1. 1 Civil Systems Planning Benefit/Cost Analysis Scott Matthews Courses: 12-706 and 73-359 Lecture 9 - 9/29/2002

2. 12-706 and 73-3592 Common Monetary Units  Often face problems where benefits and costs occur at different times  Need to adjust values to common units to compare them  Recall photo sensor example from last lecture - could look at values over several years...

3. 12-706 and 73-3593 Admin Issues  Number/size of groups? Need to plan.  Usually 4 ‘groups’ per class session, at end of course

4. 12-706 and 73-3594 General Terms  Three methods: PV, FV, NPV  FV = $X (1+i) n  X : present value, i:interest rate and n is number of periods (eg years) of interest  Rule of 72  PV = $X / (1+i) n  NPV=NPV(B) - NPV(C) (over time)  Real vs. Nominal values

5. 12-706 and 73-3595 Minimum Attractive Rate of Return  MARR usually resolved by top management in view of numerous considerations. Among these are:  Amount of money available for investment, and the source and cost of these funds (i.e., equity or borrowed funds).  Number of projects available for investment and purpose (i.e., whether they sustain present operations and are essential, or expand present operations)

6. 12-706 and 73-3596 MARR part 2  The amount of perceived risk associated with investment opportunities available to the firm and the estimated cost of administering projects over short planning horizons versus long planning horizons.  The type of organization involved (i.e., government, public utility, or competitive industry)  In the end, we are usually given MARR

7. 12-706 and 73-3597 Notes on Notation  PV = $FV / (1+i) n = $FV * [1 / (1+i) n ]  But [1 / (1+i) n ] is only function of i,n  $1, i=5%, n=5, [1/(1.05) 5 ]= 0.784 = (P|F,i,n)  As shorthand:  Future value of Present: (P|F,i,n)  So PV of $500, 5%,5 yrs = $500*0.784 = $392  Present value of Future: (F|P,i,n)  And similar notations for other types

8. 12-706 and 73-3598 Timing of Future Values  Noted last time that we assume ‘end of period’ values  What is relative difference?  Consider comparative case:  $1000/yr Benefit for 5 years @ 5%  Assume case 1: received beginning  Assume case 2: received end

9. 12-706 and 73-3599 Timing of Benefits  Draw 2 cash flow diagrams  NPV 1 = 1000 + 1000/1.05 + 1000/1.05 2 + 1000/1.05 3 + 1000/1.05 4  NPV 1 =1000 + 952 + 907 + 864 + 823 = $4,545  NPV 2 = 1000/1.05 + 1000/1.05 2 + 1000/1.05 3 + 1000/1.05 4 + 1000/1.05 5  NPV 2 = 952 + 907 + 864 + 823 + 784 = $4,329  NPV 1 - NPV 2 ~ $216  Notation: (P|U,i,n)

10. 12-706 and 73-35910 Relative NPV Analysis  If comparing, can just find ‘relative’ NPV compared to a single option  E.g. beginning/end timing problem  Net difference was $216  Alternatively consider ‘net amounts’  NPV 1 =1000 + 952 + 907 + 864 + 823 = $4,545  NPV 2 = 952 + 907 + 864 + 823 + 784 = $4,329  ‘Cancel out’ intermediates, just find ends  NPV 1 is $216 greater than NPV 2

11. 12-706 and 73-35911 Uniform Values - Theory  Assume end of period values  Stream = F/(1+i) +F/(1+i) 2 +..+ F/(1+i) n  (P|U,i,n) = $1[(1+i) -1 +(1+i) -2 +..+ (1+i) -n ]  [(1+i) -1 +(1+i) -2 +..+ (1+i) -n ] = “A”  [1+(1+i) -1 +(1+i) -2 +..+ (1+i) 1-n ] = A(1+i)  A(1+i) - A = [1 - (1+i) -n ]  A = [1 - (1+i) -n ] / i  Stream = F*(P|U,i,n) = F*[1 - (1+i) -n ] / i

12. 12-706 and 73-35912 Uniform Values - Application  Recall $1000 / year for 5 years example  Stream = F*(P|U,i,n) = F*[1 - (1+i) -n ] / I  (P|U,5%,5) = 4.329  Stream = 1000*4.329 = $4,329 = NPV 2

13. 12-706 and 73-35913 Why Finance?  Time shift revenues and expenses - construction expenses paid up front, nuclear power plant decommissioning at end.  “Finance” is also used to refer to plans to obtain sufficient revenue for a project.

14. 12-706 and 73-35914 Borrowing  Numerous arrangements possible:  bonds and notes  bank loans and line of credit  municipal bonds (with tax exempt interest)  Lenders require a real return - borrowing interest rate exceeds inflation rate.

15. 12-706 and 73-35915 Issues  Security of loan - piece of equipment, construction, company, government. More security implies lower interest rate.  Project, program or organization funding possible. (Note: role of “junk bonds” and rating agencies.  Variable versus fixed interest rates: uncertainty in inflation rates encourages variable rates.

16. 12-706 and 73-35916 Issues (cont.)  Flexibility of loan - can loan be repaid early (makes re-finance attractive when interest rates drop). Issue of contingencies.  Up-front expenses: lawyer fees, taxes, marketing bonds, etc.- 10% common  Term of loan  Source of funds

17. 12-706 and 73-35917 Sinking Funds  Act as reverse borrowing - save revenues to cover end-of-life costs to restore mined lands or decommission nuclear plants.  Low risk investments are used, so return rate is lower.

18. 12-706 and 73-35918 Borrowing  Sometimes we don’t have the money to undertake - need to get loan  i=specified interest rate  A t =cash flow at end of period t (+ for loan receipt, - for payments)  R t =loan balance at end of period t  I t =interest accrued during t for R t-1  Q t =amount added to unpaid balance  At t=n, loan balance must be zero

19. 12-706 and 73-35919 Equations  i=specified interest rate  A t =cash flow at end of period t (+ for loan receipt, - for payments)  I t =i * R t-1  Q t = A t + I t  R t = R t-1 + Q t R t = R t-1 + A t + I t  R t = R t-1 + A t + (i * R t-1 )

20. 12-706 and 73-35920 Uniform payments  Assume a payment of U each year for n years on a principal of P  R n =-U[1+(1+i)+…+(1+i) n-1 ]+P(1+i) n  R n =-U[( (1+i) n -1)/i] + P(1+i) n  Uniform payment functions in Excel  Same basic idea as earlier slide

21. 12-706 and 73-35921 Example  Borrow $200 at 10%, pay $115.24 at end of each of first 2 years  R 0 =A 0 =$200  A 1 = - $115.24, I 1 =R 0 *i = (200)(.10)=20  Q 1 =A 1 + I 1 = -95.24  R 1 =R 0 +Q t = 104.76  I 2 =10.48; Q 2 =-104.76; R 2 =0

22. 12-706 and 73-35922 Repayment Options  Single Loan, Single payment at end of loan  Single Loan, Yearly Payments  Multiple Loans, One repayment