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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

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analytical tools

Analytical Tools

Marginal analysis

Discounted cash flow

marginal analysis
Marginal Analysis

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

marginal analysis1
Marginal Analysis
  • Economic efficiency
    • Maximize net revenue, “profit”, i.e. total revenue - total costs (TR - TC)

Slope of TR curve is MR

Profit

Slope of TC curve is MC

marginal analysis2
Marginal Analysis
  • Economic efficiency
    • Profit maximized where marginal cost = marginal revenue (MC = MR)

Price (P)

Market equilibrium exists when,

MR = MC = ATC

MC

ATC

P = MR

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

Quantity (Q)

marginal analysis3
Marginal Analysis

Price (P)

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

= (P1 – P2) *Q1

MC

ATC

P1

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

P2

Quantity (Q)

Q2

Q1

advantage of marginal analysis
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
Applications of Marginal Analysis
  • Financial maturity of individual tree
  • Minimum size tree to harvest
  • Break-even analysis
biological maturity
Biological Maturity

Output

(volume)

Inflection point

Financial maturity

Biological maturity

biological vs financial 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?
5 year rates of return
5-Year Rates of Return?
  • 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
a problem
A Problem!
  • The interest rates, percentages, shown are for the change in value over 5 years.
  • Interest rates are always given as simple annual compound rates, unless stated otherwise, i.e. change over 1 year
financial maturity of tree
Financial Maturity of Tree
  • 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
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
Before solving for “i” let’s review how compounding and discounting multipliers are used
example of use of compounding multiplier
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
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
example of use of discounting multiplier1
Example of use of discounting multiplier
  • Want to have $229 worth of stock in 5 years
  • If it increases in value at a rate of 18% each year, how much do you need to invest at beginning of first year?
  • V5 = $229, V0 = ?, i = 0.18, n = 5
  • V5 = $229/ (1.18)5 = $229/2.29 = $100
solve compounding multiplier for i
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
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
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 cut1
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
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
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.

compare mc and mb
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
Minimum diameter (q) for lump sum payment for stumpage

$’s

TR

TC

Stumpage cost

Fixed cost

q

Declining cutting diameter

pay as cut contract
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 tools1
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 irr
NPV and IRR

Time line of benefit and cost flows

$ Revenue (R2)

$ Revenue (R8)

C0 C1 C2 C3 C4 C5 C6 C7 C8

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

All revenues and costs discounted to year zero

formula for npv for a given interest rate i
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 Using Summation Notation

n

NPV = [ (Rt – Ct)/(1+i)t]

t=0

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
Internal Rate of Return Using Summation Notation

n

NPV = [ (Rt – Ct)/(1+i)t]

t=1

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
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
    • Or, use “Goal Seek” tool