Analytical Tools

Marginal analysis

Discounted cash flow

Marginal Analysis

Resources are limited, therefore we want the most “bang for our bucks” -- benefits from resources allocated to a project

Marginal Analysis• Economic efficiency

– Maximize “profit”, i.e. total revenue - total costs (TR - TC)

Profit

Slope of TR curve is MR

Slope of TC curve is MC

Marginal Analysis

• Economic efficiency– Profit maximized where marginal cost =

marginal revenue (MC = MR)

MCATC

Price (P)

P = MR

Quantity (Q)

Market equilibrium exists when,

MR = MC = ATC

No “pure profit” (economic rent) to attract new firms to industry

Marginal Analysis

P1

MCATC

Quantity (Q)

Price (P)

P2

Q1Q2

Pure profit = P1*Q1 - P2*Q1, or

= (P1 – P2) *Q1

Assumes perfect competition, i.e., P = MR

Advantage of Marginal Analysis

• Don’t have to do a complete analysis of costs and revenues

• Can estimate MC directly from market price by assuming a given profit percentage

• Can estimate MR from market price and knowledge of market structure, i.e. perfectly competitive, monopolistic, or somewhere in-between.

Applications of Marginal Analysis

• Financial maturity of individual tree

• Minimum size tree to harvest

• Break-even analysis

Biological MaturityOutput

(volume)

Inflection point

Biological maturity

Biological vs. Financial Maturity

• Financial maturity is based on benefit from letting tree/stand grow for another time period compared to cost for doing so.

• Would you expect biological and financial maturity to occur at the same point in time?

Financial Maturity of Tree

Age Volume b.f.

Tree value given $/bf

Change in value

Return on investment

70 240 $120 @ $0.5 n.a. n.a.

75 310 $186@$0.6 $66 66/120 = 55%

80 360 $234@$0.65 $48 48/186 = 26%

85 400 $260@0.65 $26 26/234 = 11%

Financial Maturity of Tree

• Compare return on investment with return from other 5-year investments– If rate of return on the alternative is 10% then

don’t cut yet– If rate of return on the alternative is > 20%, then

cut at age 80

• More practical to compare with prevailing annual compound rates of interest?– How can we compute annual compound rate of

interest for changes in value over a 5-year period?

Estimating Annual Compound Rate of Interest

Use basic compounding multiplier,

Vn = V0 (1+i)n, where,

Vn = value n years in future

V0 = value in year 0 (current time)

n = number of years

i = annual compound rate of interest

Before solving for I let’s review how compounding and

discounting multipliers are used

Example of use of compounding multiplier

• Buy $100 worth of stock today

• If it increases in value at a rate of 18% each year, what will it be worth in 5 years?

• V5 = ?, V0 = $100, i = 0.18, n = 5

• V5 = $100 (1.18)5 = $100 x 2.29 = $229

Example of use of discounting multiplier

• Solve compounding multiplier for V0 by dividing both sides by (1+i)n

• Vn = V0 (1+i)n

• V0 = Vn /(1+i)n = Vn * 1/ (1+i)n

Solve Compounding Multiplier for i

Vn = V0 (1+i)n

Vn/ V0 = (1+i)n

(Vn/ V0 )1/n = ((1+i)n)1/n = (1+i)n/n = 1+i

(Vn/ V0 )1/n – 1 = i

Calculate compound rate of interest for 5 year value increments

Age 70 to 75:

i = (186/120)1/5 – 1 = (1.55)0.2 –1 = 1.09 – 1 = 0.09

Age 75 to 80:

i = (234/186)1/5 – 1 = (1.26)0.2 –1 = 1.05 – 1 = 0.05

Age 80 to 85:

i = (260/234)1/5 – 1 = (1.11)0.2 –1 = 1.02 – 1 = 0.02

When should we cut?

Age Volume b.f.

Tree value given $/bf

Change in value

Return on investment

70 240 $120 @ $0.5 n.a. n.a.

75 310 $186@$0.6 $66 9%

80 360 $234@$0.65 $48 5%

85 400 $260@0.65 $26 2%

When should we cut?

