Overview • What economic concepts do we draw on? • Capital theory • Discounting (time value of money) • Cost Benefit Analysis • What will we do? Computing Future Values and Present Values • Evaluating Investments Using Different Investment Criteria • And Examples • And Formulas • More formulas • And even more…
Example: Do you rebuild in Burns Lake? • What information do you need?
Capital Theory • Used to evaluate the value of assets and investments • Examples of assets • Durable goods (piece of equipment) • Financial Assets (stocks and bonds) • Land and natural resources (timberland or timber license) • Used to evaluate investments and alternatives-should you invest in new equipment? How much? • Can use to determine how much can you withdraw over time without compromising the value of your asset
Timberland Values • Forests serve as an asset (a store of value) and also generate income • In the standard model (private land) these values are embodied in land values (on which the timber grows) • Estimated value of timberlands as an asset class $35-$40 billion (Reid Carter, Brookfield Asset Management)(2010) • http://www.forestweb.com/Corporate/timberlandInvesting.cfm • http://www.forestlegacy.com/fundamentals/ • The Peel Commission in BC in 1991 estimated that the value of timber in BC ranged from $1 billion to $12 billion • http://www.for.gov.bc.ca/hfd/pubs/Docs/Mr/Rc/Rc001d/V4BP001.pdf
Interest Rates • Interest or rate of return • Payment expected from holding an asset • Types of interest • Simple interest • Earn interest only on principal • Compound interest • Earn interest on principal and accumulated interest • What interest rate to use? • Best alternative (opportunity cost), minimum acceptable rate; real or nominal • What perspective? Private or public?
Why do we use interest rates? • Time preference (social preferences) • Opportunity cost • Accounting for risk and chance of failure • Accounting for inflation
Compounding and Discounting • If you know value today (or series of values) you can compound them forward to some future point in time • This gives you future value • Alternatively you can convert those series of values into what they are worth today-this is present value (invert the formula)
How important is compounding and future value calculations? TFSA’soffer a good example. $5,000 invested in a tax free account today (with no further investment) at 8% and no taxes would yield $50,313 in 30 years $5k $50k if the account did incur taxes the same investment (at a 40% tax rate) would yield only $20,408 in 30 years. $5,000 invested each year for the next 30 years in a tax free account at 8% and no taxes would yield $566,416 in 30 years $150k $566k if the same investment was made in an account that did incur taxes the same investment (at a 40% tax rate) would yield only $321,008 in 30 years (44% less).
. . . . . . Vn = V0 (1 + i) (1 + i) ………………….(1 + i) n - times <=> Vn = V0 (1 + i) n Future Value of a Single Sum: V1 = V0 (1 + i) Here Vnis future value n periods in the future using compounding V2 = V0 (1 + i) (1 + i) V3 = V0 (1 + i) (1 + i) (1 + i)
Divide both sides By (1 + i) n <=> V0 = Vn / (1 + i) n Present Value of a Single Sum: Vn = V0 (1 + i) n Here we calculate the present value, given the future value
Net Present Value: Definition: Present value of revenues minus the present value of costs. Ry Cy n Σ ( ) - NPV = (1+i)y (1+i)y y=0
Conventions • Interest rates are given in yearly percentage rates of change unless otherwise stated. • Costs and revenues occur at the same time of the year. • Interest rates can be given either as a percentage (e.g. 8%) or a decimal value (e.g. 0.08). • Year 0 is now.
Why Do This? • Provides basis on which to make decisions • Evaluate investment decisions-yes or no • Compare alternative investments • Determine proper investment amounts • Establish valuations
$10,000 $2,000 $2,000 $3,000 $2,000 $2,000 $2,000 $5,000 $2,000 $2,000 $2,000 $2,000 $10,000 $5,000 $5,000 $5,000 $5,000 $5,000 Years 0 1 2 3 4 5 6 7 8 9 10 Cost and Revenue Streams: Payment stream #1 Years 0 1 2 3 4 5 6 7 8 9 10 Payment stream #2 Costs are shown in red and revenues in orange
Evaluating Payment Streams with no Discounting Here payment stream #1 is preferred (it pays off more)
Evaluating Payment Streams with a 5% Discount Rate Here payment stream #1 is still preferred (it pays off more) but not as much as in the previous example as it is discounted (note that $2,326 is the NPV)
Evaluating Payment Streams with a 10% Discount Rate Here payment stream #2 is now preferred. Note that payment stream #1 is now negative; this is because the future revenues are discounted more so the upfront costs are proportionately Greater.
