basic discounting formula: $PV = $FV / (1 + r)^t
perpetual annuity of $pmt/year: $PV = $pmt/r
Calculate the $FV of $PV = $100 invested for t = 10 years at annual growth rate
r=
0.03
r=
0.06
r=
0.1
$FV =
$FV =
$FV =
$134.39
$179.08
$259.37
Calculate the $PV of $FV = $1 million discounted at rate r = 0.06 to be received
t=
5 years from now
t=
10 years from now
t=
20 years from now
$PV =
$PV =
$PV =
$747,258
$558,395
$311,805
Calculate the implicit rate of return r on a $PV = $1 million investment that
yields $FV = $2 million
t=
8 years from now
t=
9 years from now
t=
12 years from now
IRR =
IRR =
IRR =
r = (FV/PV)^(1/t) - 1
0.0905
0.0801
0.0595
Calculate how many years t are required to increase your $PV by 50% at annual
growth rate
r=
0.04
r=
0.07
r=
0.1
Years =
Years =
Years =
t = log(FV/PV)/log(1+r)
10.34
5.99
4.25
if r =
then $PV =
…so:
(using perpetual annuity as approx)
0.045
0.05
0.055
$2,222,222
$2,000,000
$1,818,182
build
break even
don't build
A sewage treatment facility will cost $2 million to build and $100,000/year to
operate over a 50-year lifespan. The environmental benefits will be $200,000/year
over the facility's lifespan. How do you decide if this facility should be built?
A new scrubber on a coal-fired power plant will cost $600,000 today and has a 5-year
lifespan. It will cut the plant's SO2 emissions by 400 tons/year starting next year.
The plant is currently paying $200/ton in the market for SO 2 emissions permits but
expects the price of permits to rise 20% each year. Should the plant install the
scrubber or keep buying 400 SO2 permits for 5 more years?
costs:
year scrubber
permitP
permits
0 $600,000
$200
1
$240
$96,000
2
$288
$115,200
3
$346
$138,240
4
$415
$165,888
5
$498
$199,066
$714,394
year
0
1
2
3
4
5
net$PV:
if r =
0.045
(600,000)
91,866
105,492
121,139
139,107
159,740
17,345
scrubber
0.050
(600,000)
91,429
104,490
119,417
136,476
155,973
7,785
scrubber
rule of 72: t*r
0.724
0.721
0.714
0.055
(600,000)
90,995
103,502
117,727
133,908
152,312
(1,556)
permits