• Depends on what rate of return the owner is willing to accept.

• We refer to this rate as the owner’s guiding rate or, alternative rate of return.

• Rate is based on owner’s alternative uses for the capital tied-up in the trees.

When should we cut?

• If owner’s alternative rate of return is – 10% - cut at age 70– 7% - cut at age 75– 5% - cut at age 75– 1% - let grow to age 85

How does an owner select her alternative rate of return?

• Borrowing rate – if she would have to borrow money if tree wasn’t get, she could use the interest rate she would have to pay on the loan, i.e. the “borrowing rate”

• Lending rate – if owner could “lend” the revenue from cutting the tree now to someone else, she could use the rate she would get by making the “loan”

Minimum Size Tree to Cut

Logger buys cutting rights on 200 acre tract of pine pulpwood for lump sum amount of $40,000. Landowner placed no limits on what logger can cut. Logger wants to cut to maximize net revenue (profit). Should he give cutting crew a minimum size tree to cut? Answer with marginal approach.

Calculate Marginal Cost

Min. Cutting

Dia. (inches)

Total Volume Deliverd to Mill (cords)

Total Cost

(dollars)

Change in Cost

(dollars)

Change in Volume (cords)

Marginal Cost per

Cord (dollars)

20 3,300 82,965

18 4,200 92,316 9,360 900 10.40

16 5,000 101,916 9,600 800 12.00

14 5,700 111,856 9,940 700 14.20

12 6,320 122,520 10,664 620 17.20

10 6,920 135,000 12,480 600 20.80

8 7,500 150,000 15,000 580 25.86

6 8,000 165,000 15,000 500 30.00

4 8,400 179,400 14,400 400 36.00

Compare MC and MB

• If price per cord received by logger is $30, then shouldn’t cut any tree less than about 7 inches.

• If price per cord increases to $35, then cut down to 5 inches.

• If price per cord decreases to $25, then cut down to about 9 inches

Minimum diameter (q) for lump sum payment for stumpage

TR

TC

Stumpage cost

Fixed cost

Declining cutting diameter

$’s

q

Pay as cut contract

• Would minimum diameter change if logger paid for stumpage as trees were cut (log scale) instead of for lump sum amount in advance?

• Yes, stumpage now a variable cost, not a fixed cost

Analytical Tools

• Discounted cash flow– Net present value

• Discount or compound all cash flows to same point in time

• Calculate using an assumed interest rate

– Internal rate of return• Interest rate (i) that makes NPV zero, i.e.

equates PV of all costs and all benefits• “Calculate” the interest rate

NPV and IRRTime line of benefit and cost flows

Cost in each year for 8 years plus “year zero”

C0 C1 C2 C3 C4 C5 C6 C7 C8

$ Revenue (R2) $ Revenue (R8)

All revenues and costs discounted to year zero

Formula for NPV for a given interest rate ( i )

NPV0 =

- C0 - C1/(1+i)1 - C2/(1+i)2 - C3/(1+i)3 - . . . . - C8/(1+i)8 +

R2/(1+i)2 + R8/(1+i)8

Simplify by netting R’s and C’s for given year

-C0 - C1/(1+i)1 +(R2 -C2)/(1+i)2 - C3/(1+i)3 . . . .

- + (R8 - C8)/(1+i)8

NPV Using Summation Notation

NPV = [ (Rt – Ct)/(1+i)t]t=0

n

where,

NPV = unknown

n = number of years

Rt = revenue (income) in year t

Ct = cost (expense) in year t

t = index number for years

i = discount rate (alternative rate of return)

Internal Rate of Return Using Summation Notation

NPV = [ (Rt – Ct)/(1+i)t]t=1

n

where,

NPV = 0

Rt = revenue (income) in year t

Ct = expense (cost) in year t

t = index number for years

i = unknown

Finding Internal Rate of Return

• Calculate NPV using spreadsheet

• Make “i” a variable referenced to one cell

• Change “i” in that cell until NPV equals approximately zero