Mining Econ Final Review

The five types of cash flows

(1) a single cash flow (2) a uniform series (3) a linear gradient series (4) a geometric gradient series (5) an irregular series

Cash flow

(revenue - cost) over time

Effects of Inflation

- Produces distorted investment criteria if not handled properly... investment criteria must be based on monetary values that have the same purchasing power. - Inflation has a significant effect on project economics, especially on an after-tax basis (the problem is caused by depreciation, which is based on historical expenses and is not indexed. - Inflation also has an effect on debt financing... it lowers the cost of debt because interest and principal payments are remitted in current money. This effect must be corrected in the assessment of the cost of capital.

Weighted Average Cost of Capital (WACC)

- is a calculation of a firm's cost of capital in which each category of capital is proportionately weighted; - it incorporates the required rates of return of the firm's lenders and investors and the particular mix of financing sources that the firm uses; - It is the weighted average of returns required by investors and lenders; - all sources of capital, including common stock, preferred stock, bonds and any other long-term debt, are included in a WACC calculation; - a firm's WACC increases as the beta and rate of return on equity increase, as an increase in WACC denotes a decrease in valuation and an increase in risk; and - to calculate WACC, multiply the cost of each capital component by its proportional weight and take the sum of the results.

Cost of capital

- the cost of funds used for financing a business; equity capital and debt capital. - refers to the opportunity cost of making a specific investment. - it is the rate of return that could have been earned by putting the same money into a different investment with equal risk. - the cost of capital is the rate of return required to persuade the investor to make a given investment. - lenders also require certain returns which can be interpreted as a cost.

Advantages if IRR

-provides a single figure which can be used as a measure of project value -IRR is expressed as a percentage value. - the IRR is determined internally for each project and is a function of the magnitude and timing of the cash flows. -the IRR eliminates the need to have an external interest rate supplied for calculation purposes

Some common pitfalls in NPV

1) Negative cash flow seems to be favorable when discount rate is high Solution: include explicit provision for financing any losses after the initial investment, i.e., explicitly calculate the cost of financing such losses. 2) Adjusting risk factor by adding a premium to the discount rate Solution: A rigorous approach to risk requires identifying and valuing risks explicitly, e.g. by actuarial and explicitly calculating the cost of financing any losses incurred. 3) Lacks the intuitive appeal of payback 4) Doesn't capture managerial flexibility

Straight Line Depreciation Method

1. Find the Total Cost (TC) of the asset: TC = cost + shipping + installation 2. Find the depreciable value: Depreciable value = TC - salvage value 3. Find the yearly depreciation: = depreciable value ÷ years of expected life Calculated by M=(K-S)/N where N is number of years expected of the machine

Reasons for Replacement

1. The current asset, which we call the defender, has developed several deficiencies, including: • high set-up cost, • excessive maintenance expense, • declining productivity, • high energy cost, • limited capability, and • physical impairment. 2. Potential replacement asset(s), which we call the challengers, are available which have a number of advantages over the defender, including: • new technology that is quicker to set-up and easier to use, • along with having lower labor cost, • lower maintenance expense, • lower energy cost, • higher productivity, and • additional capabilities. 3. A changing external environment, including: a) changing user and customer preferences and expectations, b) changing requirements, c) new, alternative ways of obtaining the functionality provided by the defender, including the availability of leased equipment and third-party suppliers, and d) increased demand that cannot be met with the current equipment - either supplementary equipment or replacement equipment is required to meet demand.

Risk Premium

= expected return - risk-free rate

ramp down

It would be the present (value of gradient 1 equal to the same number of years as the ramp up - the present value of the ramp up) multiplied by the present value of the year before ramping down occurs.

Annuity

A = series of consecutive, equal, end-of-period amounts of money. Also A is called the annual worth (AW), annuity, and equivalent uniform annual worth (EUAW); dollars per year, euros per month A occurs with the same value in each interest period for a specified number of periods Ordinary Annuity: Payments are required at the end of each period. Annuity Due: Payments are required at the beginning of each period.

Typical capital budgeting process

A Capital Budgeting Process should: • Account for the time value of money • Account for risk • Focus on cash flow • Rank competing ​projects appropriately • Lead to investment decisions that maximize shareholders' wealth Capital budgeting decision methods essentially compare a project's net investment with its net cash flows

Capital recovery factor (CRF)

A Capital Recovery Factor (CRF) converts a present value into a stream of equal annual payments over a specified time, at a specified discount rate (interest).

