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∫ Решение определённых интегралов — Калькулятор Онлайн

Введите функцию, для которой надо найти интеграл

Калькулятор предоставляет ПОДРОБНОЕ решение определённых интегралов.

Этот калькулятор находит решение определенного интеграла от функции f(x) с данными верхними и нижними пределами.

Примеры

С применением степени
(квадрат и куб) и дроби

(x^2 - 1)/(x^3 + 1)

Квадратный корень

sqrt(x)/(x + 1)

Кубический корень

cbrt(x)/(3*x + 2)

С применением синуса и косинуса

2*sin(x)*cos(x)

Арксинус

x*arcsin(x)

Арккосинус

x*arccos(x)

Применение логарифма

x*log(x, 10)

Натуральный логарифм

ln(x)/x

Экспонента

exp(x)*x

Тангенс

tg(x)*sin(x)

Котангенс

ctg(x)*cos(x)

Иррациональне дроби

(sqrt(x) - 1)/sqrt(x^2 - x - 1)

Арктангенс

x*arctg(x)

Арккотангенс

x*arсctg(x)

Гиберболические синус и косинус

2*sh(x)*ch(x)

Гиберболические тангенс и котангенс

ctgh(x)/tgh(x)

Гиберболические арксинус и арккосинус

x^2*arcsinh(x)*arccosh(x)

Гиберболические арктангенс и арккотангенс

x^2*arctgh(x)*arcctgh(x)

Правила ввода выражений и функций

Выражения могут состоять из функций (обозначения даны в алфавитном порядке):

absolute(x)

Абсолютное значение x
(модуль x или |x|)

arccos(x)

Функция — арккосинус от x

arccosh(x)

Арккосинус гиперболический от x

arcsin(x)

Арксинус от x

arcsinh(x)

Арксинус гиперболический от x

arctg(x)

Функция — арктангенс от x

arctgh(x)

Арктангенс гиперболический от x

e

e число, которое примерно равно 2.7

exp(x)

Функция — экспонента от x (что и e^x)

log(x) or ln(x)

Натуральный логарифм от x
(Чтобы получить log7(x), надо ввести log(x)/log(7) (или, например для log10(x)=log(x)/log(10))

pi

Число — «Пи», которое примерно равно 3.14

sin(x)

Функция — Синус от x

cos(x)

Функция — Косинус от x

sinh(x)

Функция — Синус гиперболический от x

cosh(x)

Функция — Косинус гиперболический от x

sqrt(x)

Функция — квадратный корень из x

sqr(x) или x^2

Функция — Квадрат x

tg(x)

Функция — Тангенс от x

tgh(x)

Функция — Тангенс гиперболический от x

cbrt(x)

Функция — кубический корень из x

В выражениях можно применять следующие операции:

Действительные числа

вводить в виде 7.5, не 7,5

2*x

— умножение

3/x

— деление

x^3

— возведение в степень

x + 7

— сложение

x — 6

— вычитание

Другие функции:

floor(x)

Функция — округление x в меньшую сторону (пример floor(4.5)==4.0)

ceiling(x)

Функция — округление x в большую сторону (пример ceiling(4.5)==5.0)

sign(x)

Функция — Знак x

erf(x)

Функция ошибок (или интеграл вероятности)

laplace(x)

