Types Negative Base & Positive Gradient

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1 6: Trends Linear Constant amount Geometric Constant percentage 6.1 DAs of Linear Trends Basics Typically two components Base series First flow Arithmetic gradient Remainder B 50 + Base Series Linear Trend G +2 Arithmetic Gradient Types Negative Base & Negative Gradient B -50 Trend and base series start at same time Arithmetic gradient starts one period later G -2 Types Positive Base & Negative Gradient B G -2 Types Negative Base & Positive Gradient B G +2 +

2 Observations Base series First flow Arithmetic gradient Remainder Trend and base series start at same time Arithmetic gradient starts one period later Basic Logic for Linear Trend Factors Use existing series factors for base series Develop new factors for arithmetic gradient Use consistent s-r values for all factors B 50 G +2 + r s r s Base Series Arithmetic Gradient Arithmetic Gradient Present Worth Factor E r G (P G, i, s-r ) (s-r-1)g G 2G r r+1 r+2 r+3 s E r Gγ Gγ (s-r-1)gγ -(s-r) E r G { [(1+i) s-r (s-r)i 1 ] / [ i 2 (1+i) s-r ] } E G (P G, i, m s r ) r Observations E r G (P G, i, s-r ) G 2G (s-r-1)g r r+1 r+2 r+3 s Arithmetic grad present worth factor P G Find prior equivalent given an arithmetic gradient, interest rate, and number of cash flows in the linear trend (or base series) Position and Fourth Parameter Position of prior equivalent Single Flow: wherever desired Uniform Series: one before Arithmetic Gradient: same place as series s-r position of last flow position of prior eq Ex. 6.1 Equivalents of Various Linear Trends E 9 E B + E G Linear Trend 50 E G G +2 + Base Series Arithmetic Gradient

3 Step 2 E B 50 E G G +2 + Base Series Arithmetic Gradient E B $ (P A, 10%, 20-9) E G $ (P G, 10%, 20-9) Other Linear Trends Decompose Base series First flow Arithmetic gradient Remainder Position of prior equivalent for both One before first trend flow Discount two components Sum two DAs E 9 $ E B + E G Example 6.2 Equivalent of a Pure Gradient Pure gradient Base series 0 Find E 7% P G puts eq 2 periods before first grad flow! E Pure gradient Base series 0 Find E 7% Pure Gradient Step 2 P G puts eq 2 periods before first grad flow! E E E 9 10(P G, 7%, 20-9) $ Pure gradient Base series 0 Find E 7% Pure Gradient Step 3 P G puts eq 2 periods before first grad flow! E E10 E9 (F P, 7%, 10-9) $ E E 9 10(P G, 7%, 20-9) $ E E 10 Alternative Solution E G +10 E E B G B E 10 E B + E G

4 Alternative Solution Step 2 G +10 E E B G B E B $ (P A, 7%, 20-10) E G $ (P G, 7%, 20-10) 6.2 Uniform Series Equivalents of Linear Trends E 10 $ E B + E G Rounding error $0.01 ( ) Basics Objective Express linear trend as a uniform series Example of use: average maintenance cost that includes the time value of money Base series already a uniform series Need factor to convert gradient component to uniform equivalent Equivalent needs to be positioned like the base series, from r + 1 to s Arithmetic Gradient Uniform Series Factor E r (s-r-1)g E U G (A G, i, s-r ) G 2G r r+1 r+2 r+3 s r r+1 r+2 r+3 s E r G(P G, i, s-r) E U Er (A P, i, s-r) E U G (P G, i, s-r) (A P, i, s-r) E U G(A G, i, s-r) (A G, i, s-r) (1/ i ) { (s-r) / [ (1+ i ) s-r 1] } r Observations (s-r-1)g E U G (A G, i, s-r ) G 2G r+1 r+2 r+3 s r r+1 r+2 r+3 s Arithmetic grad uniform series factor or A G Find annuity given arithmetic gradient, interest rate, and number of terms in the base series (s-r) Extreme care needed on positioning equivalents and fourth argument Example 6.3 Converting Various LTs to US s Linear Trend B 50 + Base Series E U B + E U,G G +2 Arithmetic Gradient

