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Z-TransformThe Z-Transform is a mathematical tool which is used to convert the difference equations in time domain into the algebraic equations in the z-domain. Mathematically, the Z-transform of a discrete-time signal or a sequence $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is defined as −$$\mathrm{\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty }}^{\infty }\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{z}^{-\mathit{n}}}$$Properties of Z-TransformThe following table highlights some of the important properties of Z-Transform −PropertyTime-Domainz-DomainRegion of Convergence (ROC)Notation$\mathrm{\mathit{x}\mathrm{\left(\mathit{n}\right)}}$$\mathrm{\mathit{X}\mathrm{\left(\mathit{z}\right)}}$$\mathrm{\mathit{R}}$$\mathrm{\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{n}\right)}}$$\mathrm{\mathit{X}_{\mathrm{1}}\mathrm{\left(\mathit{z}\right)}}$$\mathrm{\mathit{R}_{\mathrm{1}}}$$\mathrm{\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n}\right)}}$$\mathrm{\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{z}\right)}}$$\mathrm{\mathit{R}_{\mathrm{2}}}$Linearity and Superposition$\mathrm{\mathit{a}\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{n} \right)}\:\mathrm{+}\:\mathit{b}\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n} \right)}}$$\mathrm{\mathit{a}\mathit{X}_{\mathrm{1}}\mathrm{\left(\mathit{z} \right)}\:\mathrm{+}\:\mathit{b}\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{z}\right)}}$$\mathrm{\mathit{R}_{\mathrm{1}}\:\cap \mathit{R}_{\mathrm{2}}}$Time-Shifting$\mathrm{\mathit{x}\mathrm{\left(\mathit{n-k}\right)}}$$\mathrm{\mathit{z}^{-\mathit{k}}\mathit{X}\mathrm{\left(\mathit{z}\right)}}$$\mathrm{\mathrm{same\:as\:}\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{except}\:\mathit{z}\:\mathrm{=}\:\mathrm{0}}$$\mathrm{\mathit{x}\mathrm{\left(\mathit{n\mathrm{+}\mathit{k}}\right)}}$$\mathrm{\mathit{z}^{\mathit{k}}\mathit{X}\mathrm{\left(\mathit{z}\right)}}$$\mathrm{\mathrm{same\:as\:}\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{except}\:\mathit{z}\:\mathrm{=}\:\mathrm{\infty}}$Scaling in zdomain$\mathrm{\mathit{a}^{\mathit{n}}\mathit{x}\mathrm{\left(\mathit{n}\right)}}$$\mathrm{\mathit{X}\mathrm{\left( \frac{\mathit{z}}{\mathit{a}}\right )}}$$\mathrm{\left|\mathit{a}\right|\mathit{R}_{\mathrm{1}}Read More
Discrete Time Fourier TransformThe discrete time Fourier transform is a mathematical tool which is used to convert a discrete time sequence into the frequency domain. Therefore, the Fourier transform of a discrete time signal or sequence is called the discrete time Fourier transform (DTFT).Mathematically, if $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is a discrete time sequence, then the discrete time Fourier transform of the sequence is defined as −$$\mathrm{\mathit{F}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{\omega }\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty }}^{\infty }\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{e}^{-\mathit{j\omega n}}}$$Properties of Discrete-Time Fourier TransformFollowing table gives the important properties of the discrete-time Fourier transform −PropertyDiscrete-Time SequenceDTFTNotation$\mathrm{\mathit{x}\mathrm{\left(\mathit{n}\right)}}$$\mathrm{\mathit{X}\mathrm{\left(\mathit{\omega}\right)}}$$\mathrm{\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{n}\right)}}$$\mathrm{\mathit{X}_{\mathrm{1}}\mathrm{\left(\mathit{\omega}\right)}}$$\mathrm{\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n}\right)}}$$\mathrm{\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{\omega}\right)}}$Linearity$\mathrm{\mathit{a}\mathit{x}_{\mathrm{1}}\mathrm{\left( \mathit{n}\right)}\:\mathrm{+}\:\mathit{b}\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n}\right)}}$$\mathrm{\mathit{a}\mathit{X}_{\mathrm{1}}\mathrm{\left( \mathit{\omega }\right)}\:\mathrm{+}\:\mathit{b}\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{\omega}\right)}}$Time Shifting$\mathrm{\mathit{x}\mathrm{\left(\mathit{n-k}\right)}}$$\mathrm{\mathit{e}^{\mathit{-j\omega k}}\mathit{X}\mathrm{\left(\mathit{\omega }\right)}}$Frequency Shifting$\mathrm{\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{e}^{\mathit{j\omega} _{\mathrm{0}}\mathit{n}}}$$\mathrm{\mathit{X}\mathrm{\left(\mathit{\omega -\omega _{\mathrm{0}}}\right)}}$Time Reversal$\mathrm{\mathit{x}\mathrm{\left(\mathit{-n}\right)}}$$\mathrm{\mathit{X}\mathrm{\left(\mathit{-\omega}\right)}}$Frequency Differentiation$\mathrm{\mathit{n}\mathit{x}\mathrm{\left(\mathit{n}\right)}}$$\mathrm{\mathit{j}\frac{\mathit{d}}{\mathit{d\omega}}\mathit{X}\mathrm{\left(\mathit{\omega }\right)}}$Time Convolution$\mathrm{\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{n}\right)}\:*\:\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n}\right)}}$$\mathrm{\mathit{X}_{\mathrm{1}}\mathrm{\left(\mathit{\omega }\right)}\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{\omega }\right)}}$Frequency ... Read More