Reasons for Capital Expenditure (Capex)

A capital expenditure is an amount spent to acquire or improve a long-term asset such as equipment or buildings. Usually, the cost is recorded in an account classifiedasProperty, Plantand Equipment. The cost (except for the cost of land) will then be charged to depreciation expense over the useful life of the asset. • Expansion - grow the company, add reserves • Replacement - new truck fleet, new mill equipment • Renewal - overhaul • Other - exploration, research and development, safety - strategic-e.g., get experience in a country - intangible returns

Real and current discount rates

A quoted discount rate is a current or "market" rate, r, which contains the effects of inflation, ρ, and the constant (real) discount rate, r* where roh is the effect of inflation calculated by 1-r=(1+roh)(1+r*) or... r*=(r-roh)/1+roh for continuous r*=r-roh

The Balance Sheet

A statement of the assets, liabilities, and capital of a business or other organization at a particular point in time, detailing the balance of income and expenditure over the preceding period.

Tax shields

A tax shield is the reduction in income taxes that results from taking an allowable deduction from taxable income. For example, because interest on debt is a tax- deductible expense, taking on debt creates a tax shield. The tax shield is equal to M(tax rate)+ I(tax rate) where M is Depreciation and I is interest payments. This is due to the fact that it is tax deductable.

Finding an annuity given gradient

A= (1/i) -(N/((1+i)^n)-1) where gradient G = 1/i Uniform Gradient Series (G) exists when cash flows either increase or decrease by a fixed amount in successive periods. if there is a base annuity (A') then that must be added to the equation to get the total annuity. If decreasing gradient you subtract.

Equivalent annual cost

A=P[r/(1-(1+r)^-n)] Useful in replacement analysis or selection of alternatives.

Annual Percentage Rate (APR)

APR reflects the nominal rate of interest to be repaid or earned each year. does not include the effects of compounding

Methods of project evaluation

Accounting Rate of Return Profitability Index Internal Rate of Return Net Present Value Payback Period

Advantages and disadvantages of simulation analysis

Advantages: • Reflects the probability distributions of each input. • Shows range of NPVs, the expected NPV, σ , and coefficient of variation • Gives an intuitive graph of the risk situation. Disadvantages of simulation analysis: • Decision trees and Monte Carlo simulation are powerful tools for analyzing decision making situations that involve risk. - Use of probability distributions permits the engineer to get an overall picture of risk. • A drawback of these methods is that specifying the probability distribution of outcomes can be challenging, and at times, highly subjective. • Difficult to also specify correlations. • If inputs are bad, output will be bad: "Garbage in, garbage out." • Despite this drawback, decision trees and Monte Carlo simulation are widely used.

Perpetuities

An infinite sequence of annuities (a constant stream of identical cash flows with no end) assuming infinit cash flow

Financial Statements: Glossary

Capital Costs (K): are fixed, one-time expenses incurred on the purchase of land, buildings, construction, and equipment used in the production of goods or in the rendering of services. Put simply, it is the total cost needed to bring a project to a commercially operable status. Working Capital (W): measures how much in liquid assets a company has available to build its business. The number can be positive or negative, depending on how much debt the company is carrying. Working capital is calculated as: Working Capital = Current Assets - Current Liabilities. Revenue (R): (net sales) is the income that a company receives from its normal business activities, usually from the sale of goods and services to customers. Operating Costs (C): are expenses associated with the maintenance and administration of a business on a day-to-day basis. The operating cost is a component of operating income and is usually reflected on a company's income statement. Royalties (Q): a royalty is a payment made by one party (the "licensee") to another that owns a particular asset (the "licensor") for the right to ongoing use of that asset. A royalty interest is the right to collect a stream of future royalty payments. Depreciation (M): a noncash expense that reduces the value of an asset as a result of wear and tear, age, or obsolescence. Most assets lose their value over time (in other words, they depreciate), and must be replaced once the end of their useful life is reached.

Replacement analysis

Cash Flow Approach: Treats the proceeds from sale of the old machine as a down payment toward purchasing the new machine. Opportunity Cost Approach: Treats the proceeds from sale of the old machine as the investment required to keep the old machine.

Investment in Operating Business Activities

Concerns investment opportunities encountered in an operating business. For example: • Choice of appropriate processes • Choice of appropriate equipment • Equipment overhaul or replacement • Expansion of operating activities Factors to consider • Benefits (future savings or increased revenue) • Sunk costs • Income tax effects • Planning period • Effects of technological improvements

Operating Costs

Costs experienced continually over the useful life of the project activity. They are of two broad categories: • Fixed Costs (FC) • Variable Costs (VC)

Scenario analysis vs. Sensitivity analysis

Differences from Sensitivity Analysis: • Allows you to change more than one variable at a time. • Look at a group of scenarios (best case, base case, and worst case) for example worst case - what if all variables change against us by 20%... • Includes probability estimates of ​each scenario.