Single Sum Annual Series Present or Future Value Terminating Periodic Perpetual Annual Annual Payment Periodic Meaning of Symbols: a = equal annual or periodic payment i = interest rate n = number of years or interest bearing periods t = interval between periodic payments Vo = present (initial) value Vn = future (end) value
Problem 1 • Present value of a Periodic Series (pg. 109-110 in text) p p p p 10 20 30 40 Common in forestry-recurring payments or costs as set intervals In example in book, assume $3,000 in Christmas tree revenues every 10 years and assume a 6% interest rate-what is the present value of this?
Problem 1 associated math p V0= Use formula for present value of a perpetual periodic series (#8) Substitute in the values and determine that the present value is $3,739 This means that if you can earn 6% somewhere else, this is the most you’d pay So if someone offered it to you for $3,500 you’d be interested-but not if they wanted $3,800 (1 + r)t - 1 $3,000 V0= (1 + .06)10 - 1
Problem 2 • How much will I need to make to justify my investment? • Land purchase - $400/ha • Planting cost - $200/ha • Brushing and thinning (in 10 year’s time) - $75/ha • 7% interest rate • Expected harvest in 30 years
Problem 2 associated math • Setting problem up -estimate how much revenue you will need in the future • Calculating future values Land and planting cost -600(1+.07)30 = -$4,567.35 Brushing cost -$75(1+.07)20 = - $290.23 Total revenues needed in 30 years -$4,857.58
The Power of Time Note how much greater revenues have to be the longer you wait; Also notice the reduction in revenue required if you can shorten the harvest period by only one year (you need $318 less)
Decision Rules Rt Ct n Σ ( ) - NPV = (1+r)t (1+r)t t=0 Criteria: a project is acceptable if the NPV exceeds 0. If you have multiple projects, you can rank them in preferred order by NPV (highest to lowest).
n R t n C t ∑ ∑ = 0 (1 + IRR) t (1 + IRR) t t=0 t=0 The IRR is the discount rate at which the present value of revenues minus the present value of costs is zero. Internal Rate of Return [IRR] Therefore, the IRR is unique to each project. Projects are acceptable if IRR is greater or equal to the minimum acceptable rate of return [MAR]. Projects can be ranked by their IRR (highest is best). Typically assume MAR is equal to r, the real discount rate. When IRR=MAR=r(real discount rate) then NPV=0
The benefit/cost ratio (or profitability index) is the present value of benefits divided by the present value of costs, using the investor’s MAR. Benefit/Cost Ratio [B/C Ratio] n R t ∑ (1 + MAR) t PV (Revenues) t=0 B/C Ratio = = n C t PV (Costs) ∑ (1 + MAR) t t=0 If B/C=1 then NPV=); if B/C<1 then NPV<0
Payback Period Payback Period: The payback period is the number of years it takes to recover the invested capital. Note: The payback period does not say anything about the NPV or IRR of an investment. It should therefore only be used as a secondary criterion.
Using NPV to Evaluate Projects So Project D has an NPV of $758, greater than Project N with an NPV of $713 (based on 6% real rate). Both are acceptable (NPV>0); Project D>Project N. Note that the original outlay (expenditure) is included.
Evaluation Dependent on Interest Rate If the interest rate increases to 10%, note that Project D is no longer acceptable (negative NPV of -$36) while Project N is still acceptable (positive NPV).
