ANNUITIES &
DISCOUNTED CASH
FLOW RATE OF
RETURN
ANNUITY EQUATIONS



ARE USED TO EVALUATE DIFFERENT
OPTIONS FOR FINANCING PROJECTS
THE BASE PROJECT FOR THIS CLASS
ASSUMES THAT THE PROJECT IS 100%
FUNDED BY THE COMPANY FROM
AVAILABLE FUNDS. 100% EQUITY
DEBT FUNDING - MORE TYPICAL FUNDING
IS ON THE ORDER OF 20% - 40% EQUITY
WITH THE REMAINDER AS DEBT 60% 80%.
EXAMPLE OF ANNUITY
CALCUATIONS
GIVEN :
Investment in a plant is to total $12,500,000.
WANTED :
Determine which funding option is best for the facility.
BASIS :
The funding will be based on a 30% equity position. The available
funding combinations (Term, Interest Rate, Loan Cost) are:
Case 1: 10 Years, 6.75%, 1.25%:
Case 2: 15 Years, 6.95%, 1.375%
Case 3: 20 Years, 7.15%, 1.5%
Case 4: 25 Years, 7.5%, 1.75%.
SOLUTION :
Equity  0.3
Investment Tot  12500000
Debt  1  Equity
Debt  0.70
P Inv  Debt  Investment Tot
P Inv  8750000.00
Calculate the loan principal and the payment for each case:
ANNUITY EXAMPLE
n  10
For Case 1:
P act 
i  0.0675
Pts  0.0125
P Inv
P act  8860759.49
( 1  Pts )
R 1  P act 
i  ( 1  i)
n
R 1  1247033.30
n
( 1  i)  1
n  15
For Case 2:
P act 
i  0.0695
Pts  0.01375
P Inv
P act  8871989.86
( 1  Pts )
R 2  P act 
i  ( 1  i)
n
n
( 1  i)  1
R 2  971022.77
ANNUITY EXAMPLE
n  20
For Case 3:
P act 
i  0.0715
Pts  0.015
P Inv
P act  8883248.73
( 1  Pts )
R 3  P act 
i  ( 1  i)
n
R 3  848316.57
n
( 1  i)  1
n  25
For Case 4:
P act 
i  0.075
Pts  0.0175
P Inv
P act  8905852.42
( 1  Pts )
R 4  P act 
i  ( 1  i)
n
n
( 1  i)  1
R 4  798950.00
RESULTS OF EXAMPLE
 THE RANGE OF VALUES FOR THE
REGULAR PAYMENTS IS $798,950 TO
$1,247,033 PER YEAR
 THE LOWEST PAYMENT VALUES
OCCUR WHEN THE NOTE IS PAID
OVER THE LONGEST PERIOD OF
TIME
 THIS IS ALSO ASSOCIATED WITH THE
HIGHEST INTEREST RATE
RESULTS OF EXAMPLE




THE RANGE OF THE PRESENT WORTH
VALUES IS FROM $8,860,759 TO $8,905,852
THE LOWEST PRESENT WORTH OCCURS
WHEN THE LOAN COSTS (POINTS) ARE
MINIMIZED
THIS IS ALSO ASSOCIATED WITH THE
LOWEST INTEREST RATE AND SHORTEST
TERM
SO THE BEST OPTION IS THE ONE THAT
HAS THE LOWEST PRESENT WORTH
VALUE
PRESENT WORTH ANALYSIS




PRESENT WORTH VALUE
THIS IS NET PRESENT WORTH OF ALL THE
PAYMENTS THAT WILL BE MADE TO
COMPLETE THIS LOAN
THIS METHOD PROVIDES AN OBJECTIVE
BASIS OF COMPARISON EVEN THOUGH
THE TERMS, INTEREST RATES AND LOAD
COSTS ALL VARY.
THIS IS ONE VARIATION OF THE
DISCOUNTED CASH FLOW RATE OF
RETURN (DCRR)
FORMAL DCRR
 SEE PAGE 328 FOR REFERENCE
 FORMAL VERSION OF CALCULATES THE
DCRR INTEREST RATE THAT WOULD YIELD
A NET PRESENT WORTH OF $0 FOR A
PROJECT OVER A SPECIFIED LIFETIME
 SOMETIMES CALLED INTERNAL RATE OF
RETURN, INTEREST RATE OF RETURN,
INVESTOR’S RATE OF RETURN
INVESTMENT PERIOD (DCRR)




FOR THIS CALCULATION, AN INVESTMENT
IS MADE IN A FACILITY OVER A SPECIFIED
CONSTRUCTION TIME PERIOD
THESE VALUES START AT YEAR ZERO
THEY ARE EXPRESSED IN CURRENT
(CONSTANT VALUE) DOLLARS FOR EACH
YEAR
THEY ARE CONSIDERED NEGATIVE
VALUES BECAUSE THEY ARE
EXPENDITURES
PROFIT PERIOD (DCRR)