Net Present Cost (NPC) of an alternative

Discrete cash flow NPC=K+C(P/ A,r,N) Continuous cah flow NPC= K=C[P/A,r,0,T] where K = capital cost C = annual operating costs N = service life of alternative

Effective Annual Interest Rate (EAR)

EAR accounts for the number of compounding periods and adjusts the annualized interest rate for the time value of money EAR is a more accurate measure of the rates involved in lending and investing EAR= (1+(quoted interest rate/compound periods))^compounding periods-1 used to calculate r sub e which is the effective annual rate

Economic service life

Economic Service Life (ESL) is the useful life of a Defender, or a Challenger, that results in the minimum Equivalent Annual Cost (EAC).

Sensitivity and Risk Analyses

Estimates • Estimates are best guesses or most likely values. An estimate is uncertain because: • There is incomplete information at the time of evaluation • Project parameters are realized in the future, and the investor has no control over them • Under conditions of uncertainty, investment decisions should be based on: • Expected value analysis - return that investor expects to get, based on his/her estimates • Sensitivity analysis - sensitivity of project performance to variations in estimates • Risk analysis - economic risk associated with an investment project • Economic Uncertainty - Uncertainty of financial outcome, reflecting the uncertainty of estimates • Economic Risk - Possibility of financial loss due to the possible occurrence of undesirable outcomes

Continuous compounding

F=Pe^rt (EAR) equivalent for continuous compounding is given by r sub e= e^r-1

Solving for the number of periods

FV = PV(1+i)^n ➢(FV/PV) = (1+i)^n Move PV to the left ➢ln(FV/PV) = n ln(1+i) Take the natural logs (ln) on both sides ➢n = [ln(FV/PV)]/[ln(1+i)] Move ln(1+i) to the left, switch sides

Financial Statements

Financial statements are summaries of the operating, financing, and investment activities of a firm. The financial statements of publicly-traded firms must be audited at least annually by independent public accountants.

Internal Rate of Return (IRR)

IRR is used to evaluate the attractiveness of a project or investment. In variable form IRR is shown as r* Rule: If the IRR of a new project exceeds a company's required rate of return, that project is desirable. If IRR falls below the required rate of return, the project should be rejected. Besides NPV, IRR is probably the most common evaluation technique used in the minerals industry. The IRR or yield for an investment is the discount rate that equates the present value of the expected cash outflows with the expected inflows or the interest rate that makes the NPV = 0 When borrowing, a low IRR is preferred on the loan.

Discounted Cash Flow

If future cash flow is discounted, we can have cash flow in terms of present value • DCF considers the time value of money and applies it to the inflow and outflow of money occurred in the future. • DCF is a tool that enables us to compare the future cash flow with the present value of money. • Different investment projects have different cash flows that happen at different time intervals in the future and DCF can give an assessment to decide which project is more profitable.

Impact of interest rates on PV

If the interest rate (or discount rate) is higher (say 9%), the PV will be lower. If the interest rate (or discount rate) is lower (say 2%), the PV will be higher.

Simple Interest

Interest earned only on the original investment

Interest in the tax shield

Interest, I, and principal, P, are cash paid to the lender The tax shield I (= interest paid out * tax rate) is cash

Equivalent Annual Value Method (EAV)

Involves calculating the annual cash flow of an annuity that has the same life as the project and whose present value equals the net present value of the project. Steps: • Calculate the NPV of each project over its live using the appropriate cost of capital. • Divide the NPV of each positive NPV project by the PVIFA at the given cost of capital and the project's live to get the EAV for each project. • Select the project with the highest EAV. If the projects differ only by costs only it's called the Equivalent Annual Cost (EAC) method

The Statement of Cash Flows

Is a financial statement that shows how changes in balance sheet accounts and income affect cash and cash equivalents (non-cash items). Specifically, there are two ways to calculate the cash flow statement: 1. Examine changes in the balance sheet accounts 2. Add by non-cash items to net income.

Capital Cost Allowance (CCA)

Is the means by which Canadian businesses may claim depreciation expense for calculating taxable income under the Income Tax Act (Canada). In other words, CCA is the amount of write-off on depreciable assets allowed by Canada Revenue Agency (CRA) against taxable income. Uses declining balance method The tax shield associated with CCA is =tax rate(CCA) The half-year rule consists in adding only one-half of the purchase cost of the asset to the asset class and use it to compute CCA in the year of purchase. In other words, only HALF of the depreciation value (or half of the CCA rate) is realized in the year of purchase.This rule is to discourage the last minute purchase. Regardless when the asset is actually purchased, the deduction is the same. The remaining half of the purchase cost is added to the asset class in the next year.

After-tax cash flow to equity (FTE)

K +W =E + L where W is working capital, K is capital costs, E is equity and L is Loans. cash flow is equal to f=-E+R-C-T-P-I

Simulation Analysis

Monte Carlo simulation is a further attempt to model real- world uncertainty. A Monte Carlo method is a technique that involves using random numbers and probability to solve problems. They are methods for simulation of mathematical models, in which random numbers are involved. It is a computerized version of scenario analysis which uses continuous probability distributions. Monte Carlo simulation can be very useful when analyzing complex problems characterized by multiple sources of risk. End result: Probability distribution of NPV and IRR based on sample of simulated values.

