Hi,

What are the formulas for npv for perpetuity and fixed period? I am seeing all sorts of formulas online and cannot tell which is what I need

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# Net present value calculation for both perpetuity and for fix period

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Hi,

What are the formulas for npv for perpetuity and fixed period? I am seeing all sorts of formulas online and cannot tell which is what I need

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Top answer

Hey there,

If you don't have an MBA, the likelihood of having to use the formula in an interview (at least for MBB) is close to zero. If you did, you'd probably just to have to discount a couple cash flows in the future (i.e. a manual DCF) rather than just calculating the NPV using the NPV formula. If you are applying to different firms, you should check, but again I would be surprised.

Anyways, to answer your question - for a fix period you would just discount each cash flow year by year, see below:

In perpetuity, you just divide the cash flow by the (discount rate - the growth in the cash flow):

Note that if there is no growth in cash flow, then g=0, and the formula is D/r.

0 comments

Vlad

on Nov 08, 2018
Coach

McKinsey / Accenture Alum / Got all BIG3 offers / Harvard Business School

Hi,

Check this presentation:

https://web.iit.edu/sites/web/files/departments/academic-affairs/academic-resource-center/pdfs/NPV_calculation.pdf

I believe you need to understand the concept rather than just apply the formulas. So google can help you here.

Best

Clara

on Jul 27, 2023
Coach

McKinsey | Awarded professor at Master in Management @ IE | MBA at MIT |+180 students coached | Integrated FIT Guide aut

Hello!

Precisely for the high amount of questions (1) asked by my coachees and students and (2) present in this Q&A, I created the “Economic and Financial concepts for MBB interviews”, recently published in PrepLounge’s shop (https://www.preplounge.com/en/shop/prep-guide/economic_and_financial_concepts_for_mbb_interviews).

After +5 years of candidate coaching and university teaching, and after having seen hundreds of cases, I realized that the economic-related knowledge needed to master case interviews is not much, and not complex. However, you need to know where to focus! Hence, I created the guide that I wish I could have had, summarizing the most important economic and financial concepts needed to solve consulting cases, combining key concepts theorical reviews and a hands-on methodology with examples and ad-hoc practice cases.

It focuses on 4 core topics, divided in chapters (each of them ranked in scale of importance, to help you maximize your time in short preparations):

- Economic concepts: Profitability equation, Break even, Valuation methods (economic, market and asset), Payback period, NPV and IRR, + 3 practice cases to put it all together in a practical way.
- Financial concepts: Balance sheet, Income statement/P&L and Performance ratios (based on sales and based on investment), +1 practice case
- Market structure & pricing: Market types, Perfect competition markets (demand and supply), Willingness to pay, Pricing approaches, Market segmentation and Price elasticity of demand, +1 practice case
- Marketing and Customer Acquisition: Sales funnel, Key marketing metrics (CAC and CLV) and Churn, +1 practice case

Feel free to PM me for disccount codes for the guide, and I hope it helps you rock your interviews!

