1.  Introduction

In the modern financial investment scenarios, Investors apply different approaches of valuation to estimate a company's worth [1]. Among these, The Discounted Cash Flow (DCF) model, a fundamental analysis tool extensively used in business and finance. The DCF model helps investors work out a company's intrinsic value and appraise their investments based on projected future cash - flows [2]. When applying DCF analysis, Investors can take more informed and exact investment decisions [3].

The DCF model is extremely important when it comes to estimating continuing value, a critical component in firm v. Often, the Gordon growth formula is employed to derive this value by projecting free cash flow outside the explicit forecast period [4]. Jennergren (2008) did a detailed examination of the elements that constitute continuing value, with especial emphasis on capital expenditures and tax savings arising from property depreciation, processing plant, and equipment (PPE) [5]. Further enhancing this notion, value driver formulas give a well - structured and consistent framework for estimating continuing value inside the DCF model, augmenting its practicality and reliability in corporate valuation [6].

Actual empirical examinations, for instance, the poll of UK investment analysts by Imam et al. Show that in practice, DCF is regarded as significantly more important than what previous survey evidence had indicated [7]. Moreover, The DCF model is often compared alongside other valuation m. For instance, Lundholm and O’Keefe (2001) compared the DCF and residual income models and concluded that although it cannot be claimed that one model is superior in all aspects. Equity values derived via DCF often show statistically significant divergences from book values, insinuating the need for more accuracy trials [8]. The consonance of assumptions, for instance, the correlation between growth and capital expenditure, is a significant factor having an effect on the reliability of DCF outcomes [9].

This paper exhibits a practical application of the DCF model to figure out the intrinsic value of Apple Inc., one of the biggest companies globally, selected for the reason that its cash flows are relatively stable and predictable [10]. We calculate Apple’s Free Cash Flow to Firm, which are subsequently utilized to estimate the company's present and terminal values [11]. Based on these outcomes, we gauge the intrinsic value of Apple and its commons. To raise the reliability of our valuation, we add sensitivity testing and Monte Carlo simulation to the DCF analysis. Resolving valuation uncertainty and improving the soundness of the results when cash flows are highly predictable [12].

2.  The principle and application of DCF model

The DCF model first needs the future free cash flow of the company over a defined period. Then, people can get terminal value of the company by assuming a stable growth rate of the company beyond the forecast period. After deriving the discount rate which people can get based on the calculation of weighted average cost of capital (WACC), we can discount the future free cash flow to present value and sum them to calculate the company’s intrinsic value.

The rationale of our model selection can be shown in three aspects. First, Apple’s financial ratios show low historical volatility, validating mean reversion. Second, we use the mean value of Apple’s financial parameters in our calculation, which can mitigate outlier impact. Third, the 10-year dataset is insufficient for complex models like ARIMA.

2.1.  Data collecting and pre-processing

We collect the required parameters (The parameters can be seen in Table 1) and basic data from the official financial report of Apple from 2020 to 2025, Yahoo Finance, Guru Focus, and other official sources.

2.2.  The calculation of future free cash flow to firm (2025-2029)

With the parameters needed to calculate the FCFF, we use the formula of FCFF to forecast Apple’s free cash flow from 2025 to 2029 and all parameters in the formula is the mean value over 10 years.

$$$ {FCFF}_t={Revenue}_t*EBIT Margin*\left(1-Tax Rate\right)+ {Revenue}_t*Depreciation Rate- {Revenue}_t*CapEx Rate-{Revenue}_t*∆ WC Rate$$$(1)

Then, we can get the forecast of FCFF (2025-2029). The precise number is $96.079B, $102.290B, $108.902B, $115.942B, and $123.437B.

2.3.  The calculation of weighted average cost of capital

The WACC is the company’s overall cost of capital, representing the average return expected by shareholders and lenders. To derive the WACC, we must introduce another model called capital asset pricing model (CAPM) to calculate an important value in WACC formula — Cost of equity ( $$ r_e$$ ).

$$ r_e=r_f+ \beta *MRP=4.40\%+1.20*4.33\%=9.60\%$$(2)

With cost of equity, we get the WACC of Apple through WACC formula.

$$ WACC=\left(\frac{E}{E+D}\right)*r_e+ \left(\frac{D}{E+D}\right)* r_d *(1-Tax Rate)$$(3)

$$$ WACC=\left(\frac{32800}{32800+70}\right)*0.9596+ \left(\frac{70}{32800+70}\right)* 0.045 *\left(1-0.1546\right)=0.0958$$$(4)

So, the WACC is 9.58%.