Функция Лапласа

90000 Equation of a line 90001 90002 Line is an endless line, which forms the shortest path between any of two its points. 90003 90004 Equation of a line on plane 90005 90006 General form of a line equation 90007 90002 Any equation of a line on plane can by written in the 90009 general form 90010 90003 90002 A x + B y + C = 0 90003 90002 where A and B are not both equal to zero. 90003 90016 90006 Slope intercept form of a line equation 90007 90002 The general equation of a line when B ≠ 0 can be reduced to the next form 90003 90002 y = k x + b 90003 90002 where k is the 90009 slope of the line 90010 and b is the 90009 y-intercept 90010.90009 Slope of the line 90010 is equal to the tangent of the angle between this line and the positive direction of the x-axis. The 90009 y-coordinate 90010 is the location where line crosses the y-axis. 90003 90002 k = tg φ 90003 90006 Equation of the line passing through two different points on plane 90007 90002 If the line passes through two points A (x 90038 1 90039, y 90038 1 90039) and B (x 90038 2 90039, y 90038 2 90039), such that x 90038 1 90039 ≠ x 90038 2 90039 and y 90038 1 90039 ≠ y 90038 2 90039, then 90009 equation of line 90010 can be found using the following formula 90003 90057 90058 90059 90060 x — x 90038 1 90039 90063 90064 = 90063 90060 y — y 90038 1 90039 90063 90070 90059 90060 x 90038 2 90039 — x 90038 1 90039 90063 90060 y 90038 2 90039 — y 90038 1 90039 90063 90070 90085 90086 90087 90006 Parametric equations of a line on plane 90007 90002 Parametric equation of the line can be written as 90003 90057 90058 90059 90095 90096 90063 90060 x = l t + x 90038 0 90039 90063 90070 90059 90060 y = m t + y 90038 0 90039 90063 90070 90085 90086 90002 where N (x 90038 0 90039, y 90038 0 90039) is coordinates of a point that lying on a line, a = {l, m} is coordinates of the direction vector of line.90003 90006 Canonical equation of a line on plane 90007 90002 If you know the coordinates of the point A (x 90038 0 90039, y 90038 0 90039) If you know the coordinates of the point n = {l; m}, then the equation of the line can be written in the canonical form using the following formula 90003 90002 Example 1. Find the equation of a line passing through two points A (1, 7) and B (2, 3). 90003 90002 90009 Solution. 90010 We use the formula for the equation of a straight line passing through two points 90003 90002 x — 12 — 1 = y — 73 — 7 90003 90002 From this equation, we express y in terms of x 90003 90002 x — 11 = y — 7-4 90003 90002 Find the Slope intercept form of line equation.90003 90002 Multiply equations by -4. 90003 90002 y — 7 = -4 (x — 1) 90003 90002 y = -4x + 11 90003 90002 Find the Parametric equations of this line 90003 90002 We use MN as direction vector of line. 90003 90002 MN = {2 — 1; 3 — 7} = {1; -4} 90003 90002 We use coordinates of point М in parametric equations of line 90003 90002 x = t + 1y = -4t + 7 90003 Example 2. Find the equation of a line passing through two points M (1, 3) and N (2, 3). 90002 90009 Solution. 90010 It is impossible to use Equation of the line passing through two different points, since M 90038 y 90039 — N 90038 y 90039 = 0.90003 90002 Find the Parametric equations of this line. We use MN as direction vector of line. 90003 90002 MN = {2 — 1; 3 — 3} = {1; 0} 90003 90002 We use coordinates of point М in parametric equations of line 90003 90002 x = t + 1y = 3 90003 90004 Equation of a line in space 90005 90006 Equation of the line passing through two different points in space 90007 90002 If the line passes through two points A (x 90038 1 90039, y 90038 1 90039, z 90038 1 90039) and B (x 90038 2 90039, y 90038 2 90039, z 90038 2 90039), such that x 90038 1 90039 ≠ x 90038 2 90039, y 90038 1 90039 ≠ y 90038 2 90039 and z 90038 1 90039 ≠ z 90038 2 90039, then 90009 equation of line 90010 can be found using the following formula 90003 90057 90058 90059 90060 x — x 90038 1 90039 90063 90064 = 90063 90060 y — y 90038 1 90039 90063 90064 = 90063 90060 z — z 90038 1 90039 90063 90070 90059 90060 x 90038 2 90039 — x 90038 1 90039 90063 90060 y 90038 2 90039 — y 90038 1 90039 90063 90060 z 90038 2 90039 — z 90038 1 90039 90063 90070 90085 90086 90006 Parametric equations of a line in space 90007 90002 Parametric equation of the line can be written as 90003 90057 90058 90059 90252 90096 90063 90060 x = l t + x 90038 0 90039 90063 90070 90059 90060 y = m t + y 90038 0 90039 90063 90070 90059 90060 z = n t + z 90038 0 90039 90063 90070 90085 90086 90002 where (x 90038 0 90039, y 90038 0 90039, z 90038 0 90039) is coordinates of a point that lying on a line, {l; m; n} is coordinates of the direction vector of line.90003 90006 Canonical equation of a line in space 90007 90002 If you know the coordinates of the point A (x 90038 0 90039, y 90038 0 90039, z 90038 0 90039) that lies on the line and the direction vector of the line n = {l; m; n}, then the equation of the line can be written in the canonical form using the following formula. 