Canadian Mortgage Loans:
A two-step procedure is applied to calculate the monthly rate in order to calculate the monthly
payment:
a. First, convert the APR to EAR: EAR = (1 + 6%/2)^2 – 1 = 6.09%
b. Second, calculate the monthly rate (r) with the same EAR as 6.09%: 6.09% = (1 + r)^12 – 1
c. Solve the above equation, r = 0.00494
Growing Annuities and Growing Perpetuities:
—> Growing annuity is a series of ﬁnite cash ﬂows that grow at a ﬁxed rate.
—> Growing perpetuity is a constant stream of cash ﬂows without end that is expected to rise
indeﬁnitely.
—> Note: “grow” does not necessary means “increase”: the growth rate can be positive or
negative
e.g, Assume your starting salary after graduation is $40,000 and it expects
N
to grow at 10% annually until you retire. What is the PV of all your fCtur⎪ ⎡1+g ⎤ ⎪
expected salary if you will be in employed for 40 years ant the interest rate ⎢ ⎥ ⎬
I −g ⎩ ⎣1+I ⎦ ⎭
is 6%
I=0.06 = interest rate g=0.1 = growC=$40,000 = cash payment at the ﬁrst period
Thus, PV = $3,400,203
e.g, What is the value of a growing perpetuity with the ﬁrst payment of $1,000 and the
PV =
growth rate of 4% if the interest rate is 12%? I −g
I=0.12 = interest rate g=0.04 = growth rate C=$1,000 = cash payment at the ﬁrst period
—> NOTE: we must have r>g in order to have the ﬁnite value of a perpetuity
Thus, PV = $1,000/(0.12-0.04)=$12,500
e.g, you will live for 20 years after you retire and you want to be able to spend $40,000/year.
You plan to retire in 40 years and want to contribute an equal amount of money into your saving
plan each year starting next year. What your annual contributions should be if the interest rate is
5% a. First, ﬁnd PV of your future withdrawals.
N=20; I=5%, PMT = 40,000, FV = 0 Thus, PV (at N=40) = $498,488
b. Therefore, PV (at N=0) = $ 498,488 /(1.05^40)= $70,808
The PV of your savings must be equal to the PV of your withdrawals:
PV = - $70,808; N=40; I=5%. Thus, PMT = $4,127/year.
e.g, You think, you will live for 20 years after you retire and you want to be able to spend
$40,000/year adjusted for inﬂation. Assuming the inﬂation rate is 2%/year it means that you
need $40,000*(1.02^41)=$90,088 in your ﬁrst retirement year and this ﬁgure grows with the rate
of 2%/year. You plan to retire in 40 years and want to contribute an equal percentage of your
salary into your saving plan each year starting next year. Your salary next year will be $40,000
and you expect it to grow 8% per year. What

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