Frequency of compounding

More frequent compounding will generate higher interest income for the savers if the annual interest rate is the same. if interest is given to you as a yearly rate but it is compounded every month you would divide the interest rate by 12 and multiply the number of years by 12 in the discount equation.

Variable discount rates in NPV

NPV= N sum t=0 of net cash outflow at a time/ (1+r1)(1+r2)....(1+rN)

Present value and gradient

P total= P annuity + P gradient

Present value of delayed annuity

P=A/r[((1+r)^-a)-((1+r)^-b)] where a is when the annuity payments start and b when they end.

Continuous Discounting

P=Fe^-rt

Diversification

Portfolio diversification is the investment in several different asset classes or sectors. • Diversification is not just holding a lot of assets. • For example, if you own 50 mining stocks, you are not diversified. • However, if you own 50 stocks that span 20 different industries, then you are diversified. • There are benefits to diversification whenever the correlation between two stocks is less than perfect (p < 1.0). • If two stocks are perfectly positively correlated, then there is simply a risk-return trade-off between the two securities. Principle of diversification = spreading an investment across a number of assets eliminates some, but not all of the risk

Present value

Present value calculations allow us to determine the value today of a stream of cash flows to be reconsidered /paid in the future, by taking into account the time value of money Most commonly known as P but also P is referred to as present worth (PW), present value (PV), net present value (NPV), net present worth (NPW), discounted cash flow (DCF), and capitalized cost (CC); monetary units, such as dollars.

Pros and cons of Payback Period

Pros: • It is a simple method to apply. • It identifies how long funds are committed to a project. • Payback period as a tool of analysis is often used because it is easy to apply and easy to understand for most individuals. • When used carefully or to compare similar investments, it can be quite useful. • It can be used to compare which investment is better, but may just mean less loss! Cons: • It does not properly account for the time value of money, inflation, risk, financing or other important considerations. • Inferior to discounted cash flow techniques because it fails to account for the magnitude and timing of all the project's cash flows. • An implicit assumption in the use of payback period is that returns to the investment continue after the payback period. Payback period does not specify any required comparison to other investments or even to not making an investment. • Does not consider how profitable a project will be, just how quickly outlay will be recovered.

Real Options Analysis

Real options exist when managers can influence the size and risk of a project's cash flows by taking different actions during the project's life in response to changing market conditions. • The option to wait before investing the option to delay) • The option to make follow-on investments • The option to abandon a project • The option to vary output or production methods • The option to expand a project • Two key elements in strategic options and their valuation: 1. New information arrives over time. 2. Decisions can be made after receiving new information. • An option's value is at least the increase in NPV less the option's cost • But the real option may be worth more if it also reduces project risk (e.g. abandonment)

Salvage value

S=K(1 - d)^n where d is depreciation rate, and k is asset value (initial)

The Income Statement

Shows the company's revenues and expenses during a particular period. It indicates how the revenues (money received from the sale of products and services before expenses are taken out, also known as the "top line") are transformed into the net income (the result after all revenues and expenses have been accounted for, also known as "net profit" or the "bottom line").

Activity-based depreciation

Takes into account an asset's use in terms of production, for example: • Items produced • Miles Driven • Hours operated • Number of times it has performed a specific operation Calculated by M sub j= ((K-S)/U)*u sub j where U= total output over assets life where u sub j= units produced in time j

Minimum Acceptable Rate of Return (MARR)

That is, MARR is a reasonable rate of return (percent) established for evaluating and selecting alternatives. MARR is an opportunity cost. An investment is justified economically if it is expected to return at least the MARR. • MARR is established by the financial managers of the firm. • MARR is fundamentally connected to the cost of capital. • Both types of capital financing are used to determine the WACC and the MARR. • MARR usually considers the risk(s) inherent to a project.

Measuring systematic risk

The Beta coefficient measures the relative systematic risk of an asset. • Negative beta. A beta less than 0, which would indicate an inverse relation to the market, is possible but highly unlikely. However, some investors believe that gold and gold stocks should have negative betas because they tended to do better when the stock market declines. • Beta of 0. Basically, cash has a beta of 0. In other words, regardless of which way the market moves, the value of cash remains unchanged (given no inflation). MINE 396 - Engineering Economics 77 Measuring systematic risk • Beta between 0 and 1. Companies with volatilities lower than the market have a beta of less than 1 (but more than 0). As we mentioned earlier, many utilities fall in this range. • Beta of 1. A beta of 1 represents the volatility of the given index used to represent the overall market, against which other stocks and their betas are measured. The S&P 500 is such an index. If a stock has a beta of one, it will move the same amount and direction as the index. So, an index fund that mirrors the S&P 500 will have a beta close to 1. • Beta greater than 1. This denotes a volatility that is greater than the broad-based index. Again, as we mentioned above, many technology companies on the Nasdaq have a beta higher than 1. • Beta greater than 100. This is impossible, as it essentially denotes a volatility that is 100 times greater than the market. If a stock had a beta of 100, it would be expected to go to 0 on any decline in the stock market. If you ever see a beta of over 100 on a research site it is usually the result of a statistical error, or the given stock has experienced large swings due to low liquidity, such as an over-the-counter stock. For the most part, stocks of well- known companies rarely ever have a beta higher than 4.