90003 90057 90058 90059 90060 x — x 90038 0 90039 90063 90064 = 90063 90060 y — y 90038 0 90039 90063 90064 = 90063 90060 z — z 90038 0 90039 90063 90070 90059 90313 l 90063 90313 m 90063 90313 n 90063 90070 90085 90086 90006 Straight line as crossings of two planes 90007 90002 If the line is the intersection of two planes, then the equation of line can be found as the solution of the following system of equations 90003 90057 90058 90059 90095 90096 90063 90060 A 90038 1 90039 x + B 90038 1 90039 y + C 90038 1 90039 z + D 90038 1 90039 = 0 90063 90070 90059 90060 A 90038 2 90039 x + B 90038 2 90039 y + C 90038 2 90039 z + D 90038 2 90039 = 0 90063 90070 90085 90086 90002 on condition that there has not been equality 90003 90057 90058 90059 90060 A 90038 1 90039 90063 90064 = 90063 90060 B 90038 1 90039 90063 90064 = 90063 90060 C 90038 1 90039 90063 90064.90063 90070 90059 90313 A 90038 2 90039 90063 90313 B 90038 2 90039 90063 90313 C 90038 2 90039 90063 90070 90085 90086 .90000 The Reorder Point Formula And Calculating Safety Stock 90001 90002 How little is too little? That’s exactly what your Reorder Point will help you determine. Let’s talk about the concept of reorder points, safety stock, and how to put those numbers together to make sure you have the right amount of stock on-hand. 90003 90002 Whether you’ve just started a new business or you’ve sold products for years, anyone can benefit from using the reorder point formula. This post will show you what that is, why it’s useful, and which numbers you’ll need to generate a reorder point.90003 90006 How to calculate your reorder point 90007 90002 A reorder point is not a static number. It’s based on your own purchase and sales cycles, and it varies on a per-product basis. However, once you have a handle on the patterns of a product, you’re ready to start putting the variables together. 90003 90002 The reorder point formula is daily unit sales multiplied by delivery lead time, with some safety stock for good measure. 90003 90002 90003 90006 The reorder point formula video 90007 90002 If you’re the type who likes to watch instead of read, we’ve created a video version of this post.The examples are a little different, but the formula is exactly the same. Peter covers the reorder point formula in just three minutes: 90003 90006 Calculating average daily unit sales 90007 90002 OK! Back to the post at hand. The first variable is simply the average amount of that product you sell on any given day. Let’s examine the fictional company, Archon Optical, and their Ghost glasses. Here are the sales of the Ghost over the last three months: 90003 90002 If we total those numbers, we get 180 total units sold over the past 90 days.90003 90002 That means that the average daily sales for the Ghost is 2 per day. 90003 90006 Calculating average delivery lead time 90007 90002 Average delivery lead time is the time it usually takes for product shipments to arrive. You should have a couple of purchase orders on-hand to check the numbers. Delivery times can vary based on the quantity you order (larger orders could take longer to ship). When you place the order also affects the lead time (compare orders during a busy season versus a slow season).There are different ways to calculate this, but a three-month average is a good start. 90003 90002 Here are a few Archon Optical purchase for the last three month of Ghost shipments: 90003 90002 Add the total delivery time (15 days) and divide it by the number of orders (3 orders). That’s an average lead time of five days for the product to arrive. 90003 90006 How to calculate safety stock quickly 90007 90002 Safety stock is similar to a reorder point, but it’s a surplus quantity to ensure that you do not run completely out of stock if there are delays.90003 90002 When deciding on a safety stock level you’ll want to consider: average daily sales and the daily average that product used in work orders (if applicable). Lead time is also important to safety stock. We’ll keep things simple by calculating based on two weeks of extra demand (14 days). 90003 90002 Since the average daily sales for the Ghost are 2 (as calculated earlier on this page), that means the safety stock for Ghost is about 14 x 2 = 28. 90003 90002 [We’ve also written a more detailed article on safety stock, if you’d like to calculate it based on lead-time demand.] 90003 90006 Putting the reorder point formula together 90007 90002 Now that we’ve got all of the Archon Optical numbers down, we’re ready to put together a reorder point for the Ghost. 