Net Present Value

EXERCISE 01

Pay €750,000 today to receive €885,000 in one year. Comparable investments require 20%.

A.Calculate the return and decide whether to invest.

A.

IRR=885{,}000/750{,}000-1=18\%

NPV=-750{,}000+885{,}000/1.2=-12{,}500

Reject: NPV is negative.

Net Present Value

EXERCISE 02

Buy a building for €400,000 and spend €350,000 immediately on refurbishment. One year later, sell six apartments for €200,000 each. Comparable investments require a 20% return.

A.Calculate the project return and NPV.
B.Explain the investment decision.

A.

R=\frac{1{,}200{,}000-750{,}000}{750{,}000}=60\%

NPV=-750{,}000+\frac{1{,}200{,}000}{1.20}=250{,}000

B.

Accept: the 60% return exceeds the 20% opportunity cost and the project creates €250,000 of value today.

Net Present Value

EXERCISE 03

An investment costs $2,000 and pays $672, $868 and $784 at the ends of years 1–3. The required return is 12%.

A.Calculate NPV and decide whether to invest.

A.

NPV=-2{,}000+\frac{672}{1.12}+\frac{868}{1.12^2}+\frac{784}{1.12^3}

NPV = $-150.00. Reject.

Net Present Value

EXERCISE 04

Invest $10,000 today to receive $2,500 at the end of each of the next five years. The appropriate discount rate is 9%.

A.Calculate present value and net present value.
B.Should the investment be accepted?

A.

PV=2{,}500\frac{1-(1.09)^{-5}}{0.09}

Measure	Amount ($)
Present value of inflows	9,724.13
Initial investment	(10,000.00)
Net present value	-275.87

NPV=-10{,}000+2{,}500\frac{1-1.09^{-5}}{0.09}\approx-275.87

B.

Reject it: NPV is negative. Receiving more dollars in total than the initial outlay does not guarantee value creation once timing and opportunity cost are considered.

Net Present Value

EXERCISE 05

An investment costs €10,600, pays €1,750 at the end of each of years 1–4 and €8,500 in year 5. Required return is 13.75%.

A.Calculate NPV and decide whether to invest.

A.

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

Year	0	1	2	3	4	5
Free cash flow	-10,600.00	1,750.00	1,750.00	1,750.00	1,750.00	8,500.00
Discount factor	1.0000	0.8791	0.7729	0.6794	0.5973	0.5251
Present value	-10,600.00	1,538.46	1,352.49	1,189.01	1,045.28	4,463.36

NPV=(-10{,}600)+\frac{1{,}750}{1.1375}+\frac{1{,}750}{1.1375^{2}}+\frac{1{,}750}{1.1375^{3}}+\frac{1{,}750}{1.1375^{4}}+\frac{8{,}500}{1.1375^{5}}

NPV = -1,011.40. Reject.

Net Present Value

EXERCISE 06

Amounts are in thousands of dollars. The opportunity cost of capital is 10%.

Project	Year 0	Year 1	Year 2	Year 3
A	-112	40	50	60
B	45	60	-70	-70
C	-100	-26	80	80
D	146	-70	-60	-50
E	-100	450	-550	175

A.Which independent projects create value?

A.

NPV_A=-112+\frac{40}{1.10^{1}}+\frac{50}{1.10^{2}}+\frac{60}{1.10^{3}}\approx 10.76

NPV_B=45+\frac{60}{1.10^{1}}-\frac{70}{1.10^{2}}-\frac{70}{1.10^{3}}\approx -10.9

NPV_C=-100-\frac{26}{1.10^{1}}+\frac{80}{1.10^{2}}+\frac{80}{1.10^{3}}\approx 2.58

NPV_D=146-\frac{70}{1.10^{1}}-\frac{60}{1.10^{2}}-\frac{50}{1.10^{3}}\approx -4.79

NPV_E=-100+\frac{450}{1.10^{1}}-\frac{550}{1.10^{2}}+\frac{175}{1.10^{3}}\approx -13.97

Project	NPV	Decision
A	10.76	Accept
B	-10.90	Reject
C	2.58	Accept
D	-4.79	Reject
E	-13.97	Reject

Use NPV for every cash-flow pattern. A financing-type cash-flow stream reverses the usual IRR rule, and multiple sign changes can produce multiple IRRs.

Net Present Value

EXERCISE 07

Year	0	1	2
Cash flow	-7000	5000	20000

A.Calculate NPV at 0%, 50% and 100%.
B.Find IRR.

A.

NPV(r)=-7000+\frac{5000}{1+r}+\frac{20000}{(1+r)^2}

Discount rate	NPV
0.00%	18,000.00
50.00%	5,222.22
100.00%	500.00

At 50%, the correct NPV is €5,222.22; the larger value printed in the source does not follow from these cash flows.

B.

0=-7000(1+IRR)^2+5000(1+IRR)+20000

1+IRR=\frac{5000+\sqrt{5000^2+4(7000)(20000)}}{14000}

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