1. 1 Civil Systems Planning Benefit/Cost Analysis Scott Matthews Courses: 12-706 and 73-359 Lecture 10 - 10/2/2002

2. 12-706 and 73-3592 Repayment Options  Single Loan, Single payment at end of loan  Single Loan, Yearly Payments  Multiple Loans, One repayment

3. 12-706 and 73-3593 Note on Taxes  Companies pay tax on net income  Income = Revenues - Expenses  There are several types of expenses that we care about  Interest expense of borrowing  Depreciation  These are also called ‘tax shields’

4. 12-706 and 73-3594 Depreciation  Decline in value of assets over time  Buildings, equipment, etc.  Accounting entry - no actual cash flow  Systematic cost allocation over time  Government sets dep. Allowance  P=asset cost, S=salvage,N=est. life  D t = Depreciation amount in year t  T t = accumulated (sum of) dep. up to t  B t = Book Value = Undep. amount = P - T t

5. 12-706 and 73-3595 Depreciation Example  Simple/straight line dep: D t = (P-S)/N  Equal expense for every year  $16k compressor, $2k salvage at 7 yrs.  D t = (P-S)/N = $14k/7 = $2k  B t = 16,000-2t, e.g. B1=$14k, B7=$2k

6. 12-706 and 73-3596 Accelerated Dep’n Methods  Depreciation greater in early years  Sum of Years Digits (SOYD)  Let Z=1+2+…+N = N(N+1)/2  D t = (P-S)[N-(t-1)]/Z, e.g. D1=(N/Z)*(P-S)  D 1 =(7/28)*$14k=$3,500, D 7 =(1/28)*$14k  Declining balance: D t = B t-1 r (r is rate)  B t =P(1-r) t, D t = Pr(1-r) t-1  Requires us to keep an eye on B  Typically r=2/N -aka double dec. balance

7. 12-706 and 73-3597 Ex: Double Declining Balance  Could solve P(1-r) N = S (find nth root) tDtBt 0-$16,000 1(2/7)*$16k=$4,571.43$11,428.57 2(2/7)*$11,428=$3265.31$8,163.26 3$2332.36$5,830.90 4$1,665.97$4,164.93 5$1,189.98$2,974.95 6$849.99$2,124.96 7$607.13**$1,517.83**

8. 12-706 and 73-3598 Notes on Example  Last year would need to be adjusted to consider salvage, D7=$124.96  We get high allowable depreciation ‘expenses’ early - tax benefit  We will assume taxes are simple and based on cash flows (profits)  Realistically, they are more complex

9. 12-706 and 73-3599 Tax Effects of Financing  Companies deduct interest expense  B t =total pre-tax operating benefits  Excluding loan receipts  C t =total operating pre-tax expenses  Excluding loan payments  A t =net pre-tax operating cash flow  A,B,C: financing cash flows  A*,B*,C*: pre-tax totals / all sources

10. 12-706 and 73-35910 Notes  Mixed funds problem - buy computer  Below: Operating cash flows At  Four financing options in At

11. 12-706 and 73-35911 Further Analysis (still no tax)  MARR (disc rate) equals borrowing rate, so financing plans equivalent.  When wholly funded by borrowing, can set MARR to interest rate

12. 12-706 and 73-35912 Effect of other MARRs (e.g. 10%)  ‘total’ NPV higher than operation alone for all options  All preferable to ‘internal funding’  Why? These funds could earn 10% !  First option ‘gets most of loan’, is best

13. 12-706 and 73-35913 Effect of other MARRs (e.g. 6%)  Now reverse is true  Why? Internal funds only earn 6% !  First option now worst

14. 12-706 and 73-35914 After-tax cash flows  D t = Depreciation allowance in t  I t = Interest accrued in t  + on unpaid balance, - overpayment  Q t = available for reducing balance in t  W t = taxable income in t; X t = tax rate  T t = income tax in t  Y t = net after-tax cash flow

15. 12-706 and 73-35915 Equations  D t = Depreciation allowance in t  I t = Interest accrued in t  Q t = available for reducing balance in t  So A t = Q t - I t  W t = A t -D t -I t (Operating - expenses)  T t = X t W t  Y t = A* t - X t W t (pre tax flow - tax) OR  Y t = A t + A t - X t (A t -D t -I t )

16. 12-706 and 73-35916 Simple example  Firm: $500k revenues, $300k expense  Depreciation on equipment $20k  No financing, and tax rate = 50%  Y t = A t + A t - X t (A t -D t -I t )  Y t =($500k-$300k)+0-0.5 ($200k-$20k)  Y t = $110k

17. 12-706 and 73-35917 First Complex Example  Firm will buy $46k equipment  Yr 1: Expects pre-tax benefit of $15k  Yrs 2-6: $2k less per year ($13k..$5k)  Salvage value $4k at end of 6 years  No borrowing, tax=50%, MARR=6%  Use SOYD and SL depreciation

18. 12-706 and 73-35918 Results - SOYD  D1=(6/21)*$42k = $12,000  SOYD really reduces taxable income!

19. 12-706 and 73-35919 Results - Straight Line Dep.  Now NPV is negative - shows effect of depreciation method on decision  Negative tax? Typically a credit

20. 12-706 and 73-35920 Let’s Add in Interest - Computer Again  Price $22k, $6k/yr benefits for 5 yrs, $2k salvage after year 5  Borrow $10k of the $22k price  Consider single payment at end and uniform yearly repayments  Depreciation: Double-declining balance  Income tax rate=50%  MARR 8%

21. 12-706 and 73-35921 Single Repayment  Had to ‘manually adjust’ D t in yr. 5  Note loan balance keeps increasing  Only additional interest noted in I t as interest expense

22. 12-706 and 73-35922 Uniform payments  Note loan balance keeps decreasing  NPV of this option is lower - should choose previous (single repayment at end).. not a general result