90003 90002 When the quantity on-hand for Ghost glasses hits 38, Archon Optical knows to place a purchase order for more. Because they’ve built an average delivery lead time into the reorder point, the extra Ghost glasses should arrive before Archon ever dips below the amount of safety stock. 90003 90002 Even if there are production shortages or shipping delays, Archon Optical’s safety stock ensures that they can sell Ghost glasses for two more weeks before they run out of stock.90003 90006 Set a basic reordering reminder in a spreadsheet 90007 90002 Reorder points are vital to keeping your business running smoothly, but they’ll only work if you’re prepared to reorder on time. 90003 90002 If you’re a spreadsheet user, you can use conditional formatting for the quantity value of specific cells. You can set Excel or Google Sheets so that cells turn red when they hit a reorder point. This will effectively warn you when you need to start on a new purchase order. 90003 90006 Reorder faster with inFlow 90007 90002 Unlike spreadsheets, inFlow was designed specifically for working with inventory.Quantity and reorder point fields are built into the apps. This saves our customers a lot of setup time. 90003 90002 inFlow Cloud has a 90063 Reorder Stock 90064 window, which identifies which products need reordering, and creates new purchase orders with just one click. 90003 90002 If you’d like to implement reorder points with tailored suggestions for your business, we can help with that too! 90003 90002 inFlow Cloud has a 90063 Recommended Reorder Point 90064 report that examines your sales data and recommends reorder points for your products.90003 90002 Whether you’re just starting out with reorder points or fine-tuning them, inFlow Cloud can help! 90003.90000 Discounted Cash Flow Interview Questions & Answers (Basic) 90001 90002 Beyond knowing the basics of how to construct a DCF, you also need to understand concepts such as WACC, Cost of Equity and the proper discount rates to use depending on the scenario. Interviewers also like to ask about Terminal Value — how you calculate it, advantages and disadvantages of different methods, and signs that it’s «too high.» 90003 90002 90005 1.Walk me through a DCF. 90006 90003 90002 90003 90002 «A DCF values ​​a company based on the Present Value of its Cash Flows and the Present Value of its Terminal Value.90003 90002 First, you project out a company’s financials using assumptions for revenue growth, expenses and Working Capital; then you get down to Free Cash Flow for each year, which you then sum up and discount to a Net Present Value, based on your discount rate — usually the Weighted Average Cost of Capital. 90003 90002 90003 90002 Once you have the present value of the Cash Flows, you determine the company’s Terminal Value, using either the Multiples Method or the Gordon Growth Method, and then also discount that back to its Net Present Value using WACC.90003 90002 90003 90002 Finally, you add the two together to determine the company’s Enterprise Value. «90003 90002 90003 90002 90003 90002 90005 2.Walk me through how you get from Revenue to Free Cash Flow in the projections. 90006 90003 90002 90003 90002 Subtract COGS and Operating Expenses to get to Operating Income (EBIT). Then, multiply by (1 — Tax Rate), add back Depreciation and other non-cash charges, and subtract Capital Expenditures and the change in Working Capital.90003 90002 90003 90002 90037 Note: 90038 This gets you to Unlevered Free Cash Flow since you went off EBIT rather than EBT. You might want to confirm that this is what the interviewer is asking for. 90003 90002 90003 90002 90005 3.What’s an alternate way to calculate Free Cash Flow aside from taking Net Income, adding back Depreciation, and subtracting Changes in Operating Assets / Liabilities and CapEx? 90006 90003 90002 90003 90002 Take Cash Flow From Operations and subtract CapEx — that gets you to Levered Cash Flow.To get to Unlevered Cash Flow, you then need to add back the tax-adjusted Interest Expense and subtract the tax-adjusted Interest Income. 90003 90002 90003 90002 90005 4.Why do you use 5 or 10 years for DCF projections? 90006 90003 90002 90003 90002 That’s usually about as far as you can reasonably predict into the future. Less than 5 years would be too short to be useful, and over 10 years is too difficult to predict for most companies. 90003 90002 90003 90002 90005 5.What do you usually use for the discount rate? 90006 90003 90002 90003 90002 Normally you use WACC (Weighted Average Cost of Capital), though you might also use Cost of Equity depending on how you’ve set up the DCF. 90003 90002 90003 90002 90005 6.How do you calculate WACC? 90006 90003 90002 90003 90002 The formula is: 90037 Cost of Equity * (% Equity) + Cost of Debt * (% Debt) * (1 — Tax Rate) + Cost of Preferred * (% Preferred). 