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Operating risk is associated with a company’s cost structure. It is the risk a company faces due to the level of fixed costs in the company’s operations. As the name suggests, operating risks are associated with the operations of the business. This may include risks due to failure of fixed assets or unpredictable operational risks that cannot be foreseen.Business risks are of two types − Operating risk and Sales risk.Operating risk is related to the cost structure and fixed costs of a company.Sales risk is the risks associated with the loss of revenue due to fewer goods and services sold.Fixed ... Read More
Operating leverage is a tool that measures a company’s fixed costs as a percentage of its overall costs. It is often used to evaluate the breakeven point of a business and the profit from overall sales. When expressed as the degree of operating leverage (DOL), it represents a financial ratio that calculates the sensitivity of a company’s operating income to its sales. As such, the DOL is a financial metric that shows how a change in the company’s sales will affect the company’s operating income.High Operating LeverageIn the case of high operating leverage, a large portion of a company’s costs ... Read More
Financial risk refers to a condition where a company with a certain amount of debt will fail to repay them in a given time period. In other words, financial risk means the risk of losing money by investing it in a lossmaking company.Investors usually remain averse to risky companies and hence calculating the financial risk is of paramount importance to them. In general, the more debt a company has, the more will be its financial risk.Types of Financial RisksFinancial risks can lead to loss of shareholders’ income, as the money is lost while carrying on with a loss-making company. However, ... Read More
Meaning of Capital StructureCapital Structure is the ratio of different types of securities raised by a firm as its long-term finance. Capital structure decision involves two philosophies −Type of securities to be issued in capital structures must be equity shares, preference shares, and long-term borrowings (Debentures).Relative ratio of the securities can be obtained by the process of capital gearing. On the basis of gearing, the companies are divided into two categories −Highly geared companies – The companies which have a proportion of equity capitalization that is small.Low geared companies – The companies the equity capital of which is high in ... Read More
Payback period or simply payback in capital budgeting refers to the time required for the ROI (Return on Investment) to repay the original sum of investment.Payback is a preferred tool because it is easy to understand and apply, irrespective of whether the manager is aware of financial calculations or not.Payback is an effective tool to derive the worth of an investment when similar projects are compared.The payback method is a simple tool to measure the months or years it takes to repay the initial investment of a project.The payback method doesn’t have any specific criteria for the evaluation of investments ... Read More
Fourier TransformThe Fourier transform of a continuous-time function $x(t)$ can be defined as, $$\mathrm{X(\omega)=\int_{−\infty}^{\infty}x(t)e^{-j\omega t}dt}$$Modulation Property of Fourier TransformStatement – The modulation property of continuous-time Fourier transform states that if a continuous-time function $x(t)$ is multiplied by $cos \:\omega_{0} t$, then its frequency spectrum gets translated up and down in frequency by $\omega_{0}$. Therefore, if$$\mathrm{x(t)\overset{FT}{\leftrightarrow}X(\omega)}$$Then, according to the modulation property of CTFT, $$\mathrm{x(t)\:cos\:\omega_{0}t\overset{FT}{\leftrightarrow}\frac{1}{2}[X(\omega-\omega_{0})+X(\omega+\omega_{0})]}$$ProofUsing Euler’s formula, we get, $$\mathrm{cos\:\omega_{0}t=\left [\frac{e^{j\omega_{0} t}+e^{-j\omega_{0} t}}{2} \right ]}$$Therefore, $$\mathrm{x(t)\:cos\:\omega_{0}t=x(t)\left [ \frac{e^{j\omega_{0} t}+e^{-j\omega_{0} t}}{2}\right ]}$$Now, from the definition of Fourier transform, we have, $$\mathrm{F[x(t)]=X(\omega)=\int_{−\infty}^{\infty}x(t)e^{-j\omega_{0} t} \:dt}$$$$\mathrm{\Rightarrow\:F[x(t)\:cos\:\omega_{0} t]=\int_{−\infty}^{\infty}x(t)\:cos\:\omega_{0} t\:e^{-j\omega_{0} t}dt}$$$$\mathrm{\Rightarrow\:F[x(t)\:cos\:\omega_{0} t]=\int_{−\infty}^{\infty}x(t)\left [ \frac{e^{j\omega_{0} t}+e^{-j\omega_{0} t}}{2}\right ]e^{-j\omega t}dt}$$$$\mathrm{\Rightarrow\:F[x(t)\:cos\:\omega_{0} ... Read More