Sensitivity Analysis: Benefits and Shortcomings

The benefits of sensitivity graphs: • it can be used to select key parameters in an economic analysis. • it is easy to understand and communicates a lot of information in a single diagram. • useful for determining the importance of parameters. The shortcomings of sensitivity graphs: • they are valid only over the range of parameter values in the graph. • they do not consider the interaction between two or more parameters (ignores relationships among variables). • nice idea, but it really states the obvious, especially if the model is linear. • it assumes that the parameters can vary independently of each other, which is usually never the case. • it does not consider all combinations of input parameters. • often parameters are correlated (e.g., copper and gold prices may be correlated).

Consumer Price Index (CPI)

The current cost of the bundle of goods and services is compared to the cost of the same bundle in a base year. 100 * ((current $) /($ in base year))

Scenario analysis

The determination of what happens to NPV estimates when we ask 'what if' questions. • Examines several possible situations, usually: worst (or "pessimistic") case, most likely (or "expected") case, and best case (or "optimistic") outcome. • Provides a range of possible outcomes. • The best-case scenario will incorporate future parameters that are more favorable to the project success than they appear at the time of evaluation. • The worst-case scenario is the analysis with less favorable values such as low unit sales, low unit price, high variable cost per unit, high fixed cost etc. • The most-likely-case scenario (base case) is the analysis with the most likely inputs and outputs from the point of view of the project evaluator. • The NPV under the worst and the best conditions are then calculated and compared with the expected, or the base case, NPV.

depreciation

The fact of taking away part of the value of a piece of machinery every year to express its loss of value. In effect, depreciation is the transfer of a portion of the asset's cost from the Balance Sheet to the Income Statement during each year of the asset's life. The allocation of asset cost is done by deducting Depreciation (M) from net revenue to reduce taxes. Depreciation M is NOT cash.

What is Financial Analysis

The main objective of the financial analysis is to look into the financial benefits, costs and profitability of the project from the investor's perspective. includes taxes, duties, subsidies and does not include environmental impacts and their costs

Weird notation

The notation (F/P, N, r) or (F/P, t, r) is a functional style which is interpreted as "a factor that gives the future value for a unit present value for a given interest rate and time period."

the replacement problem

The question is: when to replace the asset? Shall the Defender be replaced now or be maintained for one or more periods?

Security Market Line

The security market line (SML) is a line drawn on a chart that serves as a graphical representation of the capital asset pricing model (CAPM), which shows different levels of systematic, or market, risk of various marketable securities plotted against the expected return of the entire market at a given point in time.

The Statement of Retained Earnings

The statement explains the changes in a company's retained earnings over the reporting period. They break down changes in the owners' interest in the organization, and in the application of retained profit or surplus from one accounting period to the next. Line items typically include profits or losses from operations, dividends paid, issue or redemption of stock, and any other items charged or credited to retained earnings.

Sinking fund factor (SFF)

This factor determines how much must be deposited each period in a uniform series, A, for n periods at i interest per period to yield a specified future sum. The SFF is typically used to determine how much money (annuity) must be set aside each period in order to meet a future monetary obligation.

Present worth factor

This factor finds the equivalent present value, P, of a series of end-of-period payments, A, for n periods at i interest per period.

NPV Discount Rate

Two ways to determine the right Discount Rate: (A) by the firm's Weighted Average Cost of Capital (e.g. bank lending rate) (B) by deciding the rate which the capital needed for the project could return if invested in an alternative venture (or call reinvestment rate) • When analyzing projects in a capital constrained environment, it may be appropriate to use the reinvestment rate rather than the firm's weighted average cost of capital as the discount factor. • It reflects the opportunity cost of investment, rather than the possibly lower cost of capital.

Discounting

discounting brings a future sum of money to the present value present value= future value/((1+discount rate)^number of periods) assume periods are annual if not specified

WACC (weighted average cost of capital)

WACC= E/(E+D)×r_e+D/(E+D)×(1-t)×r_d E - Market value of equity D - market value of debt Re - cost of equity Rd - cost of debt T - marginal tax rate

Importance of WACC

Weighted average cost of capital is the discount rate used in calculation of net present value (NPV) and other valuations models such as free cash flow valuation model. It is the hurdle rate in the capital budgeting decisions. WACC represents the average risk faced by the organization. It would require an upward adjustment if it has to be used to calculate NPV of projects which are riskier than the company's average projects and a downward adjustment in case of less risky projects. Further, WACC is after all an estimation. Different models for calculation of cost of equity may yield different values.