90038 90003 90002 90003 90002 In all cases, the percentages refer to how much of the company’s capital structure is taken up by each component.90003 90002 90003 90002 For Cost of Equity, you can use the Capital Asset Pricing Model (CAPM — see the next question) and for the others you usually look at comparable companies / debt issuances and the interest rates and yields issued by similar companies to get estimates. 90003 90002 90003 90002 90005 7.How do you calculate the Cost of Equity? 90006 90003 90002 90003 90002 90037 Cost of Equity = Risk-Free Rate + Beta * Equity Risk Premium 90038 90003 90002 90003 90002 The risk-free rate represents how much a 10-year or 20-year US Treasury should yield; Beta is calculated based on the «riskiness» of Comparable Companies and the Equity Risk Premium is the% by which stocks are expected to out-perform «risk-less» assets.90003 90002 90003 90002 Normally you pull the Equity Risk Premium from a publication called Ibbotson’s. 90003 90002 90003 90002 90037 Note: 90038 This formula does not tell the whole story. Depending on the bank and how precise you want to be, you could also add in a «size premium» and «industry premium» to account for how much a company is expected to out-perform its peers is according to its market cap or industry . 90003 90002 90003 90002 Small company stocks are expected to out-perform large company stocks and certain industries are expected to out-perform others, and these premiums reflect these expectations.90003 90002 90003 90002 90005 8.How do you get to Beta in the Cost of Equity calculation? 90006 90003 90002 90003 90002 You look up the Beta for each Comparable Company (usually on Bloomberg), un-lever each one, take the median of the set and then lever it based on your company’s capital structure. Then you use this Levered Beta in the Cost of Equity calculation. 90003 90002 90003 90002 For your reference, the formulas for un-levering and re-levering Beta are below: 90003 90002 90003 90002 90037 Un-Levered Beta = Levered Beta / (1 + ((1 — Tax Rate) x (Total Debt / Equity))) Levered Beta = Un-Levered Beta x (1 + ((1 — Tax Rate) x (Total Debt / Equity))) 90038 90003 90002 90003 90002 90005 9.Why do you have to un-lever and re-lever Beta? 90006 90003 90002 90003 90002 Again, keep in mind our «apples-to-apples» theme. When you look up the Betas on Bloomberg (or from whatever source you’re using) they will be levered to reflect the debt already assumed by each company. 90003 90002 90003 90002 But each company’s capital structure is different and we want to look at how «risky» a company is regardless of what% debt or equity it has. 90003 90002 90003 90002 To get that, we need to un-lever Beta each time.90003 90002 90003 90002 But at the end of the calculation, we need to re-lever it because we want the Beta used in the Cost of Equity calculation to reflect the true risk of our company, taking into account its capital structure this time. 90003 90002 90003 90002 90005 10.Would you expect a manufacturing company or a technology company to have a higher Beta? 90006 90003 90002 90003 90002 A technology company, because technology is viewed as a «riskier» industry than manufacturing.90003 90002 90003 90002 90005 11.Let’s say that you use Levered Free Cash Flow rather than Unlevered Free Cash Flow in your DCF — what is the effect? 90006 90003 90002 90003 90002 Levered Free Cash Flow gives you Equity Value rather than Enterprise Value, since the cash flow is only available to equity investors (debt investors have already been «paid» with the interest payments). 90003 90002 90003 90002 90005 12.If you use Levered Free Cash Flow, what should you use as the Discount Rate? 90006 90003 90002 90003 90002 You would use the Cost of Equity rather than WACC since we’re not concerned with Debt or Preferred Stock in this case — we’re calculating Equity Value, not Enterprise Value.90003 90002 90003 90002 90005 13.How do you calculate the Terminal Value? 90006 90003 90002 90003 90002 You can either apply an exit multiple to the company’s Year 5 EBITDA, EBIT or Free Cash Flow (Multiples Method) or you can use the Gordon Growth method to estimate its value based on its growth rate into perpetuity. 90003 90002 90003 90002 The formula for Terminal Value using Gordon Growth is: 90037 Terminal Value = Year 5 Free Cash Flow * (1 + Growth Rate) / (Discount Rate — Growth Rate).90038 90003 90002 90003 90002 90005 14.Why would you use Gordon Growth rather than the Multiples Method to calculate the Terminal Value? 