Taking Time Into Account and Making Investment Decisions

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?

$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).

TFSA’s offer a good example.

Future Value of a Single Sum:

V1 = V0 (1 + i)

V2 = V0 (1 + i) (1 + i)

V3 = V0 (1 + i) (1 + i) (1 + i)

Vn = V0 (1 + i) n

<=>

Vn = V0 (1 + i) (1 + i) ………………….(1 + i)

n - times

.

.

.

.

.

.

Here Vn is future value n periods in the future using compounding

Present Value of a Single Sum:

Vn = V0 (1 + i) n

<=>

V0 = Vn / (1 + i) n

Divide both sides

By (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.

NPV = Σ ( ) Ry

(1+i)y

Cy

(1+i)y-

y=0

n

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

Cost and Revenue Streams:

0 1 2 3 4 5 6 7 8 9 10

Years$10,

000

$5,0

00

$2,0

00

$2,0

00

$2,0

00

$2,0

00

$2,0

00

$2,0

00

$2,0

00

$2,0

00

$3,0

00

$2,0

000 1 2 3 4 5 6 7 8 9 10

Years$10,

000

$5,0

00

$5,0

00

$5,0

00

$5,0

00

$5,0

00

Payment stream #1

Payment stream #2

Costs are shown in red and revenues in orange

Evaluating Payment Streams with no Discounting

0 -$10,0001 $2,000 -$10,0002 $2,000 $5,0003 $2,0004 $2,000 $5,0005 $2,0006 -$5,000 $2,000 -$5,000 $5,0007 $2,0008 $2,0009 $2,000

10 $3,000 $5,000-$15,000 $21,000 -$15,000 $20,000

$6,000 $5,000

Payment Stream 1 Payment Stream 2

Here payment stream #1 is preferred (it pays off more)

Evaluating Payment Streams with a 5% Discount Rate

Interest rate0.05

0 -$10,000 $01 $1,905 -$9,5242 $1,814 $4,5353 $1,7284 $1,645 $4,1145 $1,5676 -$3,731 $1,492 -$3,731 $3,7317 $1,4218 $1,3549 $1,289

10 $1,842 $3,070-$13,731 $16,057 -$13,255 $15,449

$2,326 $2,194

Payment Stream 2Payment Stream 1

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

Interest rate0.1

0 -$10,0001 $1,818 -$9,0912 $1,653 $4,1323 $1,5034 $1,366 $3,4155 $1,2426 -$2,822 $1,129 -$2,822 $2,8227 $1,0268 $9339 $848

10 $1,157 $1,928-$12,822 $12,675 -$11,913 $12,297

-$148 $384

Payment Stream 1 Payment Stream 2

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.

TerminatingPresent or Future Value

Series

Single Sum

Annual Payment

Annual

Periodic

Annual

Periodic

Perpetual

Meaning of Symbols:a = equal annual or periodic paymenti = interest raten = number of years or interest bearing periodst = interval between periodic paymentsVo = present (initial) valueVn = future (end) value

non i)(1V V

nno i)(1

1V V

1

1- i)(1 a V

n

n

n

n

o i)(1 i

1-i)(1 a V

1 - i)(1

1 - i)(1 a V

t

t*n

n

t*nt

t*n

o i)1)(1i)((1

1- i)(1 a V

i

aVo

1i)(1

1aV

to

1i)(1

iVa

nn

1i)(1

i)i(1Va

n

n

o

Formula Formula Name

Future Value Future Value of a Single Sum 1

Present Value Present Value of a Single Sum 2

Future Value Future Value of a Terminating Annual Series 3

Present Value Present Value of a Terminating Annual Series 4

Future Value Future Value of a Terminating Periodic Series 5

Present Value Present Value of a Terminating Periodic Series 6

Present Value Present Value of a Perpetual Annual Series 7

Present Value Present Value of a Perpetual Periodic Series 8

to accumulate a future amount

Sinking Fund Formula 9

to pay off an original investment

Installment Payment (capital Recovery) Formula 10

Problem 1

Present value of a Periodic Series (pg. 109-110 in text)

pp 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

V0=p

(1 + r)t - 1

V0=$3,000

(1 + .06)10 - 1

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

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

-600(1+.07)30 = -$4,567.35

-$75(1+.07)20 = - $290.23

Land and planting costBrushing cost

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)

Years to harvestEstablishment Costs 29 30 50 80

-600 -$4,269 -$4,567 -$17,674 -$134,541-75 -$271 -$290 -$1,123 -$8,549

-$4,540 -$4,858 -$18,797 -$143,090

Decision Rules

NPV = Σ ( ) Rt

(1+r)t

Ct

(1+r)t-

t=0

n

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).

The IRR is the discount rate at which the present value of revenues

minus the present value of costs is zero.

∑ ∑ R t C t

(1 + IRR) t (1 + IRR) t

n n

t=0 t=0= 0

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

Internal Rate of Return [IRR]

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.