Fourier TransformFor a continuous-time function $x(t)$, the Fourier transform can be defined as, $$\mathrm{X(\omega)=\int_{−\infty}^{\infty}x(t)e^{-j\omega t}dt}$$Linearity Property of Fourier TransformStatement − The linearity property of Fourier transform states that the Fourier transform of a weighted sum of two signals is equal to the weighted sum of their individual Fourier transforms.Therefore, if$$\mathrm{x_{1}(t)\overset{FT}{\leftrightarrow}X_{1}(\omega)\:\:and\:\:x_{2}\overset{FT}{\leftrightarrow}X_{2}(\omega)}$$Then, according to the linearity property of Fourier transform, $$\mathrm{ax_{1}(t)+bx_{2}(t)\overset{FT}{\leftrightarrow}aX_{1}(\omega)+bX_{2}(\omega)}$$Where, a and b are constants.ProofFrom the definition of Fourier transform, we have, $$\mathrm{F[x(t)]=X(\omega)=\int_{−\infty}^{\infty}x(t)e^{-j \omega t}dt}$$$$\mathrm{\Rightarrow\:X(\omega)=F[ax_{1}(t)+bx_{2}(t)]=\int_{−\infty}^{\infty}[ax_{1}(t)+bx_{2}(t)]e^{-j \omega t}dt}$$$$\mathrm{\Rightarrow\:X(\omega)=\int_{−\infty}^{\infty}ax_{1}(t)e^{-j \omega t} dt+\int_{−\infty}^{\infty}bx_{2}(t)e^{-j \omega t}dt}$$$$\mathrm{\Rightarrow\:X(\omega)=a\int_{−\infty}^{\infty}x_{1}(t)e^{-j \omega t} dt+b\int_{−\infty}^{\infty}x_{2}(t)e^{-j \omega t}dt}$$$$\mathrm{\Rightarrow\:X(\omega)=aX_{1}(\omega)+bX_{2}(\omega)}$$$$\mathrm{\therefore\:F[ax_{1}(t)+bx_{2}(t)]=aX_{1}(\omega)+bX_{2}(\omega)}$$Or, it can also be written as, $$\mathrm{ax_{1}(t)+bx_{2}(t)\overset{FT}{\leftrightarrow}aX_{1}(\omega)+bX_{2}(\omega)}$$Frequency Shifting Property of Fourier TransformStatement – Frequency ... Read More
Fourier SeriesIf $x(t)$ is a periodic function with period $T$, then the continuous-time exponential Fourier series of the function is defined as, $$\mathrm{x(t)=\displaystyle\sum\limits_{n=−\infty}^\infty\:C_{n}\:e^{jn\omega_{0}t}\:\:\:… (1)}$$Where, $C_{n}$ is the exponential Fourier series coefficient, which is given by, $$\mathrm{C_{n}=\frac{1}{T}\int_{t_{0}}^{t_{0}+T}X(t)e^{-jn\omega_{0}t}\:dt\:\:… (2)}$$Linearity Property of Continuous Time Fourier SeriesConsider two periodic signals $x_{1}(t)$ and $x_{2}(t)$ which are periodic with time period T and with Fourier series coefficients $C_{n}$ and $D_{n}$ respectively. If$$\mathrm{x_{1}(t)\overset{FS}{\leftrightarrow}C_{n}}$$$$\mathrm{x_{2}(t)\overset{FS}{\leftrightarrow}D_{n}}$$Then, the linearity property of continuous-time Fourier series states that$$\mathrm{Ax_{1}(t)+Bx_{2}(t)\overset{FS}{\leftrightarrow}AC_{n}+BD_{n}}$$Proof By the definition of Fourier series of a periodic function, we get, $$\mathrm{FS[Ax_{1}(t)+Bx_{2}(t)]=\frac{1}{T}\int_{t_{0}}^{t_{0}+T}[Ax_{1}(t)+Bx_{2}(t)]e^{-jn\omega_{0}t}\:dt}$$$$\mathrm{\Rightarrow\:FS[Ax_{1}(t)+Bx_{2}(t)]=A\left ( \frac{1}{T}\int_{t_{0}}^{t_{0}+T}\:x_{1}(t)\:e^{-jn\omega_{0}t}\:dt\right )+B\left ( \frac{1}{T}\int_{t_{0}}^{t_{0}+T}\:x_{2}(t)\:e^{-jn\omega_{0}t}\:dt \right )\:\:… (3)}$$On comparing equations (2) & (3), ... Read More
What is GIBBS Phenomenon?The GIBBS phenomenon was discovered by Henry Wilbraham in 1848 and then rediscovered by J. Willard Gibbs in 1899.For a periodic signal with discontinuities, if the signal is reconstructed by adding the Fourier series, then overshoots appear around the edges. These overshoots decay outwards in a damped oscillatory manner away from the edges. This is known as GIBBS phenomenon and is shown in the figure below.The amount of the overshoots at the discontinuities is proportional to the height of discontinuity and according to Gibbs, it is found to be around 9% of the height of discontinuity irrespective ... Read More
Fourier TransformThe Fourier transform of a continuous-time function can be defined as, $$\mathrm{X(\omega)=\int_{−\infty }^{\infty}\:X(t)e^{-j\omega t}\:dt}$$Differentiation in Frequency Domain Property of Fourier TransformStatement − The frequency derivative property of Fourier transform states that the multiplication of a function X(t) by in time domain is equivalent to the differentiation of its Fourier transform in frequency domain. Therefore, if$$\mathrm{X(t)\overset{FT}{\leftrightarrow}X(\omega)}$$Then, according to frequency derivative property, $$\mathrm{t\cdot x(t)\overset{FT}{\leftrightarrow}j\frac{d}{d\omega}X(\omega)}$$ProofFrom the definition of Fourier transform, we have, $$\mathrm{X(\omega)=\int_{−\infty }^{\infty}x(t)e^{-j\omega t}\:dt}$$Differentiating the above equation on both sides with respect to ω, we get, $$\mathrm{\frac{d}{d\omega}X(\omega)=\frac{d}{d\omega}\left [ \int_{−\infty }^{\infty}x(t)e^{-j\omega t}\:dt \right ]}$$$$\mathrm{\Rightarrow\:\frac{d}{d\omega}X(\omega)=\int_{−\infty }^{\infty} x(t)\frac{d}{d\omega}\left [e^{-j\omega t} \right ]dt}$$$$\mathrm{\Rightarrow\:\frac{d}{d\omega}X(\omega)=\int_{−\infty }^{\infty} x(t)(-jt)e^{-j\omega ... Read More