Discounted Cash Flow Model
Principles, Analysis, and Implementation
Corrado Botta, PhD
Bocconi University
May 16, 2025
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Outline
What is a Discounted Cash Flow Model?
Components of DCF Model
Mathematical Framework of DCF
Terminal Value Calculation
Analytical Derivation of DCF
R Implementation
Applications and Takeaways
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What is a Discounted Cash Flow Model?
Discounted Cash Flow (DCF) is a valuation method used to esti-
mate the value of an investment based on its expected future cash
flows:
▶ Foundational valuation technique in finance
▶ Based on the time value of money principle
▶ Estimates intrinsic value rather than market value
DCF serves as a comprehensive valuation tool that:
▶ Quantifies the present value of future cash flows
▶ Enables comparison between different investment opportunities
▶ Provides a simple formula: Value = nt=1 (1+r
CFt TV
P
)t + (1+r )n
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Components of DCF Model
A complete DCF model consists of three essential components:
1. Cash Flow Projections: Forecasting future free cash flows
▶ Based on revenue growth, margins, capital expenditures, and
working capital
▶ Typically projected for 5-10 years explicitly
2. Terminal Value: Capturing value beyond the forecast period
▶ Perpetuity growth method: TV = FCF n+1
r −g
▶ Exit multiple method: TV = FCFn × Multiple
3. Discount Rate: Reflecting the time value of money and risk
▶ Often uses Weighted Average Cost of Capital (WACC): WACC =
E D
V × re + V × rd × (1 − t)
▶ Must reflect the specific risk profile of cash flows
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Mathematical Framework of DCF
The general formula for a DCF model is:
n
X CFt TV
Value = t
+ (1)
(1 + r ) (1 + r )n
t=1
Where:
▶ CFt is the expected cash flow in period t
▶ r is the discount rate (typically WACC)
▶ n is the forecast period
▶ TV is the terminal value
Properties
▶ Higher discount rates lead to lower present values
▶ Cash flows further in the future have less impact on present value
▶ Terminal value often represents 60-80% of total value
▶ Sensitivity to discount rate and growth assumptions increases with time
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Terminal Value Calculation
Terminal value captures all cash flows beyond the explicit forecast
period:
FCFn+1 FCFn × (1 + g )
TV = = (2)
r −g r −g
Key considerations for terminal value:
▶ Growth rate (g ) must be sustainable in perpetuity
▶ Typically limited to long-term GDP growth (2-3%)
▶ Assumes the company reaches steady state
▶ For stable businesses, g should not exceed r
Limitations:
▶ High sensitivity to small changes in inputs
▶ Assumes going concern with infinite life
▶ Challenging to estimate for cyclical businesses
▶ May overvalue companies with unsustainable returns
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Analytical Derivation of DCF - Part 1
Starting Point: Present Value Formula
FV
PV = (3)
(1 + r )t
Step 1: Extending to Multiple Cash Flows
CF1 CF2 CFn
PV = 1
+ 2
+ ... +
(1 + r ) (1 + r ) (1 + r )n
n
X CFt
= (4)
t=1
(1 + r )t
Step 2: Incorporating Terminal Value
n
X CFt
PVtotal = + PVterminal
t=1
(1 + r )t
n
X CFt TV
= + (5)
t=1
(1 + r )t (1 + r )n
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Analytical Derivation of DCF - Part 2
Step 3: Deriving the Terminal Value
For the perpetuity growth model, we start with an infinite sum:
CFn+1 CFn+2 CFn+3
TV = 1
+ 2
+ + ... (6)
(1 + r ) (1 + r ) (1 + r )3
If cash flows grow at a constant rate g , then:
CFn+t = CFn+1 × (1 + g )t−1 (7)
Step 4: Deriving the Gordon Growth Model
CFn+1 CFn+1 (1 + g ) CFn+1 (1 + g )2
TV = + + + ...
(1 + r )1 (1 + r )2 (1 + r )3
∞
X (1 + g )t−1
= CFn+1 ×
t=1
(1 + r )t
1
= CFn+1 × (for r > g ) (8)
r −g
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Analytical Derivation of DCF - Part 3
Step 5: Putting It All Together
n
X CFt TV
Value = +
t=1
(1 + r )t (1 + r )n
n
X CFt 1 CFn+1
= t
+ n
×
t=1
(1 + r ) (1 + r ) r −g
n
X CFt CFn × (1 + g )
= + (9)
t=1
(1 + r )t (1 + r )n × (r − g )
Step 6: Sensitivity Analysis
The sensitivity of value to the discount rate:
n
X −t × CFt
∂Value n × TV CFn+1
= t+1
− n+1
− n × (r − g )2
(10)
∂r t=1
(1 + r ) (1 + r ) (1 + r )
This shows the high sensitivity of DCF to discount rate changes, particu-
larly for the terminal value component.