90006 90003 90002 90003 90002 In banking, you almost always use the Multiples Method to calculate Terminal Value in a DCF. It’s much easier to get appropriate data for exit multiples since they are based on Comparable Companies — picking a long-term growth rate, by contrast, is always a shot in the dark. 90003 90002 90003 90002 However, you might use Gordon Growth if you have no good Comparable Companies or if you have reason to believe that multiples will change significantly in the industry several years down the road.For example, if an industry is very cyclical you might be better off using long-term growth rates rather than exit multiples. 90003 90002 90003 90002 90005 15.What’s an appropriate growth rate to use when calculating the Terminal Value? 90006 90003 90002 90003 90002 Normally you use the country’s long-term GDP growth rate, the rate of inflation, or something similarly conservative. 90003 90002 90003 90002 For companies in mature economies, a long-term growth rate over 5% would be quite aggressive since most developed economies are growing at less than 5% per year.90003 90002 90003 90002 90005 16.How do you select the appropriate exit multiple when calculating Terminal Value? 90006 90003 90002 90003 90002 Normally you look at the Comparable Companies and pick the median of the set, or something close to it. 90003 90002 90003 90002 As with almost anything else in finance, you always show a range of exit multiples and what the Terminal Value looks like over that range rather than picking one specific number.90003 90002 90003 90002 So if the median EBITDA multiple of the set were 8x, you might show a range of values ​​using multiples from 6x to 10x. 90003 90002 90003 90002 90005 17.Which method of calculating Terminal Value will give you a higher valuation? 90006 90003 90002 90003 90002 It’s hard to generalize because both are highly dependent on the assumptions you make. In general, the Multiples Method will be more variable than the Gordon Growth method because exit multiples tend to span a wider range than possible long-term growth rates.90003 90002 90003 90002 90005 18.What’s the flaw with basing terminal multiples on what public company comparables are trading at? 90006 90003 90002 90003 90002 The median multiples may change greatly in the next 5-10 years so it may no longer be accurate by the end of the period you’re looking at. This is why you normally look at a wide range of multiples and do a sensitivity to see how the valuation changes over that range. 90003 90002 90003 90002 This method is particularly problematic with cyclical industries (e.g. semiconductors). 90003 90002 90003 90002 90005 19.How do you know if your DCF is too dependent on future assumptions? 90006 90003 90002 90003 90002 The «standard» answer: if significantly more than 50% of the company’s Enterprise Value comes from its Terminal Value, your DCF is probably too dependent on future assumptions. 90003 90002 90003 90002 In reality, almost all DCFs are «too dependent on future assumptions» — it’s actually quite rare to see a case where the Terminal Value is less than 50% of the Enterprise Value.90003 90002 90003 90002 But when it gets to be in the 80-90% range, you know that you may need to re-think your assumptions … 90003 90002 90003 90002 90005 20.Should Cost of Equity be higher for a $ 5 billion or $ 500 million market cap company? 90006 90003 90002 90003 90002 It should be higher for the $ 500 million company, because all else being equal, smaller companies are expected to outperform large companies in the stock market (and therefore be «more risky»).Using a Size Premium in your calculation would also ensure that Cost of Equity is higher for the $ 500 million company. 90003 90002 90003 90002 90005 21.What about WACC — will it be higher for a $ 5 billion or $ 500 million company? 90006 90003 90002 90003 90002 This is a bit of a trick question because it depends on whether or not the capital structure is the same for both companies. If the capital structure is the same in terms of percentages and interest rates and such, then WACC should be higher for the $ 500 million company for the same reasons as mentioned above.90003 90002 90003 90002 If the capital structure is not the same, then it could go either way depending on how much debt / preferred stock each one has and what the interest rates are. 90003 90002 90003 90002 90005 22.What’s the relationship between debt and Cost of Equity? 90006 90003 90002 90003 90002 More debt means that the company is more risky, so the company’s Levered Beta will be higher — all else being equal, additional debt would raise the Cost of Equity, and less debt would lower the Cost of Equity.90003 90002 90003 90002 90005 23.Cost of Equity tells us what kind of return an equity investor can expect for investing in a given company — but what about dividends? Should not we factor dividend yield into the formula? 