Discounted Cash Flow Analysis

BUSI 721: Data-Driven Finance I

Kerry Back, Rice University

Cash Flows

• Cash inflow = positive cash flow
• Cash outflow = negative cash flow
• Cash inflows to a project can be redistributed elsewhere in a company or paid to shareholders or bondholders.
• Cash outflows must be financed by running down cash balances, using cash generated elsewhere in the company, or issuing new equity or debt.

Timing

• Years are periods between dates.
  • Date 0 is project inception (or first couple of months)
  • Date 1 is end of first year
  • etc.
• Put first year's cash flows at date 1, second year's at date 2, etc.
• Most assets and liabilities should be down to zero at project's end.
  • Collect receivables, pay payables, draw down inventory,
  • Might sell, or dispose of PP&E.
  • If it stays in the company and has value, take a credit for it in cash flows (terminal value).

Net Present Value

• Suppose a project's cash flows are $-100, -20, 50, 50, 50, 80$ at years 0, 1, 2, 3, 4, 5.
• Consider making investments yourself of $100$ and then $20$. Would you be able to withdraw $50, 50, 50$, and $80$?
• We can answer this on a PV basis - compare PVs of deposits and withdrawals.

import numpy as np

r = 0.08
pv_investments = np.sum(
    np.array([100, 20]) * (1+r)**np.arange(0, -2, -1)
)
pv_withdrawals = np.sum(
    np.array([50, 50, 50, 80]) * (1+r)**np.arange(-2, -6, -1)
)
print(f"PV of investments is ${pv_investments:.2f}")
print(f"PV of withdrawals is ${pv_withdrawals:.2f}")

PV of investments is $118.52
PV of withdrawals is $173.76

Above Market Projects

• The answer is no: we couldn't withdraw (50, 50, 50, 80) with investments of (100, 20).
• If the company can do this, then it has an "above market" project.
• It will benefit investors because it generates positive cash flows with lower investments than investors can do on their own.
• This assumes we're using the right discount rate (interest rate). More on this later.

Positive NPV Projects

• To quickly calculate whether it is an above market project, we can put all of the cash flows together in a sequence (negative for outflows) and compute the PV of the sequence.
• Positive PV means withdrawals are high relative to investments $\rightarrow$ good project.
• This is called the Net Present Value (NPV). It is the PV of the cash inflows net of the PV of the cash outflows.

cash_flows = np.array(
    [-100, -20, 50, 50, 50, 80]
)

NPV = np.sum(
    cash_flows * (1+r)**np.arange(0, -6, -1)
)
print(f"The NPV is ${NPV:.2f}")

The NPV is $55.24

Alternate Calculation

• We can use the npv function from numpy-financial.
• It assumes the first cash flow is at date 0.
• Warning: The Excel NPV function works differently. It assumes the first cash flow is one period away.
• In Excel, use -100 + NPV(0.08, (-20, 50, 50, 50, 80))
• See npv.xlsx

import numpy_financial as npf

NPV = npf.npv(0.08, cash_flows)
print(f"The NPV is ${NPV:.2f}")

The NPV is $55.24

Internal Rate of Return

• Call a project standard if cash flows are negative early and positive late.
• Consider any standard project.
  • NPV > 0 means later positive cash flows have greater PV than negative early cash flows.
  • If we raise the discount rate, the NPV will fall.
  • The discount rate at which NPV=0 is called the Internal Rate of Return (IRR)
  • NPV > 0 if and only if IRR > discount rate.
• Calculate with
  • npf.irr((-100, -20, 50, 50, 50, 80))
  • or, in Excel, IRR(-100, -20, 50, 50, 50, 80)