Finding IRR

You set the NPV equation equal to zero and find the rate of return or r*= (payoff over one time period/ initial investment)-1 The IRR is compared to the required rate of return (k): • If IRR > k, the project should be accepted. • If IRR < k, the project should be rejected. • If IRR = k, indifferent to such a project. found graphically by the x-intercept of the NPV vs discount rate graph

Present value during fast and slow ramp up

You would first calculate the present value where the gradient is increasing either fast (.5) or slow (lower than 5) then calculate the rest of the present value using a gradient of 1 for the range of years where it is constant.

Inflation adjustment in NPV

You would multiply the top and bottom of the NPV equation by (1+ inflation rate)^time period Inflation should not change project value To make this so: • Discount real cash flows at the real discount rate. • Discount current cash flows at the current (market) discount rate.

Modified Internal Rate of Return (MIRR)

a capital budgeting method that converts a project's cash flows using a more consistent reinvestment rate prior to applying the IRR decision rule • Assumes that the project's cash flows are reinvested at the cost of capital, not at the IRR. • This slight difference, makes the MIRR more accurate than the IRR. • Controls for some problems with IRR • A project's Modified Internal Rate of Return (MIRR) is the interest rate equating a project's investment costs with the terminal value of the project's net cash flows • The present value of a project's investment costs is called the Beginning Value (BV) • The future value of a project's net cash flows is called the Terminal Value (TV) is given by: BV=TV/(1+MIRR)^n

Benefit-Cost Ratio (aka: Profitability Index)

calculated by dividing the present value of the future net cash flows by the initial cash outlay. That is, it is the ratio of the sum of the present value of future benefits to the sum of the present value of the future capital expenditures and costs. the decision rule they follow is to invest in the project when the index is greater than 1.0

required return

calculated by risk free rate + beta(expected return-free rate)

Geometric gradient series

cash flows that increase or decrease by a fixed percentage Pg = A1{1- [(1+g)/(1+i)]n}/(i-g) if first anuity includes gradient then multily A1 by (1+g) if g = i, Pg = A1n/(1+i) where A1 is cash flow in the first period and g is the rate of increase or decrease It is common for annual revenues and annual costs such as maintenance, operations, and labor to go up or down by a constant percentage

What is Economical analysis

estimates project benefits and returns from the perspective of national economy and assesses the effects that the project will have on the overall economy of the country includes social and environmental costs and benefits of the project while excluding transfer payments

Finding interest rate for continuous compounding

i=(e^r)-1

Investment Costs

include initial costs and replacement costs

Project Cycle

includes the following steps i. Identification ii. Preparation iii. Appraisal iv. Negotiation/Approval v. Implementation, and vi. Evaluation

Compound Interest

interest earned on both the principal amount and any interest already earned

Market risks

interest rates, metal prices, and inflation. aka: systematic risks

Net Present value

is simply the difference between the PV of the net cash flows (NCF) from an investment, discounted at the required rate of return, and the initial investment outlay NPV is an indicator of how much value an investment or project adds to the value of the firm. If your first cash flow occurs at the beginning of the first period, the first value must be added to the NPV result, not included in the values arguments. given by n sum t=1 of (net cash outflow at a time/(1+ required rate of return))-initial cash outlay on project outlay only occurs at the beginning of the project

Opportunity Cost

is the amount lost by not using a resource (labor, capital or any factor of production) in its best alternate use

Accounting Rate of Return (ARR)

is the average after-tax income from a project divided by the average investment in the project. Ignores the time value of money. If the ARR is less than the required minimum, the investment is rejected. Accept the project only if its ARR is equal to or greater than the required accounting rate of return

Payback Period

measures the time it will take to recoup, in the form of expected future cash flows, the initial investment in a project. Payback period = Amount invested/ Expected annual net cash inflows • If the project's payback period is less than the maximum acceptable payback period, accept the project except in the case of mutually exclusive projects in which case the project with the shortest payback is chosen. • If the project's payback period is greater than the maximum acceptable payback period, reject the project

Capital Asset Pricing Model

stock return=risk free rate+beta(expected rate of return-risk free rate) R_e-R_f=β×[R_m-R_f ] Re = stock return Rm = market return Rf = risk free rate

Non-market risks

strikes, war, results of mining exploration, political risks, technical or operational failures, construction problems. aka: unsystematic risks Unsystematic risk is essentially eliminated by diversification, so a portfolio with many assets has almost no unsystematic risk

Constrained optimization

the process of optimizing an objective function with respect to some variables in the presence of constraints on those variables.

