adjusted present value

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


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


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


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


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)


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


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


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


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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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


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


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


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

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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

adjusted present value

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