For ECON 03C TPE#4
Cash Flow Future Value (FV) The value of cash flows at a given future date Present Value (PV) The value of cash flows today (time zero) r (Discount rate) The rate of return an investor would require
PV vs FV Compounding @ r 0*(+r) 5 00*(+r) 4 00*(+r) 3 00*(+r) 2 00*(+r) 00 Future Value Present Value 0 00/(+r) 00/(+r) 2 00/(+r) 3 00/(+r) 4 00/(+r) 5 Discounting @ r
PV vs FV For a lump sum payment (F) at the end of T time periods with the discount rate r PV= F/(+r) T FV=PV*(+r) T = F/(+r) T *(+r) T =F For an annuity which pays A at the end of each time period and continues for N time periods with the discount rate r PV = A r + r N FV=PV*(+r) N = A r +r N * + r N = A r + r N
PV with Changing r and a Lump Sum Payment Present Value +r 2 * +r 2 2 * 00 +r 3 Discounting @ r, r 2 and r 3 In this example, PV = + r 2 + r 2 00 2 + r 3 In the general case, PV = + r t + r 2 t 2 F + r 3 t 3
PV with Changing r and Annuity Present Value 0 00/(+ r ) 00/(+ r ) 2 00/(+ r ) 3 +r 3 *00/(+ r 2) +r 3 *00/(+ r 2) 2 ] Discounting @ r and r 2 In this example, In the general case, PV = 00 r +r 3 + +r 3 * 00 r 2 [ +r 2 2 ] PV = A r + * A [ +r n +r n r 2 +r 2 n2 ]
How PV changes with parameters? () For a lump sum payment (F) at the end of T time periods with the discount rate r PV= F* +r T =F*(+r) -T dpv dr = F T ( + r) T <0 dpv dt = F +r T ln ( +r ) <0 because r>0 => ( +r ) < => ln ( +r ) <0 => T/r increases, PV decreases
How PV changes with parameters? (2) For an annuity which pays A at the end of each time period and continues for N time periods with the discount rate r PV = A r +r N = A r +r N dpv dr = A A +r N r2 + - A N * (-N) * + r r 2 r = A r 2 + r N + r N+ + + r + N r < 0 because + r N+ = N + r => r increases, PV decreases dpv = A N ln ( dn r +r dpv = > 0 da r +r N +r => N/A increases, PV increases N+ N 2 r 2 ) > 0 because ln ( +r ) <0
Expensed vs Capitalized Expenditures No matter which method is chosen, the cash is paid at time zero. => The PVs of pre-tax cash outflow for the investment are same in two cases. But the PVs of taxes saved are different. e.g. Q. 3-4 At time zero: $M is paid for both methods. The PV of tax saved (t represents tax rate): Expensed expenditures: PV=t*M since expenditure is recognized as expense at time zero. Capitalized expenditures: PV= 5 k= t (0.2M) (5 k) = = t (0.2M) r +r 5 < t*m => For the tax payer, the PV of tax saved with expensed expenditures is higher than that with capitalized expenditures.
Depreciation How to calculate annual depreciation Straight-line method Annual depreciation expense = (cost of the asset the book value at the end of the expected life of the asset )/years e.g. Q. 5 Cost of the asset=00; B=5;Years=0 Annual depreciation expense = (00-5)/0=9.5
The Effect of Depreciation on Tax Payments Annual depreciation expense Reduce annual tax payment (positive cash flow): tax rate * annual depreciation expense The market price (S) and B. B reduces tax payment (positive cash flow) S induces tax payment (negative cash flow) => Net effect on tax = tax rate * B - tax rate * S if the result is positive (capital loss) => tax rebate if the result is negative (capital gain) => tax due
The Effect of Depreciation on Tax Payment (Q.5) Annual depreciation expense Reduce annual tax payment (positive cash flow) = 0.2* 9.5=.9 (annuity) At the termination: the market price (S) and B. Net effect = 0.2* 5 0.2 * 0 = - (tax due) (a lump sum ) PV of all the tax implications with r=5% (positive means cash inflow -> PV represents the PV of reduced tax payment related to this asset) PV =.9 0.5 +0.5 0 + +0.5 0 = 9.29 Note: From the tax perspective, higher positive PV is better when you compare different deprecation methods
Incremental Cash Flow Analysis Without Depreciation and Tax Q.2 () The initial outlay of cash (negative incremental cash flow) -00 Annual cash flow changes. With the equipment: net cash flow=total cash inflow cash outflow = 70-25=45 Without the equipment: net cash flow=45-25=20 => The equipment increases annual net cash flow by 45-20=25 (positive incremental cash flow, i.e. more cash inflow)
Incremental Cash Flow Analysis without depreciation and tax Q.2 (2) The life of the equipment 5 years NPV of the of all incremental changes in cash flow associated with the project. NPV = 25 0.0 +0.0 5 00 = 94.77-00< 0
Incremental Cash Flow Analysis with Depreciation and Tax () Background information: Q.23- Q.27 with one differences: the tax rate (t) = 40%, instead of 0% The initial net after-tax cash flow: -60.6 = -60.6+0 After-tax cash outflow (two parts): -60.6 = -60-0.6 Costs will be capitalized no tax involved at time zero: -60 The investment cost: -55 Part of the startup costs: -5 Costs which are expensed at time zero: The expensed startup costs: -*(-0.4)=-0.6 After-tax cash inflow: $0
Incremental Cash Flow Analysis with Depreciation and Tax (2) Annual after-tax incremental cash inflow during the life of the project ( for 0 years):.2=9+2.2 After-tax net incremental operating cash inflow: 5*(-0.4)=9 After-tax cash inflow induced by depreciation: 0.4*(55+5-5)/0=2.2 The after-tax net cash flow at the termination: 2.6=-3+5.6 After-tax cash outflow: -3 Clean-up costs : -5*(-0.4)= -3 After-tax cash inflow: 5.6=2+3.6 By the book value of the asset at the termination: 0.4*5=2 By the sell of the asset at the termination: (-0.4)*6=3.6
Incremental Cash Flow Analysis with Depreciation and Tax (3) The incremental after-tax cash flows for the project
Incremental Cash Flow Analysis with Depreciation and Tax (4) The NPV of the incremental after-tax cash flows for the project @ r=5% NPV = 60.6 +.2 0.05 +0.05 9 + 3.8 +0.05 0 = 60.6 +.2 0.05 +0.05 0 + 2.6 +0.05 0 = 27.48