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Advanced Finance 2007 03 APV |104/10/23
The Adjusted Present Value Rule
• The most straightforward. Permits the user to see the sources of value in the project, if it's accepted
• Procedure:
– (1) Compute the base-case NPV using a discount rate that employs all equity financing (rA), applied to the project's cash flows
– (2) Then, adjust for the effects of financing which arise from:
• Flotation costs
• Tax Shields on Debt Issued
• Effects of Financing Subsidies
» APV = NPV + NPVF
Advanced Finance 2007 03 APV |204/10/23
Advanced Finance 2007 03 APV |304/10/23
APV - Example
• Data
– Cost of investment 10,000
– Incremental earnings 1,800 / year
– Duration 10 years
– Discount rate rA 12%
• NPV = -10,000 + 1,800 x a10 = 170
• (1) Stock issue:
• Issue cost : 5% from gross proceed
• Size of issue : 10,526 (= 10,000 / (1-5%))
• Issue cost = 526
• APV = + 170 - 526 = - 356
Advanced Finance 2007 03 APV |404/10/23
APV calculation with borrowing
• Suppose now that 5,000 are borrowed to finance partly the project
• Cost of borrowing : 8%
• Constant annuity: 1,252/year for 5 years
• Corporate tax rate = 40%
Year Balance Interest Principal Tax Shield
1 5,000 400 852 160
2 4,148 332 920 133
3 3,227 258 994 103
4 2,223 179 1,074 72
5 1,160 93 1,160 37
• PV(Tax Shield) = 422
• APV = 170 + 422 = 592
Advanced Finance 2007 03 APV |504/10/23
Discounting Safe, Nominal Cash Flows
“The correct discount rate for safe, nominal cash flows is your company’s after-tax, unsubsidized borrowing rate” (Brealey and Myers sChap19 – 19.5)
• Discounting
– after-tax cash flows
– at an after-tax borrowing rate rD(1-TC)
• leads to the equivalent loan (the amount borrowed through normal channels)
• Examples:
– Payout fixed by contract
– Depreciation tax shield
– Financial lease
Advanced Finance 2007 03 APV |604/10/23
APV calculation with subsidized borrowing
• Suppose now that you have an opportunity to borrow at 5% when the market rate is 8%.
• What is the NPV resulting from this lower borrowing cost?
• (1) Compute after taxes cash flows from borrowing
• (2) Discount at cost of debt after taxes
• (3) Subtract from amount borrowed
• The approach developed in this section is also applicable for the analysis of leasing contracts (See B&M Chap 25)
Advanced Finance 2007 03 APV |704/10/23
Subsidized loan
• To understand the procedure, let’s start with a very simple setting:
• 1 period, certainty
• Cash flows after taxes: C0 = -100 C1 = + 105
• Corporate tax rate: 40%, rA=rD=8%
• Base case: NPV0= -100 + 105/1.08 = -2.78 <0
• Debt financing at market rate (8%)
• PV(Tax Shield) = (0.40)(8) / 1.08 = 2.96
• APV = - 2.78 + 2.96 = 0.18 >0
Advanced Finance 2007 03 APV |804/10/23
NPV of subsidized loan
• You can borrow 100 at 5% (below market borrowing rate -8%). What is the NPV of this interest subsidy?
Net cash flow with subsidy at time t=1: -105 + 0.40 × 5 = -103
• How much could I borrow without subsidy for the same future net cash flow?
• Solve: B + 8% B - 0.40 × 8% × B = 103
• Solution:
• NPVsubsidy = +100 – 98.28 = 1.72
28.98048.1
103
)40.01%(81
103
B
Net cash flow
After-tax interest rate
PV(Interest Saving)=(8 – 5)/1.048 = 2.86 + PV(∆TaxShield)
=0.40(5 – 8)/1.048 = -1.14
Advanced Finance 2007 03 APV |904/10/23
APV calculation
• NPV base case NPV0 = - 2.78
• PV(Tax Shield) no subsidy PV(TaxShield) = 2.96
• NPV interest subsidy NPVsubsidy = 1.72
• Adjusted NPV APV = 1.90
• Check After tax cash flows
• t = 0 t = 1
• Project - 100 + 105
• Subsidized loan +100 - 103
• Net cash flow 0 + 2
• How much could borrow today against this future cash flow?