If B/C=1 then NPV=); if B/C<1 then NPV<0

∑

∑

R t

C t

(1 + MAR) t

(1 + MAR) t

n

n

t=0

t=0

B/C Ratio = PV (Revenues)

PV (Costs)

=

Benefit/Cost Ratio [B/C Ratio]

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.

Payback Period

Comparing Two Different Potential Investments

YearCash Flow for

DCash Flow for

N0 -$400 -$4005 -$100 -$1008 $1,200

15 $20030 $6,600 $2,500

Using NPV to Evaluate Projects

YearCash Flow for

D PV for DCash Flow for

N PV for N0 -$400 -$400 -$400 -$4005 -$100 -$74.73 -$100 -$74.738 $1,200 $752.89

15 $200 $83.4530 $6,600 $1,149.13 $2,500 $435.28

$758 $713

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

B/C =

6,600

(1.06)30

200

(1.06)15+

100

(1.06)5

+ 400

= 2.60

Here it is acceptable since B/C > 1 (benefits exceed costs)

Payback period

YearCash Flow for

DCash Flow for

N0 -$400 -$4005 -$100 -$1008 $1,200

15 $20030 $6,600 $2,500

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

Using Criteria

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 ($)

NPV

Present Value (Costs)

Present Value (Revenues)

Interest Rate (%)

NPV

Interest Rate (%)

Project D:Costs:Revenues:

Interest Rate (%)

Net Present Value ($)

NPV Project N

NPV Project D

IRR for D IRR for N

Which project is better depends on criteria

NPV for D=NPV for N at this interest rate (or MAR)

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

Interest Rate (%)

Present Value ($)

B/C > 1

Present Value (Costs)

Present Value (Revenues)

NPV

NPV < 0NPV > 0B/C < 1

IRR= 9.68%

6%

NPV =$758

B/C= $1233/$475 =2.60

Pvrevenues =$1233

PVcosts = $475

Project D: Showing NPV, B/C, and IRR

Would accept if MAR chosen is less than IRR

What You Choose Depends on Your Criteria

If MAR is less than 9.7%, had enough capital, and projects were independent, you could do both (positive NPV, and IRR is equal or greater than MAR)

If not, how do you choose? If MAR is greater than 6.3% Project N wins based on IRR and NPV

If MAR is less than 6.3%, Project D maximizes NPV

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

Ranking 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

Project Initial Cost LifeAnnual

Revenue NPV @ 10%NPV per

cost

1 $20,000 14 $3,396.50 $5,020.95 0.251

2 $20,000 6 $5,615.92 $4,458.80 0.223

3 $20,000 14 $3,236.54 $3,842.58 0.192

4 $100,000 10 $18,947.41 $16,423.63 0.164

5 $50,000 20 $6,319.89 $3,804.79 0.076

6 $40,000 12 $6,015.07 $984.83 0.025

7 $30,000 6 $6,293.87 $(2,588.55) -0.086

8 $35,000 10 $4,755.38 $(5,780.25) -0.165

First check for capital requirements;Then check to see whether or not unequal lives

$100,000 budget, non-repeating projects-which should you select?

Should You Replant or Let it Regenerate40 hectares, and crop matures in 55 years

Replant Natural RegenCost/ha $400Return (m3/ha) 475 375Value ($/m3) 25 15

Value w/ planting @ 3% $93,464Value w/o planting @ 3% $44,273Increase in value $49,192

Increase in Value $49,192Cost $16,000NPV $33,192

B/C 3.07

Calculating IRRFuture ValuePlanting $475,000Regen $225,000Increase $250,000

So $16,000 compounded 55 years5.12%

Examples from Pearse

Should You Space or Not Space?60 hectares, and crop matures in 50 years

Space No SpaceCost/ha $550Return (m3/ha) 550 425Value ($/m3) 25 15

Value w/ spacing @ 3% $188,188Value w/o spacing @ 3% $87,251Increase in value $100,937

Increase in Value $100,937Cost $33,000NPV $67,937

B/C 3.06

Calculating IRRFuture ValueSpacing $825,000No spacing $382,500Increase $442,500

So $33,000 compounded 50 years5.33%

Should You Spray or Not Spray?100 hectares, and crop would be harvested in 15 years

Spray No SprayCost/ha $350Return (m3/ha) 350 262.5Value ($/m3) 15 15

Value w/ spraying @ 3% $336,978Value w/o spraying @ 3% $252,733Increase in value $84,244

Increase in Value $84,244Cost $35,000NPV $49,244

B/C 2.41

Calculating IRRFuture ValueSpraying $525,000No spraying $393,750Increase $131,250

So $35,000 compounded 15 years9.21%

Planting Spacing SprayingNPV $33,192 $67,937 $49,244Benefit/Cost 3.07 3.06 2.41IRR 5.12% 5.33% 9.21%

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.