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Net Present Value - NPV

Net Present Value (NPV) – Definition, Examples, And How to Calculate ItDefinition of the Net Present Value (NPV)The net present value (NPV) allows you to evaluate future cash flows based on the present value of money. It is the sum of present values of money in different future points in time. The present value (PV) determines how much future money is worth today. Based on the net present valuation, we can compare a set of projects/ investments with different cash flows over time. This enables us to quantitatively assess a business' attractiveness using a benchmarking of NPVs.Time Value of Money – An Intuitive and Financial ExplanationThe closer future cash flows are to the present, the more valuable your money is. The concept is also known as the time value of money and we provide two explanations below:An intuitive explanation: People will prefer money at present because of risk aversion. Would you rather have $100 today or in a year? Obviously today, because there is a risk that you may not get that $100 in a year. In addition, once you have the money, you again have a decision to either spend right away or wait to spendA financial explanation: Imagine you have $100. How much is it worth in a year? If you do not leave the money in your pocket, you usually have the option to put the money in your bank account at low and almost negligible risk. You will earn interest but may lose value due to inflation. However, the inflation-adjusted interest rate may be 2%, in absolute terms $2. In total your $100 is worth $102 after one year. Now, you can calculate backwards: If you have a future value of $102 in a year, how much is it worth today? It is $102 divided by 1.02 which results in $100 again.How to Calculate The Net Present Value (NPV) PV is the present valueFV is the future valueit is the decimal value of the interest rate for a specific periodn is the number of periods between present and futureThe following is the calculation of the above PV example with $102 future value at an interest rate of 2%, Below, you can find a slightly different version of the above example, in which you receive $102 in two years instead of next year. The two-year investment earns you a theoretical interest two times, which is why you discount twice. The Net Present Value of those two $102 payments in one and two years is simply its sum. Apply NPV Shortcuts to Succeed in Case SituationsIt’s unlikely that you will need to calculate a complex NPV during a case interview because the calculations tend to get overly complicated. But in some cases you can apply some shortcuts as discussed below:1. Perpetuity: the NPV for infinite cash flows (meaning business will generate profits for an infinite period of time)For infinite cash flows, there is a simplified formula: Imagine you have to value a company in a case interview. A common approach is to define the value of a company as the sum of all its discounted future profits. If you assume that a company will have the same profits every year for an indefinite time horizon, you just divide the future value of all profits by the respective discount rate. For instance, if you expect the company to yield $100 every year, the company is worth $2500 (at a discount rate of 4%). To make this more pragmatic, you could assume that the company's profits will grow every year at a certain rate, g. Especially for short-term horizons, defining expected growth is difficult. An approximated growth rate for profits that are far into the future is often around 2%. After a while, every business or product life cycle ends up in a competitive market environment and simply grows at the same rate as the overall economy. The above example recalculated with a continuous growth rate of 2% results in a net present value of $5000 for the company. Notice that the value is twice the value compared to the calculations without a growth. NPV calculations are very sensitive towards changes in inputs. Therefore, a sensitivity analysis is conducted in most cases. To do so, you need to create a range of possible NPVs by using a range of possible growth and discount rates.2. Find the right interest rateFinding the correct discounting factor for NPV calculations is the business of entire banking departments. In general, there is one basic rule: the bigger the risk, the higher the discount rate.The rationale behind this rule is simple: The less you can be sure about receiving future earnings, the less you value them. By increasing the discount rate, the NPV of future earnings will shrink. Discount rates for quite secure cash-streams vary between 1% and 3%, but for most companies, you use a discount rate between 4% - 10%, and for speculative start-up investment, the applied interest rate could reach up to 40%. In case interviews, you could ask for the discount rate directly or estimate it at 10% for most scenarios if the interviewer requests you to approximate it. Why Is The NPV Used By Companies?The NPV is a strong indicator for companies to determine whether a project is profitable or not. As it considers the interest rate and inflation rate (which are usually equal to one another), the real value of money at every year of the project can be considered. If the calculated NPV is positive, the company can argue that the project or investment will be profitable since the earnings generated exceed the costs, all calculated in the present value of dollars. On the other side, if the calculated NPV is negative, it is an indicator for a company to step back from an investment o project since it will result in a net loss. Thus, the NPV can be a valuable tool for companies to evaluate whether to invest in a project or not. Advantages of The Net Present ValueNet Present Value (NPV) offers many advantages, all of which relate to the accuracy of the calculation in determining the actual value of a future amount of money under present conditions. The measurement is based on the time value of money theory, which indicates that a given amount of money is worth more now than the same amount of money in the future. Inflation rates and the rates at which invested money can grow in the future combine to form a discount rate that devalues future money. Because the discount rate is considered in the calculation, one of the main advantages of NPV is that it allows important financial decisions to be made. Business decisions are difficult to make at any scale, from deciding on a key purchase to deciding on a costly new project. It can be foolish to make these decisions without first considering the impact of time on money. For this reason, a calculation that takes into account the relationship between time and money, which is one of the main advantages of NPV, is critical. The perhaps most important advantage of the present value is its usefulness in a business decision. Once the NPV is calculated, the company making the decision needs to only compare current costs with the NPV. For example, a company that has the opportunity to purchase a new factory for $100,000 (USD) should only proceed with the purchase if the present value of future cash flows is greater than $100,000. Otherwise, it would be better for the company to invest the money elsewhere.Limitations of The Net Present ValueIn practice, the concept of net present value is common and is used oftentimes. Nevertheless, there are disadvantages and limitations of this key figure. For example, the calculated value is based on various assumptions. If one or more assumptions do not materialize in practice, the net present value and the actual benefit of an investment may differ. At the same time, the concept is susceptible to influence by the valuer. Individual assumptions can theoretically be changed until the result matches the expectations of the valuer. Consequently, the net present value has the highest informative value when an investor calculates it himself.Example: A company examines the benefits of investing in a new production line. The company assumes that the machine will be used for ten years and consequently plans with ten periods to determine the net present value. However, after seven years, customers no longer demand the product that has been produced. The production line has to be discarded in favor of a new machine. The originally calculated net present value is significantly lower than originally assumed due to the change in the utilization period.Although the net present value is a comprehensive concept, the secondary effects of an investment are not taken into account. The cash flow series used relates directly to the investment examined. The net present value cannot measure synergy effects in other parts of the company or, for example, an improved corporate image.Key TakeawaysUse NPVs to evaluate future cash-flows in today’s time value of moneyBy calculating risk-adjusted NPVs, you can quantitatively compare different investmentsNPVs are used to value a company based on its future profitsThe NPV does not take into account secondary effects of an investment*box-open*Solve the Gravestone case, an interviewer led case which includes a company valuation*box-close*