2.4.  Discounting future cash flows back to present value

Using the discount factor based on WACC, we calculate the present value of future cash flow, and compare it with FCFF. The comparison between FCFF and its present value can be seen in Figure 1.

$$ {PV}_{FCFF,t}=\frac{{FCFF}_t}{{\left(1+WACC\right)}^t}=416.27$$(5)

2.5.  Calculation of terminal value

Terminal value is also a very important parameter in DCF model. We calculate the terminal value and discount it to present value.

$$ TV2029=\frac{FCFF2029*\left(1+g\right)}{WACC-g}$$(6)

$$ TV=\frac{123.437*(1+0.03)}{0.958-0.03}=1932.22 billion USD$$(7)

$$ {PV}_{TV}=\frac{1932.22}{{\left(1+0.0958\right)}^t}=\frac{1321.39}{1.670}=1222.93 billion USD$$(8)

Final results

After we get all parameters that are required for company’s intrinsic value, we calculate the enterprise value, equity value, and intrinsic value per share as our results of DCF model.

$$$ Enterprise value EV=discounted FCFF sum+dicounted terminal value$$$

$$$ EV=412.34+1222.93=1635.27 billion USD$$$

$$$ Equity value=EV-Net Debt$$$

$$$ Equity value=1635.27-70=1565.27 billion USD$$$

$$$ Intrinsic value per Share=\frac{Equity value}{Shares Outstanding}$$$

$$ Intrinsic value per Share=\frac{1565.27}{14.94}=105.033 Dollar per Share$$(9)

3.  Analysis of DCF model results

To estimate the accuracy of our results, we have taken two methods to assess our model: Sensitivity Analysis and Monte Carlo Stimulation, which is two common methods to estimate the results of model.

3.1.  The steps of application of DCF model

Sensitivity Analysis tests model output response to changes in key inputs, assessing uncertainty and robustness. We use grid method varying WACC (6%-12%) and g (2-4.5%) to compute EV ranges. The result of sensitivity analysis can be seen in Table 2.

By conducting modeling to calculate the average relative change rate:

WACC ±1% (g fixed): ∓17.6%

g ±1% (WACC fixed): ±18.8%

WACC ±1% + g ±1% (same direction): ±1.2%

WACC ±1% + g ∓1% (opposite direction): ∓36.4%

Based on our sensitivity analysis, enterprise value is highly sensitive to WACC and g changes, with baseline EV around $1639B and EV ranges $1131B to $6842B.

Our conclusion based on sensitivity analysis is that WACC dominates downside risk, g drives upside potential, and investors should watch interest rate volatility and growth uncertainty, implying Apple’s current market cap may be overvalued.

3.2.  The steps of application of DCF model

Monte Carlo Simulation is a probabilistic method running model multiple times (10000) with random input samples, generating output distributions to quantify uncertainty. Our simulation assume growth ~ Normal (6.46%, 12.19%), WACC ~ Normal (9.58%, 1%), and g ~ Normal (3%, 0.5%). It computes enterprise value per share distributions. The results of Monte Carlo Simulation can be seen in Figure 2.

The mean of Monte Carlo Simulation is $105.691(red line), CI is $62.769-$162.873(green line), and SD is $25.493. The distribution is right-skewed (higher downside risk).

4.  Conclusion

Based on our overall valuation process, the model uses historical mean reversion for FCFF prediction, 9,58% WACC, yielding Apple’s intrinsic value that is $105 per share, well below market price 214$, implying overvaluation. The sensitivity analysis emphasizes interest rate risk dominates downside. The Monte Carlo simulation is right-skewed, which shows higher downside risk and support conservative valuation and capturing growth/WACC uncertainty. The model is conservative and sensitive to assumptions, monitor macro factors. If the overvaluation can be confirmed through the following estimation, investors should consider selling.