Different Criteria Give Different Ranking

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.

No One Rule

Generally avoid IRR especially when r is much lower since it can lead to inconsistencies

Where budget is unlimited maximize Net benfit (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

What is inflation?

Inflation is defined as a general rise in the price level

Interested in it for several reasons First, we want to be able to separate out changes in prices for specific

goods from overall increases Second, we want to be able to compare changes between different sectors

in the economy and classes of goods (for example commodities, food, or energy) or understand changes in people’s purchasing power

Third, while inflation is accepted, as it increases it can trigger changes in behaviour with negative economic consequences. For example, if inflation increases beyond a certain point (in Canada this would be 2-3% today) investors would demand higher interest rates to protect themselves from erosion in their asset values; workers and suppliers might start asking for more as they expect prices to start rising

Measuring Inflation

All based on a group of products All will involve the selection of a baseline (base

year) Consumer Price Index (CPI)

based on goods consumers buy (food, transportation, housing, electronics…)

Producer Price Index (PPI) based on inputs producers use

Specific Price Indices Based on a group of products (commodities, energy,

food…)

Choice of index depends on type of analysis and perspective (as a consumer? Producer?)

The real price of oil

Interest Rates and the Inflation Rate in Canada, 1981-2007

0

2

4

6

8

10

12

14

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

0

2

4

6

8

10

12

Govt. Bond YieldsInflation Rate (CPI)

Accounting for the Effect of Inflation

Current prices are measured in nominal dollars

So increases in prices from year to year incorporate the effects of inflation

In order to measure changes in real values we need to take out this effect For example, if the GDP grew 3% in one year, but inflation

was 2%, the real rate of growth was only 1% (approximately)

So we need to be able to go back and forth between nominal values (the prices and returns we see today or in the future) and real values (those changes in prices and returns without the distorting effects of inflation)

Deflating

Current dollar value, year n

(1 + f)n (1 + f)n

In= Vn=

You can either deflate (discount) by the rate of inflation:

Or by using a price index:

Constant dollar value (Vn) =

Current dollar value (In)

CPI/100

Example (SPF Lumber Prices)

YearCanadian$ (nominal)

Canadian Price ($2002)

1995 $381 $4351996 $514 $5781997 $523 $5791998 $463 $5071999 $548 $5902000 $420 $4402001 $426 $4362002 $411 $4112003 $375 $3652004 $501 $4782005 $419 $3922006 $329 $3012007 $269 $2412008 $190 $170

Note that in nominal terms, the 2008 price is half of 1995 price-but adjusted for inflation, the price today is nearly a third of what it was back then

Real lumber prices (Cdn$), 1995 - 2008

$0

$100

$200

$300

$400

$500

$600

$700

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

Cdn$2002=

100

Nominal and real rates of return

Rate of return usually expressed in current dollars (nominal terms)

Interested in real return (how much of your purchasing power you retain) or what you get back V0 is what you lend In is what you want back

In nominal dollars r stands for real return f stands for the inflation rate n is the number of periods

In = V0(1 + r)n (1 + f)n

Calculating How Much Your Investment Should Return (taking inflation into account)

So if inflation is 4% and you want a 5% real return after one year, how much do you want back after investing $100?

In = V0(1 + r)n (1 + f)n

In = $100 (1 + .05)1 (1 + .04)1

In = $109.20

Selecting an Interest Rate

Selection of interest rate starts with evaluating appropriate framework Will it be nominal? Is it risk-free? Is it from the private

or public perspective?

http://www.economica.ca/ew06_4p1.htm

Real Interest Rates

Converting nominal interest rates into real by either taking out core inflation or expected inflation (2%)

And don’t forget taxes…

Before you decide to buy or invest you also need to remember that you need to pay taxes… Income taxes Property taxes Capital gain taxes

You can deduct annual expenses from ongoing activities

And for investments (in an asset or piece of equipment) the rules will vary about how quickly that can be charged off So tax rates and timing will matter

Valuing a Timber License

Discounted Cash Flow Analysis-matching up revenues and costs over the appropriate periods of time

What is the value of this license if it were for a 10-year term?

annual volume 60,000

price/m3 $94.00fixed cost/m3 $20.00variable cost/m3 $67.00total cost/3 $87.00stumpage $3.00profit $4.00

Valuing the License

Can use the terminating annual series to come up with the value

Using 6% yields a market value of $1.77 million

annual volume 60,000

price/m3 $94.00fixed cost/m3 $20.00variable cost/m3 $67.00total cost/3 $87.00stumpage $3.00profit $4.00

value $1,766,421

tax=40%after-tax profit $2.40

after tax value $1,059,853

Do you rebuild in Burns Lake?

What information do you need? Is it profitable? (different

ways to measure) How much to rebuild? (key) Will insurance cover it?

(maybe not relevant) Interest rates? (key) Price forecasts? (key) Log supply? (key)