Valuation

This type of case can either be a subset of an M&A case, in which you need to know a company's worth before purchasing, or a standalone case (rare). For instance, “How much is Pfizer worth today?” In strategy consulting, these questions are rather rarely seen. However, cases where you do need to value something usually start with “How much would you pay for company XY?” The Most Common Methods of Valuing Are the Discounted Cash Flow (DCF) and the Industry Multiple MethodAs these are still case studies meant to fit in an interview round, the interviewer will very likely not ask you to perform an exact and comprehensive valuation analysis. Instead, you may be required to estimate the worth of a product, patent, or a service. You may also have to judge if an offered price is reasonable. Discounted Cash Flow methodThe first valuation method is the Discounted Cash Flow method. This method shows how much money you would have in your savings account at a certain interest rate in order to provide you with the same annual cash flow generated by the company that is being evaluated.Here, you simply divide projected annual cash flows by a discounted rate (or interest rate). Of course, the discount rate of your savings account will be much lower than that of an investment in a company. This is so because the risk you take putting your money in a savings account is much lower than the risk of investing in a company.👉 For more details on how to use the Discounted Cash Flow (DCF) method, have a look at our Net Present Value (NPV) lesson Industry Multiple MethodThe DCF method is limited since it does not take into account additional dimensions other than money (unless you quantify those dimensions into the future cash flows).Football teams, for instance, are often overvalued compared to their generated returns. For such cases, there is another method called the industry multiple method.This method allows you to value a firm by using a metric known of this company and multiplying it by the associated industry multiple. This can be done for similar players in the industry to assess their relative valuations using benchmarking.An example of a multiple ratio is the price-to-book ratio (P/B). This multiple is the ratio of the actual firm valuation (based for example on M&A deals) and the book value of the same firm (value of its assets, which can be found in the balance sheet). If a firm’s assets added up to 200 million, and it was sold for 100 million, the ratio is 0.5 (100 million/200 million).Do this for a set of representative industry players, take the average and you get the average industry multiple. Finally, you multiply the industry multiple with the value of the assets.Other commonly used ratios are the price-earnings ratio (P/E ratio or PER) and the EBITDA ratio.Since you will not be required to calculate the value of an investment on too high a level of detail, it is not necessary to learn values for different interest rates or industry multiples by heart. However, to give you an idea about orders of magnitude:A good guess for an industry multiple is EBITDA*10.Good guesses for interest rates would range from 3% (inflation) to up to 20% for highly speculative investments. Key takeawaysUse the Discounted Cash Flow (DCF) method to value a firm based solely on its expected profits.Use the industry multiple method to double-check if the DCF valuation is reasonable. Sometimes other aspects need to be factored in like brand value, customer loyalty, liabilities, etc.There are several types of industry multiples to choose from. For more precise valuation, choose more types of industry multiples. Practice M&A cases to optimize your valuation-techniques:👉 tkMC Case: Portfolio optimization of a holding company👉 Bain Case: Old Winery

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