paid every 3 months. Since all interests earned are reinvested, the effective annual rate (EAR) is not necessarily the same as the APR.
The EAR is given by
EAR = ( 1 + ( r / m ))m - 1
where r is the APR, and m is the number of compounding periods per year. Thus, a 5% annual interest rate compounded quarterly has an EPR of
EAR = ( 1 + ( 0.05 / 4 ))4 - 1 = 0.0509 = 5.09%