Adjusted Present Value

EXERCISE 01

A three-year project has unlevered cost of capital 10.00%. It borrows €1,000.00, with annual interest 2.00% and all principal repaid at the end of year 3. Corporate tax is 35.00%. Discount the certain, fully usable interest tax shields at the debt rate.

Year	0	1	2	3
Operating free cash flow	-1200	200	850	850

A.Calculate unlevered NPV and the financing benefit.

A.

APV=NPV_U+PV(TS)

NPV=\sum_{t=0}^{N}\frac{FCF_t}{(1+r)^t}

Year	0	1	2	3
Free cash flow	-1,200.00	200.00	850.00	850.00
Discount factor	1.0000	0.9091	0.8264	0.7513
Present value	-1,200.00	181.82	702.48	638.62

NPV=(-1{,}200)+\frac{200}{1.1}+\frac{850}{1.1^{2}}+\frac{850}{1.1^{3}}

NPV = 322.92. Accept.

TS_t=0.35(1000)(0.02)=7

PV(TS)=7\frac{1-1.02^{-3}}{0.02}=20.1872

APV=322.9151+20.1872

APV = €343.10. Accept.

The initial debt proceeds and principal repayment are financing transfers, not extra operating value.

Adjusted Present Value

EXERCISE 02

A three-year project has unlevered cost of capital 8.00%. It borrows €800.00, with annual interest 3.00% and all principal repaid at the end of year 3. Corporate tax is 30.00%. Discount the certain, fully usable interest tax shields at the debt rate.

Year	0	1	2	3
Operating free cash flow	-1000	300	500	800

A.Calculate unlevered NPV and the financing benefit.

A.

APV=NPV_U+PV(TS)

NPV=\sum_{t=0}^{N}\frac{FCF_t}{(1+r)^t}

Year	0	1	2	3
Free cash flow	-1,000.00	300.00	500.00	800.00
Discount factor	1.0000	0.9259	0.8573	0.7938
Present value	-1,000.00	277.78	428.67	635.07

NPV=(-1{,}000)+\frac{300}{1.08}+\frac{500}{1.08^{2}}+\frac{800}{1.08^{3}}

NPV = 341.51. Accept.

TS_t=0.3(800)(0.03)=7.2

PV(TS)=7.2\frac{1-1.03^{-3}}{0.03}=20.3660

APV=341.5130+20.3660

APV = €361.88. Accept.

The initial debt proceeds and principal repayment are financing transfers, not extra operating value.

Adjusted Present Value

EXERCISE 03

Apples & Oranges is evaluating a 5-year project. Equipment costs 15,000.00, annual straight-line depreciation is 1,875.00, and expected terminal sale proceeds are 6,500.00. The corporate tax rate is 30.00%. Operating working capital equals 5.00% of sales, is invested at each year-end and fully recovered in the last year. Tax losses can offset other taxable profits immediately. Annual rent is 500. The 3,000 already spent on research is sunk. A further deductible 2,000 is paid at launch, with an immediate 600 tax benefit. The unlevered beta is 1.5, risk-free rate 5% and market return 10.5%, giving RU = 13.25%. For the financing alternative, a five-year bullet loan of 15,000 costs 7.5%.

Year	1	2	3	4	5
Revenue	12,000.00	13,200.00	14,520.00	15,972.00	17,569.20
Cash operating costs	6,000.00	6,600.00	7,260.00	7,986.00	8,784.60
Additional incremental costs	500.00	500.00	500.00	500.00	500.00

A.Calculate the discount rate and evaluate the project.
B.Calculate adjusted present value with the bullet loan.

A.

The immediate net investment is 15,000 + 2,000(1−0.30) = 16,400. Equipment tax basis remains 15,000. RU = 5% + 1.5(10.5%−5%) = 13.25%.

