Annuity: PVt = [payment * (1 - (1 / ((1 + r)^n)))] / r
n payment ($ PMT)
1st payment starts at t =1, r is interest rate for a period between 2 consecutive payments
we can use this formula too that shows the exact same thing! FVn = PVt (1 + r)^n
FV for annuity due = { [payment * (1 - (1 / ((1 + r)^n)))] / r } * (1 + r)^(n+1)
-for pv and fv in calculator, one of them must be -‘ve as to not get an error
e.g, Suppose we now have $500,000 and the interest rate is 4.5%. How long will it take to grow
to $1 million?
$1 million = $500,000 (1 + 0.045)^N
$1 million / $500,000 = (1.045)^N
2 = (1.045)^N
Ln(2) / Ln(1.045) = N
15.7473 years = N
calculator steps: 4.5, I/Y, -500, PV, 1000, FV, CPT, N
e.g, Suppose you are currently 21 years old, and can earn 10 percent on your money. How
much must you invest today in order to accumulate $1 million by the time you reach age 65 (i.e.,
in 44 years)? ANS: PV = $1 million/(1.10) 44= $15,091.
e.g, What if you are currently 31 y.o. (i.e., you have only 34 years to retirement)?
34
ANS: PV = $1 million/(1.10) = $39,142.
e.g, Suppose you deposit $5000 today in an account paying r percent per year. If you will get
$10,000 in 10 years, what rate of return are you being offered?
ANS: $5000 = $10,000/(1 + r)^10
$10,000/$5,000 = (1 + r)^10
(2.00)^(1/10) - 1 = r e.g, Imprudential, Inc. has an unfunded pension liability of $425 million that must be paid in 20
years. To assess the value of the ﬁrm’s stock, ﬁnancial analysts want to discount this liability
back to the present. If the relevant discount rate is 8 percent, what is the present value of

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