Inflation

the rate at which the general level of prices for goods and services is rising and, consequently, the purchasing power of currency is falling over a period of time. • High inflation is caused by an increase in the money supply. • Low to moderate inflation can be caused by changes in demand or scarcity of supply as well as changes in money supply. Consumer Price Index (CPI) - a weighted average of a set of goods and services a "typical" consumer might buy normalized to a particular year. Used to track inflation Producer Price Index (PPI) - weighted averages of prices for specific commodities, services, for goods at stages of production, and prices paid in specific industries

Book value

the undepreciated value of the asset (i.e., what's left after depreciation is applied). Difference between book value and market value: • The book value of an asset is its original purchase cost, adjusted for any subsequent changes, such as for impairment or depreciation. • Market value is the price that could be obtained by selling an asset on a competitive, open market.

Future value

value or amount of money at some future time. Also F is called future worth (FW) and future value (FV); dollars.

interest rate

what a lender charges for use of money. It's the cost of borrowing money. It is the rate of gain received from an investment referred to the Minimum Attractive Rate of Return (MARR), the discounted rate, the expected return, the required rate of return (RRR), and the minimum rate of return for the investment.

Real options valuation

when managers can influence the size and risk of a project's cash flows by taking different actions during the project's life in response to changing market conditions. Includes expansion of a project, cutting losses and delaying projects.

Advantages and Disadvantages of Profitability Index

• Advantages - Closely related to NPV, generally leading to identical decisions • Considers all CFs • Considers TVM - Easy to understand and communicate - Useful in capital rationing • Disadvantages - May lead to incorrect decisions in comparisons of mutually exclusive investments (can conflict with NPV)

Sensitivity of Economic Service Life

• For an asset with non-increasing operating cost, keep the asset as long as it lasts. • If every thing remains the same, a higher interest rate will tend to extend the Economic Service Life (or defer the replacement decision).

Financial Analysis under Uncertainty

• Generally, there are several procedures to analyze the project under uncertainty (methods of describing project risk), such as: • Sensitivity Analysis - a means of identifying the project variables which, when varied, have the greatest effect on project acceptability. • Break-Even Analysis - a means of identifying the value of a particular project variable that causes the project to exactly break even. • Scenario Analysis - a means of comparing a "base case" to one or more additional scenarios, such as best and worst case, to identify the extreme and most likely project outcomes.

Decision tree analysis

• Graphically represent the alternatives available to us in each period and the likely consequences of our actions. • Decision Tree Analysis - develops a probability distribution of NPV given the probabilities of certain events within the project • This graphical representation helps to identify the best course of action. There are four main components in a decision tree: 1. A decision node represents a decision to be made by the decision maker. It is denoted by a square. 2. A chance node represents an event whose outcome is uncertain. It is denoted by a circle. 3. The branches of a tree are the lines connecting nodes from left to right, depicting the sequence of possible decisions and chance events. 4. Finally, the leaves indicate the values, or payoffs, associated with each terminal (rightmost) branch of the decision tree.

Sign changes in the Cash Flows

• IRR evaluates a project correctly when there is an initial negative cash flow, followed by a series of positive ones (-+++). • If the signs are reversed (+---), that will change the accurateness of the IRR calculation. • If there are multiple sign changes in the cash flows (+-+-+) or (-+-+-), your calculation would result in multiple IRRs, also making the project very difficult to evaluate. • According to "Descartes' rule of signs" there can be as many different IRRs as there are changes in the sign of the flows ("- + -" is two sign changes). With more sign changes, there could be more IRRs. • Due to the complexity and uncertainty of which IRR to use, NPV would be the best method of valuation in these cases.

MIRR versus IRR

• MIRR correctly ​assumes reinvestment at opportunity cost = WACC • MIRR avoids the multiple IRR problems • Managers like the rate of return comparisons, and MIRR is better for this than IRR

Winners and losers from inflation

• Nominal interest rate - the interest rate as stated in the contract (i.e., a car loan with 7% interest) • Real interest rate - interest rate after adjustment for inflation • If the inflation rate is higher than expected, borrowers win! The money they pay back has lower real value (less purchasing power) than the money they borrowed. • If the inflation rate is lower, lenders win! The money they receive back from the borrower is worth more than the money they lent.