5 Step 2 E G +2 U,G E U,G $8.66 2(A G, 7%, 20-9) E U $ E U,G General In general, the logic is to: Split trend into base and arith grad Convert grad to equivalent uniform series s-r position of last flow position of prior eq Sum base and equivalent series Example 6.4 Timing of Equivalent Series Trend gives E 9 50(P A, 7%, 20-9) + 2(P G, 7%, 20-9) E U needs E 10 (A P,7%, 20-10) Extra steps needed! Not same time! E U Step 1 E 9 50(P A, 7%, 20-9) + 2(P G, 7%, 20-9) Step 2 Step 3 E 10 E 9 (F P, 7%, 10-9) E 10 E U E 10 (A P, 7%, 20-10) E 9 $67.01

6 Getting Fancy! Alternative Step 2 Getting Fancy! Alternative Step 3 E 9 E 20 E 9 (F P, 7%, 20-9) E 20 E U E 20 (A F, 7%, 20-10) $ All equivalence operations produce same final balance Explicitly derived for that purpose Same answers, except for round-off Some more intuitive than others Basics 6.3 Discounted Amounts of Geometric Trends Cash flows change by a constant percentage Flows can be positive or negative Percentage can be positive or negative Not a base series plus a gradient Entire series is the gradient Positive Flow and Positive Percentage Negative Flows and Positive Percentage c j c j-1 (1+10%) c j c j-1 (1+10%)

7 Positive Flows and Negative Percentage Negative Flows and Negative Percentage c j c j-1 (1-10%) c j c j-1 (1-10%) Geometric Series DA E r c r+1 (P A, g, i, s-r ) c r+1 (1+g) s-r-1 c c r+1 (1+g) c r+1 (1+g)2 r+1 r r+1 r+2 r+3 s One before E r c γ -1 r+1 + c r+1 (1+g)γ -2 + c r+1 (1+g)2γ c r+1 (1+g) s-r-1 γ -(s-r) Geometric Series Present Worth Factor E r c r+1 (P A, g, i, s-r ) r c c r+1 (1+g) c r+1 (1+g)2 r+1 r+1 r+2 r+3 s E r c r +1 { [1 (1+g)s-r (1+i) -(s-r) ] / [ i g ] }, i g E r c r +1 { (s-r)(1+i)-1 }, i g c r+1 (1+g) s-r-1 E r c r +1 (P A, g, i, m s r) No tables too many combinations of i and g E r c r+1 (P A, g, i, s-r ) Helpful Hints c r+1 (1+g) s-r-1 c c r+1 (1+g) c r+1 (1+g)2 r+1 r r+1 r+2 r+3 s Find prior given annuity w/ geometric change Prior equivalents for trends and series always one before first flow of trend or series Arith grad component two flows before s-r Last Prior For all series and trends Example 6.5 DAs of Various Geometric Trends E c j c j-1 (1+g) g c j / c j-1 1 g 10% 110 / E 9 100(P A, 10%, i, 20-9) i 7% E {[1 (1.10) 20-9 (1.07) -(20-9) ]/[.07.10]} $1,184.98

8 Step 2 In general,... E c j c j-1 (1+g) See if cash flows positive or negative g c j / c j-1 1 Use ratio of first two flows to determine g Position equiv one period before first flow g 10% 110 / Write equiv equation using factor such as E 9 100(P A, 10%, i, 20-9) E 9 100(P A, 10%, i, 20-9) i 10% Use correct formula E { (20-9) (1.10) -1 } Cannot use wrong one for i g or divide by 0 $1, Example 6.6 US Equivalent of Geometric Trend No single factor, so use P A, g then A P E E U Trendy, yet fiscally wise... E 9 100(P A, 10%, i, 20-9), as before E U E 9 (A P, i, 20-9)

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