Calculating the Internal Rate of Return for Project D Accept if IRR is greater than your Minimum Acceptable Rate of Return (MAR)
Calculating the Benefit/Cost Ratio 6,600 200 + (1.06)30 (1.06)15 = 2.60 B/C = 100 + 400 (1.06)5 Here it is acceptable since B/C > 1 (benefits exceed costs)
Payback period For project D, you do not recover outlays until Year 30; for Project N, that happens in Year 8 So payback period for D is 30 years; For N 8 years Not as useful a criteria as it does not tell you about rate of return, or NPV-just when you can recover your expenditures
ApplyingCriteria • Criteria can be used to accept/reject projects • NPV > 0 • B/C > 1 • IRR> MAR • Criteria can also be used to compare and rank • But in some cases ranking might vary depending on criteria • But will also need to take other factors into account…
Present Value ($) Project D: Costs: Revenues: Present Value (Costs) Present Value (Revenues) Interest Rate (%) NPV NPV Interest Rate (%)
Criteria for Project D NPV: Project D has an NPV of $758 (based on 6% real rate). IRR: 9.68% (discount rate where PVrevenues=PVcosts Benefit/Cost ratio (based on 6% real rate): PVrevenues =$1233, Pvcosts=$475; So B/C=2.60
NPV > 0 NPV < 0 IRR= 9.68% Present Value ($) B/C= $1233/$475 =2.60 Project D: Showing NPV, B/C, and IRR B/C > 1 B/C < 1 NPV Pvrevenues =$1233 NPV =$758 Present Value (Costs) PVcosts = $475 Present Value (Revenues) Interest Rate (%) 6% Would accept if MAR chosen is less than IRR
Net Present Value ($) Which project is better depends on criteria IRRN>IRRD but for low interest rates (i<Î) NPVD>NPVN NPV Project D NPV Project N Interest Rate (%) IRR for N IRR for D NPV for D>NPV for N below thisinterest rate -Î
Is One Criteria Preferred? • Turns out that generally NPV, B/C, and IRR agree • But can have inconsistencies between all three • If independent and unlimited budget choose all projects that produce favourable NPV, B/C, IRR • But choice may be influenced by: • Perspective (what are you trying to maximize) • Capital budget • Time period • Type of investment
What method do you (sawmills) use to evaluate investment decisions? • 21% of respondents used more than one method • Payback period was most noted method with <22 months as average period (ranged from 12 – 36 months) Attracting Head Office Investment • 57% noted cost reduction • 19% noted econ return • Many had multiples
Different Criteria Give Different Ranking Depending on rule, you may prefer Spacing (greatest NPV); Planting (Best B/C ratio); or Spraying (highest IRR). But other factors enter into your decision-if you had a limited capital budget (under $20,000), this would affect your choice here: The cost of planting was $16,000-and others were $33,000-$35,000 -so you’d plant.
No One Rule • Generally avoid IRR especially when r is much lower since it can lead to inconsistencies • Where budget is unlimited maximize net benefit (NPV) • Where limited budget maximize B/C but… • “Ideal would be some foolproof guideline like “Choose projects in order of decreasing NPV or IRR”, but no single approach applies to all situations. Even when conditions make the NPV guideline or NPV/Co seem appropriate, analysts should always give the investor other project measures, such as IRR, payback period, and capital requirements over time…To some extent, capital budgeting is an art that can’t always be boiled down to a simple decision rule.” • Klemperer, p.188
How Would You Choose Here? Maximizing benefits to society as a whole would lead you to use benefit-cost ratio (but generally assumes resources are relatively unlimited and we are in the world of perfect competition); Maximizing return to land would lead you to use NPV; If maximizing return to capital investors would favour IRR (generally favours projects with earlier return of capital) Generally NPV favoured as theoretically cleaner-but may use all to evaluate. NPV method generally favours larger projects. Circumstances of decision will affect decision. For example, in earlier three examples (planting/spacing/spraying), capital expenditures varied between $16,000 to $35,000 and if funds were limited that would change decision.
How You Rank Projects • Ranking important where you need to make investment choices (limited capital budget) or the nature of the investment affects the decision • Exclusivity (e.g. plant one kind of species versus another) • Divisibility (are you adding hectares to a silvicultural treatment or building a new pulp mill?)
How to proceed First check for capital requirements; Then check to see whether or not unequal lives $100,000 budget, non-repeating projects-which should you select?
Capital Budgeting Flowchart for evaluating multiple investment opportunities (from Klemperer 2003)