Inverse Z-TransformThe inverse Z-transform is defined as the process of finding the time domain signal $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ from its Z-transform $\mathit{X}\mathrm{\left(\mathit{z}\right)}$. The inverse Z-transform is denoted as −$$\mathrm{\mathit{x}\mathrm{\left(\mathit{n}\right)}\:\mathrm{=}\:\mathit{Z}^{-\mathrm{1}}\mathrm{\left[\mathit{X}\mathrm{\left(\mathit{z}\right)}\right]}}$$Partial Fraction Expansion Method to Find Inverse Z-TransformIn order to determine the inverse Z-transform of $\mathit{X}\mathrm{\left(\mathit{z}\right)}$ using partial fraction expansion method, the denominator of $\mathit{X}\mathrm{\left(\mathit{z}\right)}$ must be in factored form. In this method, we obtained the partial fraction expansion of $\frac{\mathit{X}\mathrm{\left(\mathit{z}\right)}}{\mathit{z}}$ instead of $\mathit{X}\mathrm{\left(\mathit{z}\right)}$. This is because the Z-transform of time-domain sequences have Z in their numerators.The partial fraction expansion method is applied only if $\frac{\mathit{X}\mathrm{\left(\mathit{z}\right)}}{\mathit{z}}$ is a proper rational function, i.e., the order ... Read More
A balance sheet is made up of assets and liabilities and hence the balance sheet approach of evaluating a firm shows the values of the assets of a company.Book Value of Assets is the Minimum Value of a FirmWhen the values are un-adjusted, the balance sheet approach indicates the claims of investors over the assets of the company. That is, the book value of equity funds and debt funds represent the value of the firm the investors have ownership on. Therefore, the minimum value of a firm is equal to the book value of assets.Value of a Firm is Worth ... Read More
The Free Cash Flow approach using WACC for the evaluation of investment projects has certain limitations −Cash Flow PatternsThe original WACC is based on an assumption that cash flow patterns are perpetual. In fact, there is no such behavior in case of cash flow patterns. However, WACC works in all types of cash flows.Business RisksWACC assumes that a project or a business has the same risks as the existing assets of the company. This may be true in case of a small expansion in assets but for completely different types of businesses, this may not be applicable.The evaluation of a ... Read More
The Adjusted Present Value of a project takes the Net Present Value (NPV) of a project and adds this with the cost of debt, including financing effects, such as interest tax shield, issue costs, costs of distress, and subsidies etc. The APV is used instead of NPV for evaluating an investment project for various reasons. Here's why APV is used more frequently than other methods of evaluation of a project.The Effect of Debt and EquityThe use of all-equity financing may be debilitating for the health of a company's financials. In some situations, the NPV of such project turn positive due ... Read More
Loan-to-Value RatioThe loan-to-value ratio (LTV) is a ratio of loan one wants to borrow to the appraisal value of property he or she can produce as a collateral.LTV is a measure of the capability of handling a loan and repay the interest and the principle in theoretical terms.Higher LTV value means more risk as the loan amount goes up but the repayment capability remains the same.LTV shows how much property a borrower of the loan actually owns to the real value of the property that was charged while the borrower bought the property.Lenders usually determine the risk associated with the ... Read More
Under the comparative firms approach of valuation, companies are valued depending on groups formed with the key relationships of the companies. The groups of companies are formed with similar companies or similar transactions to determine the value of a firm. By deciding the group of company, the general trends are applied to each company of a group. Since the valuation is done by comparison, the approach is known as comparative firms approach.A Simple Approach in Evaluating a CompanyThe comparative firms approach is based on the fact that similar companies should have the same value and should sell for similar prices.It ... Read More
In the Adjusted Present Value (APV) approach, the after-tax subsidy is applied on after-tax cost of debt. That is, the company availing the financial subsidy gets a tax relief on their after-tax cost of loans. The debt of a company directly affects its value and hence the after-tax cost of debt also affects the company's finances. In fact, the companies get both savings in the tax paid as well as on the interest tax shield.Subsidized Financing Increases the Value of a ProjectA company paying 15 percent tax and receiving 5 percent subsidy will have to pay the interest at 10 ... Read More
When we consider fixed debt ratio and debt rebalancing, both the interest shields and Free Cash Flows are discounted at the opportunity cost of capital of the project to determine the Adjusted Present Value (APV). So, one can combine these two flows and discount them by the opportunity cost of capital.Under Fixed Capital StructureSince FCFs plus interest tax shields equal the Capital Cash Flows (CCF), the CCF and APV approaches under fixed capital structure are the same. Under the assumption of fixed capital structure, CCFs, FCFs and APVs are all equal.The FCF value is widely used to determine the valuation ... Read More