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R Implementation - Example
# Example company parameters
initial_revenue <- 1000 # £1,000M initial revenue
projection_years <- 5 # 5-year projection
growth_rates <- c(0.10, 0.08, 0.06, 0.05, 0.04) # Declining growth
ebitda_margins <- c(0.15, 0.16, 0.17, 0.17, 0.18) # Improving margins
capex_percent <- c(0.08, 0.07, 0.07, 0.06, 0.06) # CapEx as % of revenue
nwc_percent <- c(0.10, 0.10, 0.10, 0.10, 0.10) # NWC as % of revenue
tax_rate <- 0.25 # 25% tax rate
# DCF parameters
discount_rate <- 0.10 # 10% discount rate (WACC)
terminal_growth <- 0.02 # 2% terminal growth rate
# Generate financial projections
financials <- simulate_financials(
initial_revenue, projection_years,
growth_rates, ebitda_margins,
capex_percent, nwc_percent, tax_rate
)
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R Implementation - Example
# Calculate DCF valuation
dcf_result <- dcf_valuation(financials$FCF, terminal_growth, discount_rate)
print(paste("PV of Projected FCF:", round(dcf_result$pv_cf, 2)))
[1] ”PV of Projected FCF: 140.78”
print(paste("PV of Terminal Value:", round(dcf_result$pv_terminal, 2)))
[1] ”PV of Terminal Value: 774.62”
print(paste("Enterprise Value:", round(dcf_result$enterprise_value, 2)))
[1] ”Enterprise Value: 915.4”
print(paste("Terminal Value Percentage:", round(dcf_result$pv_terminal_percent
[1] ”Terminal Value Percentage: 84.62”
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Sensitivity Analysis Visualization
DCF Sensitivity Analysis
0.030
Enterprise
Terminal Growth Rate
0.025
Value ($M)
1400
0.020 1200
1000
800
0.015
0.010
0.08 0.09 0.10 0.11 0.12
Discount Rate (WACC)
▶ The heatmap shows enterprise value sensitivity to changes in discount rate and
growth rate
▶ The white dot represents the base case scenario (discount rate = 10%, growth
rate = 2%)
▶ Values increase (lighter colors)) with lower discount rates and higher growth rates
▶ The contour lines represent equal enterprise value levels
▶ Note the non-linear relationship: values increase exponentially as the growth rate
approaches the discount rate
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Scenario Analysis Visualization
Enterprise Value by Scenario
1600
$1487M
1200
Value ($M)
$915M
800
$511M
400
0
Bear Base Bull
Component PV of Terminal Value PV of Cash Flows
▶ The chart shows enterprise value breakdown for three scenarios: Bear, Base, and
Bull
▶ In all scenarios, terminal value (light blue) represents the majority of enterprise
value
▶ The Bear case shows 52% decrease in value from Base case due to the combined
impact of higher discount rate, lower growth rate, and lower cash flows
▶ The Bull case shows 73% increase in value from Base case due to the combined
impact of lower discount rate, higher growth rate, and higher cash flows
▶ Note the exponential relationship: small changes in inputs create large valuation
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differences
Monte Carlo Simulation Analysis
Monte Carlo Simulation of Enterprise Value
1,000 simulations with varying inputs
0.0020
0.0015
5th
percentile
$590
Median
$867
95th
percentile
$1284
Frequency Density
0.0010
0.0005
0.0000
400 800 1200 1600
Enterprise Value ($M)
▶ Monte Carlo simulation with 1,000 runs, varying growth rates, discount rates,
and cash flow levels
▶ The distribution is right-skewed, reflecting the non-linear relationship between
inputs and valuation
▶ The median value of $867M is less than the mean value of $894M due to this
skewness
▶ 95% confidence interval: $590M to $1284M
▶ Wide distribution emphasizes the uncertainty inherent in DCF valuation and the
importance of range estimates
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Applications and Takeaways
Key Applications
▶ Investment Valuation: Determining intrinsic value of stocks and
businesses
▶ M&A Analysis: Supporting pricing decisions in acquisition scenarios
▶ Capital Budgeting: Evaluating potential investment projects
▶ Equity Research: Informing buy/sell recommendations
Takeaways
▶ DCF provides a theoretically sound approach to valuation based on
future cash generation
▶ Quality of inputs significantly affects reliability of results
▶ Sensitivity analysis is crucial due to high impact of small changes in
key inputs
▶ Monte Carlo simulation offers a more robust way to incorporate un-
certainty
▶ DCF should be used alongside other valuation methods, not in isola-
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