Download Analytical Tools - Purdue Agriculture

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 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 Analysis
• Economic efficiency
– Profit maximized where marginal cost =
marginal revenue (MC = MR)
Price (P)
ATC
Market equilibrium
exists when,
MC
MR = MC = ATC
P = MR
Quantity (Q)
No “pure profit”
(economic rent) to
attract new firms to
industry
Marginal Analysis
Price (P)
Pure profit = P1*Q1 - P2*Q1, or
ATC
= (P1 – P2) *Q1
MC
P1
P2
Assumes perfect
competition, i.e., P = MR
Quantity (Q)
Q2 Q1
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 inbetween.
Applications of Marginal
Analysis
• Financial maturity of
individual tree
• Minimum size tree
to harvest
• Break-even analysis
Biological Maturity
Inflection
point
Financia Biological
l
maturity
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 a Tree
Age
Volume Unit
(b.f.) Value
($/b.f.)
Tree
value
Change Return on
in value investment
70
240
$0.50
$120
n.a.
n.a.
75
310
$0.60
$186
$66
80
360
$0.65
$234
$48
85
400
$0.65
$260
$26
$66/$120
= 55%
$48/$186
= 26%
$26/$234
= 11%
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!
• 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
• 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
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 = $100
Solve Compounding Multiplier
for i
Vn = V0 (1+i)n
Vn/ V0 = (1+i)n
(V
1/n = ((1+i)n)1/n = (1+i)n/n = 1+i
/
V
)
n
0
(Vn/ V0 )1/n – 1 = i
Calculate compound rate of interest
for 5 year value increments
Age 70 to 75:
i = (186/120)1/5 – 1 = (1.55)0.2 –1 = 1.09 – 1 = 0.09
Age 75 to 80:
i = (234/186)1/5 – 1 = (1.26)0.2 –1 = 1.05 – 1 = 0.05
Age 80 to 85:
i = (260/234)1/5 – 1 = (1.11)0.2 –1 = 1.02 – 1 = 0.02
Age Volume
b.f.
70
240
75
310
80
360
85
400
When should tree be harvested?
Age
Volume
(b.f.)
Unit Price Tree Change Return on
value in value investment
($/b.f.)
70
240
$0.50
$120
n.a.
n.a.
75
310
$0.60
$186
$66
9%
80
360
$0.65
$234
$48
5%
85
400
$0.65
$260
$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
Delivered
to Mill
(cords)
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
Total Cost Change in Change in
Cost
Volume
(dollars)
(dollars)
(cords)
Marginal
Cost per
Cord
(dollars)
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
$’s
TR
TC
Stumpage
cost
Fixed
cost
q
Declining cutting diameter
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 IRR
Time line of benefit and cost flows
$ Revenue (R2)
$ Revenue (R8)
C0
C1
C2
C3
C4
C5
C6
Cost in each year for 8 years plus “year zero”
All revenues and costs
discounted to year zero
C7
C8
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
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
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
• 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