The facts of life for equipment

• Operating costs per period increase as the number of periods increases - Direct result of obsolescence and/or aging • Capital costs per period decrease as number of periods increases - Think in terms of distributing the capital cost over an increasing number of periods • The value that can be obtained for the equipment (the salvage value) decreases as the number of periods increases

What should the discount rate cover

• Opportunity costs • Transaction costs • Compensate for risk • Cover anticipated inflation

Unequal project lives

• Repeated lives [Chain replication approach] - an alternative can be repeated in the future with the same revenues and costs (and EAV, NPV). A way to compare projects with unequal lives by finding a time horizon into which all the project lives under consideration divide equally, and then assuming each project repeats until it reaches this horizon. - use least common multiple of project lives - the most common approach • Study life [Equivalent Annual NPV (EANPV) approach] - (used if alternatives do not or cannot repeat) choose a length of time for which forecasts are relatively certain. A way to compare projects by finding the NPV of the individual projects, and then determining the amount of an annual annuity that is economically equivalent to the NPV generated by each project over its respective time horizon. - study life may be the time to the end of the shorter-life project and a salvage value may be assumed for the longer-life project - could also consider costs to extend life of the shorter-lived project

Sensitivity analysis

• Sensitivity analysis develops a better understanding of how uncertainty affects the outcome of the evaluation. • A sensitivity analysis reveals how much the NPV or IRR will change in response to a given change in an input variable. • This kind of analysis determines the effect on the NPV with variations in the input variables such as revenues, operating cost, and salvage value etc. • Measures the effect of variations in project parameters on investment decision criteria.

Sensitivity graphs

• Sensitivity graphs are used to assess the effect of one-at-a-time changes in key parameter values on a performance measure. - begin with the "base case" where all the estimated parameters values are used to evaluate the project. - then vary parameters above and below the base case one at a time, holding all other variables fixed. • A graph of the changes in a performance measure brought about by these changes is called a sensitivity graph. • The slopes of the lines show how sensitive the NPV is to changes in each of the inputs. • The steeper the slope, the more sensitive the NPV is to a change in a particular variable. Sensitivity graphs identify the crucial variables that affect the final outcome most. Why is sensitivity analysis useful? • Examines several possible situations, usually worst case, most likely case, and best case. • Provides a range of possible outcomes.

NPV vs. IRR

• Since NPV is an absolute measure, it will rank a project adding more dollar value higher regardless of the original investment required. • IRR is a relative measure, and it will rank projects offering best investment return higher regardless of the total value added. The NPV method is the superior method for mutually exclusive projects. IRR and NPV are fine for Independent investments. Timing and scale problems can cause NPV and IRR methods to rank projects differently In these cases, calculate the IRR of the incremental project - Cash flows of large project minus cash flows of small project - Cash flows of long-term project minus cash flows of short-term project if IRR exceeds the cost of capital then - Accept the larger project - Accept the longer term project

Two Reasons NPV Profiles Cross

• Size(scale)differences. - Smaller project frees up funds sooner for investment. - The higher the opportunity cost, the more valuable these funds, so high discount rate favors small projects. • Timing differences. - Project with faster payback provides more CF in early years for reinvestment. - If the discount rate is high, early CF especially good.

Straight line vs Declining Balance

• Straight line and activity-based depreciation methods are used to make estimates of the true value of assets on a balance sheet (due to their simplicity to calculate). • Which method is applied and how it is applied depend on the assets in question. • Declining balance depreciation methods are used for income tax purposes. Their goal is to depreciate the asset quickly so that more cash is available to the asset owner.

WACC and Beta

• WACC increases as the beta and the rate of return on the equity securities increases (all else constant) • WACC is used as the discount rate in DCF models • Therefore, increasing WACC reduces the firms valuation to reflect the increase in risk

Declining Balance (DB)

• Yearly double declining rate = yearly straight line depreciation rate x 2 • Yearly 150% declining depreciation rate = yearly straight line depreciation rate x 1.5 • First year's depreciation = total cost x yearly depreciation rate • For all other years: Year's depreciation = previous end-of-year book value x yearly depreciation rate For example, if an asset that costs $1,000 is depreciated at 25% each year, the deduction is $250.00 [=$1,000 * 0.25] in the first year; $187.50 [=($1,000 - $250) * 0.25] in the second year; $140.63 [=($750 - $187.50) * 0.25] in the third year; and so forth. • In declining-balance depreciation, the depreciation for the first year is based on the total cost of the asset. • Do not subtract the salvage value from the total cost to find the depreciation in the first year. • The end-of-year book value for any year cannot drop below the salvage value of the asset. • If the depreciation would cause it to drop below the salvage value, the year's ending value will be the salvage value.

Types of replacement decisions

• kept without major change • retired(removed)without replacement •overhauled to improve performance • replaced with another asset

Variable costs

• labor • materials • fuel cost • tax.

Fixed costs

• maintenance • insurance • lease rentals • interest payment on invested capital • certain administrative expense, and • research.

Different uses of depreciation

•Amortization usually refers to an ​intangible asset(e.g.,patent,trademarks,customer lists, etc.). • Depreciation refers a tangible asset (e.g., equipment, truck, warehouse, etc.). • Depletion refers to natural resources.