Suppose that over the past 20 years the average annual return on investment has been 10.7 percent. For each dollar invested at the beginning of the period, how much money would investors have at the end? What if they had kept the investment for only 10 years? For 30 year?

Problem 2:
Suppose that over the past 20 years the average annual return on investment has been 10.7 percent. For each dollar invested at the beginning of the period, how much money would investors have at the end? What if they had kept the investment for only 10 years? For 30 year?

Solution:
Payment (C) = $1
Interest Rate (r) = 10.7% or 0.107 per year

FV20 =? When Time (t) = 20 years
FV10 =? When Time (t) = 10 years
FV30 =? When Time (t) = 30 years

This is the case of annuity. Annuity is the cash flow stream and payment stream where the cash flows or payments are constant or equal in amount. There are both future value annuity and present value annuity. Also, if the payments or cash flows are occurring at the end of each period (as by default) then annuity is called as Ordinary Annuity, and if the payments or cash flows are occurring at the beginning of each period then the annuity is called as Annuity Due. This is the case of Future Value Annuity Due:

The formulas for different annuities are given as under:
PV Ordinary Annuity = C * (1 – 1 / (1+r) t) / r
FV Ordinary Annuity = C * ((1+r) t – 1) / r

The difference between ordinary annuity and annuity due is just of (1+r).
Annuity Due = Ordinary Annuity * (1+r), this is because as we talk about future value annuity and the payments are occurring at the beginning of each year, then they will be compounded one time more as compared to if payments occur at the end of each period. So, the formulas for Annuity Due for both PV and FV becomes
PV Annuity Due = C * (1 – 1 / (1+r) t) / r (1+r)
FV Annuity Due = C * ((1+r) t – 1) / r * (1+r)

This question is of Future Value Annuity Due. Let’s solve how much amount will be at the end of 20, 10 and 30 years.

FV after 20 years:
FV Annuity Due = C * ((1+r) t – 1) / r * (1+r) (when t=20 years)
FV Annuity Due = 1 * ((1+0.107) 20 – 1) / 0.107 * (1+0.107)
FV Annuity Due = 1 * ((1.107) 20 – 1) / 0.107 * (1.107)
FV Annuity Due = 1 * (7.6375 – 1) / 0.107 * (1.107)
FV Annuity Due = 1 * (6.6375) / 0.107 * (1.107)
FV Annuity Due = 6.6375 / 0.107 * (1.107)
FV Annuity Due = 62.0327 * (1.107)
FV Annuity Due = $68.67

Excel Hint:
=FV (rate, nper, pmt, [pv], [type])
=FV (10.7%, 20, -1, 0, 1)
Here,
PMT is payment and is considered as $-1, just because that payment is made from us or from investor to any bank is the cash outflow, and outflow is always considered as with negative sign and inflow with positive sign.
TYPE is the type of annuity, either ordinary annuity or annuity due, 0 for ordinary annuity and 1 for annuity due.

FV after 10 years:
FV Annuity Due = C * ((1+r) t – 1) / r * (1+r) (when t=10 years)
FV Annuity Due = 1 * ((1+0.107) 10 – 1) / 0.107 * (1+0.107)
FV Annuity Due = 1 * ((1.107) 10 – 1) / 0.107 * (1.107)
FV Annuity Due = 1 * (2.7636 – 1) / 0.107 * (1.107)
FV Annuity Due = 1 * (1.7636) / 0.107 * (1.107)
FV Annuity Due = 1.7636 / 0.107 * (1.107)
FV Annuity Due = 16.4822 * (1.107)
FV Annuity Due = $18.25

Excel Hint:
=FV (rate, nper, pmt, [pv], [type])
=FV (10.7%, 10, -1, 0, 1)

FV after 30 years:
FV Annuity Due = C * ((1+r) t – 1) / r * (1+r) (when t=30 years)
FV Annuity Due = 1 * ((1+0.107) 30 – 1) / 0.107 * (1+0.107)
FV Annuity Due = 1 * ((1.107) 30 – 1) / 0.107 * (1.107)
FV Annuity Due = 1 * (21.1071 – 1) / 0.107 * (1.107)
FV Annuity Due = 1 * (20.1071) / 0.107 * (1.107)
FV Annuity Due = 20.1071 / 0.107 * (1.107)
FV Annuity Due = 187.9168 * (1.107)
FV Annuity Due = $208.02

Excel Hint:
=FV (rate, nper, pmt, [pv], [type])
=FV (10.7%, 30, -1, 0, 1)

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