THE RETURN IS CALCULATED FROM THE
PROFIT EARNED DURING OPERATIONS
THESE VALUES START IN THE FIRST YEAR
AFTER CONSTRUCTION
THEY ARE EXPRESSED IN CURRENT
DOLLARS, OVER THE LIFE OF THE
FACILITY
THESE ARE CONSIDERED POSITIVE
VALUES BECAUSE THEY REPRESENT NET
PROFITS
DCRR CALCULATION
 BOTH THE INVESTMENT AND THE
PROFIT RETURN ARE DISCOUNTED
BACK TO A COMMON TIME AT YEAR
ZERO FOR THE OVERALL PERIOD j
WHICH IS THE SUM OF THE
CONSTRUCTION AND OPERATION
PERIODS
 FOR EACH YEAR THE CALCULATION
COULD BE BASED ON THE FORMULA
Pn  F n 1  i 
N
(8  6 )
DCRR INTEREST CALCULATION
 THE DCRR IS THE VALUE OF i WHEN
 Pn
0
j
 WHERE j IS THE LIFETIME OF THE
PROJECT
DCRR EXAMPLE
GIVEN :
Investment in a plant.
WANTED :
Determine the DCRR for this project.
BASIS :
The investment in the plant (in current $) will be $5,000,000 PER
YEAR expended over a three year period. The plant is expected to
operate for a period of 20 years and produce a profit (in current $)
of $2,500,000 each year of operation
SOLUTION :
The investment costs can be calculated using equation 7.24 over a
three year period:
n c  3
R c   5000000
NPW Inv
Rc 
( 1  i)
nc
i  ( 1  i)
The return can be calculated using equation 7.24 for the 20 year
period from year 4 to year 24 using 7.24:
n o  20
R o  2500000
 1
nc
DCRR EXAMPLE CALCULATION
The easiest way to complete this calculation is to first calculate a
NPW based on then end of construction as year 0, which would
actually be a future worth in year 4:
S4
Ro 
( 1  i)
no
 1
i  ( 1  i)
no
and then discount this value back to the start of construction:
NPW Ret
S4
( 1  i)
4
The DCRR occurs for i when:
NPW Inv  NPW Ret
This is a trial and error solution.
Assume :
i  0.1
0
DCRR TRIAL & ERROR
NPW Inv  R c 
( 1  i)
nc
i  ( 1  i)
S 4  R o 
( 1  i)
no
i  ( 1  i)
NPW Ret 
 1
nc
 1
no
S4
( 1  i)
4
NPW Proj  NPW Inv  NPW Ret
NPW Inv   12434260
S 4  21283909
NPW Ret  14537196
NPW Proj  2102936
DCRR EXAMPLE RESULTS
Completing the Trial and Error Calculation:
f ( i )  R c 
( 1  i)
nc
i  ( 1  i)
Ro 
1
nc
For an initial guess:
root ( f ( i )  i )  0.1185

( 1  i)
no
i  ( 1  i)
( 1  i)
1
no
4
i  0.15
DCRR  root ( f ( i )  i )
DCRR  11.85 %
ANALYSIS OF DCRR RESULTS
 THE RESULTS INDICATE A DCRR OF
11.85%
 IN THEORY, IF THE PLANT WERE
100% FINANCED, A LOAN AT A RATE
OF 11.85% COULD BE PAID BACK
OVER THE LIFE OF THE PROJECT
DCRR APPLICATION



THE EXAMPLE CAN BE USED TO
DEMONSTRATE THE ADVANTAGES OF
DEBT FINANCING
THE CALCULATION CAN BE REPEATED
WITH AN ASSUMPTION OF 25% EQUITY
FINANCING AND REDUCING THE PROFIT
EACH YEAR TO ACCOUNT FOR INTEREST
PAYMENTS
THE RESULT SHOWS THE DCRR
INCREASES TO 30% FOR THE DEBT
FUNDING APPROACH
COMPARISON OF ALTERNATES


THE RESULTS OF THE REVISED DCRR
CALCULATION SHOW THAT A PROJECT
THAT HAS 100% FUNDING MIGHT HAVE A
RELATIVELY SMALL DIFFERENCE ABOVE
CURRENT INTEREST RATES AND NOT BE
ATTRACTIVE
THE SAME PROJECT WITH DEBT FUNDING
MAY HAVE A RETURN COMFORTABLY
ABOVE THE CURRENT INTEREST RATES
OTHER COMPARISONS
 THE SIGNIFICANT VALUE TO THIS
TYPE OF CALCULATION IS BASED ON
OBJECTIVE COMPARISON OF
VARIOUS TYPES OF PROJECTS
AND/OR VARIOUS CONFIGURATIONS
OF ONE PROJECT
 INDEPENDENT OF PROJECT LIFE
 INDEPENDENT OF CURRENT
INTEREST RATES