90006 90003 90002 90003 90002 Trick question. Dividend yields are already factored into Beta, because Beta describes returns in excess of the market as a whole — and those returns include dividends. 90003 90002 90003 90002 90005 24.How can we calculate Cost of Equity WITHOUT using CAPM? 90006 90003 90002 90003 90002 There is an alternate formula: 90003 90002 90003 90002 90037 Cost of Equity = (Dividends per Share / Share Price) + Growth Rate of Dividends 90038 90003 90002 90003 90002 This is less common than the «standard» formula but sometimes you use it for companies where dividends are more important or when you lack proper information on Beta and the other variables that go into calculating Cost of Equity with CAPM.90003 90002 90003 90002 90005 25.Two companies are exactly the same, but one has debt and one does not — which one will have the higher WACC? 90006 90003 90002 90003 90002 This is tricky — the one without debt will have a higher WACC up to a certain point, because debt is «less expensive» than equity. Why? 90003 90002 90003 90002 • Interest on debt is tax-deductible (hence the (1 — Tax Rate) multiplication in the WACC formula). 90003 90002 • Debt is senior to equity in a company’s capital structure — debt holders would be paid first in a liquidation or bankruptcy.90003 90002 • Intuitively, interest rates on debt are usually lower than the Cost of Equity numbers you see (usually over 10%). As a result, the Cost of Debt portion of WACC will contribute less to the total figure than the Cost of Equity portion will. 90003 90002 90003 90002 However, the above is true only to a certain point. Once a company’s debt goes up high enough, the interest rate will rise dramatically to reflect the additional risk and so the Cost of Debt would start to increase — if it gets high enough, it might become higher than Cost of Equity and additional debt would increase WACC.90003 90002 90003 90002 It’s a «U-shape» curve where debt decreases WACC to a point, then starts increasing it. 90003 90002 90003 90002 90005 26.Which has a greater impact on a company’s DCF valuation — a 10% change in revenue or a 1% change in the discount rate? 90006 90003 90002 90003 90002 You should start by saying, «it depends» but most of the time the 10% difference in revenue will have more of an impact. 90003 90002 90003 90002 That change in revenue does not affect only the current year’s revenue, but also the revenue / EBITDA far into the future and even the terminal value.90003 90002 90003 90002 90005 27.What about a 1% change in revenue vs. a 1% change in the discount rate? 90006 90003 90002 90003 90002 In this case the discount rate is likely to have a bigger impact on the valuation, though the correct answer should start with, «It could go either way, but most of the time …» 90003 90002 90003 90002 90005 28.How do you calculate WACC for a private company? 90006 90003 90002 90003 90002 This is problematic because private companies do not have market caps or Betas.In this case you would most likely just estimate WACC based on work done by auditors or valuation specialists, or based on what WACC for comparable public companies is. 90003 90002 90003 90002 90005 29.What should you do if you do not believe management’s projections for a DCF model? 90006 90003 90002 90003 90002 You can take a few different approaches: 90003 90002 • You can create your own projections. 90003 90002 • You can modify management’s projections downward to make them more conservative.90003 90002 • You can show a sensitivity table based on different growth rates and margins and show the values ​​assuming managements ‘projections and assuming a more conservative set of numbers. 90003 90002 90003 90002 In reality, you’d probably do all of these if you had unrealistic projections. 90003 90002 90003 90002 90005 30.Why would you not use a DCF for a bank or other financial institution? 90006 90003 90002 90003 90002 Banks use debt differently than other companies and do not re-invest it in the business -they use it to create products instead.Also, interest is a critical part of banks ‘business models and working capital takes up a huge part of their Balance Sheets — so a DCF for a financial institution would not make much sense. 90003 90002 90003 90002 For financial institutions, it’s more common to use a dividend discount model for valuation purposes. 90003 90002 90003 90002 90005 31.What types of sensitivity analyses would we look at in a DCF? 90006 90003 90002 90003 90002 Example sensitivities: 90003 90002 • Revenue Growth vs.Terminal Multiple 90003 90002 • EBITDA Margin vs. Terminal Multiple 90003 90002 • Terminal Multiple vs. Discount Rate 90003 90002 • Long-Term Growth Rate vs. Discount Rate 90003 90002 90003 90002 And any combination of these (except Terminal Multiple vs. Long-Term Growth Rate, which would make no sense). 