FIN 300 – Chapter 6: Discounted Cash Flow Valuation
6.1 Future & Present Values of Multiple Cash Flows
In chapter 5, a basic simple cash flow was introduced, but in reality, investments usually consist
of multiple cash flows.
Future Value with Multiple Cash Flows
Drawing a time line is useful to give a visual representation of the cash flows and how they will
Two ways to calculate FV for multiple cash flows:
1. Compound the accumulated balance forward one year at a time (or)
2. Calculate the FV of each cash flow and hen add them up.
However, this gets difficult for more complicated questions.
Present Value with Multiple Cash Flows
Two ways to calculate PV for multiple cash flows:
1. Discount back one period at a time (or)
2. Calculate the PV individually and add them up
The PV of a series of future cash flows is simply the amount that you would need today in order
to exactly duplicate those future cash flows (for a discount rate).
Important: Always check the amount of periods because those are significant to the answer.
A Note on Cash Flow Timing
Cash flow timing is critically important in PV and FV problems.
o Implicitly assumed that the cash flows occur at the end of each period.
6.2 Valuing Level Cash Flows: Annuities and Perpetuities
Multiple cash flows usually have payments of the same amount.
Generally, a series of constant or level cash flows that occur at the end of each period for some
fixed number of periods is called an ordinary annuity.
Annuity – A level stream of cash flows for a fixed period of time.
Present Value for Annuity Cash Flows
You can use the basic PV system to calculate PV or FV, but when the number of cash flows is
quite large, that approach becomes tedious and often results in mathematical errors.
Since annuities have the same cash flow for a period of time, an equation can be created.
o PV of an annuity of C dollars per period for t periods when the rate of return or interest
rate is r is given by the following equation:
The 1/(1+r) is called the present value interest factor for annuities and
Future Values for Annuities
The equation for FV is similar to the PV because they require the same information to finish the
A Note on Annuities Due
Annuity Due – An annuity for which the cash flows occur at the beginning of the period An example would be a lease, where the payments are due at the beginning of the month.
An annuity due is calculated by multiplying the ordinary annuity value by the interest factor:
This formula works for both PV and FV. Calculating the value of annuity due involves two steps:
1. Calculate the PV or FV as though it was an ordinary annuity, and then
2. Multiply your answer by (1 + r)
Perpetuity – An annuity in which the cash flows continue forever.
o Annuity with a level stream of cash flows that continue forever.
Consol – A type of perpetuity.
Perpetuity PV × Rate = Cash Flow
o PV × r = C
If you are given a cash flow, C, and a rate of return, r, we can compute the PV:
o Perpetuity PV = C/r = C × (1/r)
Annuity for PV is also very simple to determine:
o Annuity PV Factor = (1 – PV Factor) / r
o Annuity PV Factor = (1/r) × (1 – PV Factor)
Growing Perpetuity – A constant stream of cash flows without end that is expected to rise
C is the cash flow to be received one period hence, g is he rate of growth per
period, and r is the interest rate.
Formula for Present Value of Growing Perpetuity:
There are three important points concerning the growing perpetuity formula:
o The Numerator – It is the cash flow one period hence, not at date 0.
o The Interest Rate and the Growth Rate – The interest rate, r, must be greater than the
growth rate, g, for the formula to work. The PV is undefined if g is greater than r.
o The Timing Assumption – Assumed that cash flows are received and disbursed at