CF_{sale}=6500-0.30(6500-(15000-5(1875)))=6237.50

Year	1	2	3	4	5
EBIT	3625	4225	4885	5611	6409.6
OCF	4412.5	4832.5	5294.5	5802.7	6361.72
Change in NWC	600	60	66	72.6	-798.6

NPV=\sum_{t=0}^{N}\frac{FCF_t}{(1+r)^t}

Year	0	1	2	3	4	5
Free cash flow	-16,400.00	3,812.50	4,772.50	5,228.50	5,730.10	13,397.82
Discount factor	1.0000	0.8830	0.7797	0.6885	0.6079	0.5368
Present value	-16,400.00	3,366.45	3,721.08	3,599.67	3,483.45	7,191.89

NPV=(-16{,}400)+\frac{3{,}812.50}{1.1325}+\frac{4{,}772.50}{1.1325^{2}}+\frac{5{,}228.50}{1.1325^{3}}+\frac{5{,}730.10}{1.1325^{4}}+\frac{13{,}397.82}{1.1325^{5}}

NPV = 4,962.54. Accept.

B.

APV=NPV_U+PV(TS)

PV(TS)=15000(0.075)(0.30)a_{5|0.075}=1{,}365.49

APV = 6,328.02.

Adjusted Present Value

EXERCISE 04

A debt-free company can buy a machine for 2,400, depreciated straight-line over four years, with terminal sale proceeds 250. Annual EBITDA is 850, tax 35%, no further investment or working capital is needed, and the unlevered required return is 13%. A four-year loan of 1,600 costs 5%. The wording “constant instalments” can mean equal total payments; also evaluate the equal-principal convention.

A.Calculate unlevered NPV.
B.Calculate annual tax shields and APV with equal principal repayments.
C.Recalculate for equal total annual payments and discuss the valuation method.

A.

Dep=2400/4=600,\quad OCF=(850-600)(0.65)+600=762.50

CF_{sale}=250(1-0.35)=162.50

NPV=\sum_{t=0}^{N}\frac{FCF_t}{(1+r)^t}

Year	0	1	2	3	4
Free cash flow	-2,400.00	762.50	762.50	762.50	925.00
Discount factor	1.0000	0.8850	0.7831	0.6931	0.6133
Present value	-2,400.00	674.78	597.15	528.45	567.32

NPV=(-2{,}400)+\frac{762.50}{1.13}+\frac{762.50}{1.13^{2}}+\frac{762.50}{1.13^{3}}+\frac{925}{1.13^{4}}

NPV = -32.30. Reject.

B.

Principal_t=1600/4=400,\quad TS_t=T_cR_DD_{t-1}

Year	Opening debt	Interest	Tax shield
1	1600	80	28
2	1200	60	21
3	800	40	14
4	400	20	7

PV(TS)=\frac{28}{1.05}+\frac{21}{1.05^2}+\frac{14}{1.05^3}+\frac{7}{1.05^4}=63.57

APV = 31.27. Accept with this financing; reject on an unlevered basis.

C.

Payment=\frac{1600}{a_{4|0.05}}=451.218932

Year	Opening debt	Payment	Interest	Principal	Tax shield
1	1,600.00	451.22	80.00	371.22	28.00
2	1,228.78	451.22	61.44	389.78	21.50
3	839.00	451.22	41.95	409.27	14.68
4	429.73	451.22	21.49	429.73	7.52

PV of shields = 65.04; APV = 32.74.

APV handles the changing debt balance directly. A correctly constructed time-varying WACC approach can also work; a single constant WACC is generally inappropriate for this amortising loan.

Adjusted Present Value

EXERCISE 05

A.Why is APV useful when leverage changes over the life of a project?

A.

APV=NPV_U+PV(\mathrm{financing\ effects})

APV separates the value of unlevered operating cash flows from financing effects, including interest tax shields. It is particularly useful when the debt schedule makes a constant WACC inappropriate. Changing operating risk still requires appropriate time-specific operating discount rates; APV does not remove that requirement.