90003 90002 90003 90002 90005 32. A company has a high debt load and is paying off a significant portion of its principal each year. How do you account for this in a DCF? 90006 90003 90002 90003 90002 Trick question.You do not account for this at all in a DCF, because paying off debt principal shows up in Cash Flow from Financing on the Cash Flow Statement — but we only go down to Cash Flow from Operations and then subtract Capital Expenditures to get to Free Cash Flow. 90003 90002 90003 90002 If we were looking at Levered Free Cash Flow, then our interest expense would decline in future years due to the principal being paid off — but we still would not count the principal repayments themselves anywhere.90003.90000 Beyond Accuracy: Precision and Recall | by Will Koehrsen 90001 90002 A typical ROC curve is shown below: 90003 Receiver Operating Characteristic Curve (Source) 90002 The black diagonal line indicates a random classifier and the red and blue curves show two different classification models. For a given model, we can only stay on one curve, but we can move along the curve by adjusting our threshold for classifying a positive case. Generally, as we decrease the threshold, we move to the right and upwards along the curve.With a threshold of 1.0, we would be in the lower left of the graph because we identify no data points as positives leading to no 90005 true positives 90006 and no 90005 false positives 90006 (TPR = FPR = 0). As we decrease the threshold, we identify more data points as positive, leading to more true positives, but also more false positives (the TPR and FPR increase). Eventually, at a threshold of 0.0 we identify all data points as positive and find ourselves in the upper right corner of the ROC curve (TPR = FPR = 1.0). 90003 90002 Finally, we can quantify a model’s ROC curve by calculating the total Area Under the Curve (AUC), a metric which falls between 0 and 1 with a higher number indicating better classification performance. In the graph above, the AUC for the blue curve will be greater than that for the red curve, meaning the blue model is better at achieving a blend of precision and recall. A random classifier (the black line) achieves an AUC of 0.5. 90003 90002 We’ve covered a few terms, none of which are difficult on their own, but which combined can be a little overwhelming! Let’s do a quick recap and then walk through an example to solidly the new ideas we learned.90003 90002 90015 Four Outcomes of Binary Classification 90016 90003 90018 90019 90015 True positives: 90016 data points labeled as positive that are actually positive 90022 90019 90015 False positives: 90016 data points labeled as positive that are actually negative 90022 90019 90015 True negatives: 90016 data points labeled as negative that are actually negative 90022 90019 90015 False negatives: 90016 data points labeled as negative that are actually positive 90022 90035 90002 90015 Recall and Precision Metrics 90016 90003 90018 90019 90015 Recall: 90016 ability of a classification model to identify all relevant instances 90022 90019 90015 Precision: 90016 ability of a classification model to return only relevant instances 90022 90019 90015 F1 score: 90016 single metric that combines recall and precision using the harmonic mean 90022 90035 90002 90015 Visualizing Recall and Precision 90016 90003 90018 90019 90015 Confusion matrix: 90016 shows th e actual and predicted labels from a classification problem 90022 90019 90015 Receiver operating characteristic (ROC) curve: 90016 plots the true positive rate (TPR) versus the false positive rate (FPR) as a function of the model’s threshold for classifying a positive 90022 90019 90015 Area under the curve (AUC): 90016 metric to calculate the overall performance of a classification model based on area under the ROC curve 90022 90035 90002 Our task will be to diagnose 100 patients with a disease present in 50% of the general population.We will assume a black box model, where we put in information about patients and receive a score between 0 and 1. We can alter the threshold for labeling a patient as positive (has the disease) to maximize the classifier performance. We will evaluate thresholds from 0.0 to 1.0 in increments of 0.1, at each step calculating the precision, recall, F1, and location on the ROC curve. Following are the classification outcomes at each threshold: 90003 Outcome of model at each threshold 90002 We’ll do one sample calculation of the recall, precision, true positive rate, and false positive rate at athreshold of 0.5. First we make the confusion matrix: 90003 Confusion Matrix for Threshold of 0.5 90002 We can use the numbers in the matrix to calculate the recall, precision, and F1 score: 90003.