• X + 8% X - (0.40)(8%) X = 2 → X = 2/1.048 = 1.90
Advanced Finance 2007 03 APV |1004/10/23
A formal proof
• Ct : net cash flow for subsidized loan
• r : market rate
• D : amount borrowed with interest subsidy
• B0 : amount borrowed without interest subsidy to produce identical future net cash flows
• Bt : remaining balance at the end of year t
• For final year T: CT = BT-1 + r(1-TC) BT-1
• (final reimbursement + interest after taxes)
• 1 year before: CT-1 = (BT-2 - BT-1) + r(1-TC) BT-2
• (partial reimbursement + interest after taxes)
• At time 0:
• NPVsubsidy = D – B0
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Advanced Finance 2007 03 APV |1104/10/23
Back to initial example
DataMarket rate 8%Amount borrowed 5,000Borrowing rate 5%Maturity 5 yearsTax rate 40%Annuity 1,155
Net Cash Flows CalculationYear Balance Interest Repayment TaxShield Net CF 1 5,000 250 905 100 1,055 2 4,095 205 950 82 1,073 3 3,145 157 998 63 1,092 4 2,147 107 1,048 43 1,112 5 1,100 55 1,100 22 1,133
B0 = PV(NetCashFlows) @ 4.80% = 4,750NPVsubsidy = 5,000 - 4,750 = + 250
APV calculation:NPV base case NPV0 = + 170PV Tax Shield without subsidy PV(TaxShield) = + 422NPV Subsidy NPVsubsidy = + 250APV = + 842
Advanced Finance 2007 03 APV |1204/10/23
Financial lease
• A source of financing:Extends over most of the economic life of the assetCannot be canceledSimilar to a secured loan
• 2 parties:Lessor: legal owner of the leased assset
Receives rental income (taxable)Uses the depreciation tax shield
Lessee: user of the the leased assetLease payment tax deductible
Advanced Finance 2007 03 APV |1304/10/23
Lease versus borrow
DataCost of equipment 10,000 Linear dep. 5 years Tax rate 34%Before-taxe operating savings 6,000 Cost of debt 7.58%Lease payment 2,500 After-tax cost of debt 5.00%
Lease minus buy 0 1 2 3 4 5Cost of equipment 10,000Lost depreciation tax shield -680 -680 -680 -680 -680Lease payment -2,500 -2,500 -2,500 -2,500 -2,500Tax shield of lease payment 850 850 850 850 850Net cash flow of lease minus buy 10,000 -2,330 -2,330 -2,330 -2,330 -2,330
Advanced Finance 2007 03 APV |1404/10/23
Calculating NPV of lease versus buy
• Discount after-tax cash flow at the after-tax interest rate.
= Cost of asset – Equivalent loan
• Example: After-tax interest rate = 7.58% * (1-0.34) = 5%
• NPV = 10,000 – 10,087.68 = -87.68 => buy and borrow
N
tt
CD Tr0 ))1(1(
flowcash leaseNet -asset ofCost NPV
Equivaleur loan calculation 10,087.68 Amount that could be borrowed against the same net cash flowsAmount borrowed year-beginning 10,087.68 8,262.06 6,345.17 4,332.43 2,219.05Interest -764.22 -625.91 -480.69 -328.21 -168.11Interest tax shield 259.83 212.81 163.44 111.59 57.16Interest after-tax -504.38 -413.10 -317.26 -216.62 -110.95Principal repaid 1,825.62 1,916.90 2,012.74 2,113.38 2,219.05Net cash flow -2,330.00 -2,330.00 -2,330.00 -2,330.00 -2,330.00Amount borrowed year-end 8,262.06 6,345.17 4,332.43 2,219.05 0.00