Problem 3:
Compute the present value of a $100 investment made six months, five years, and ten years from now at 4 percent interest.

Answer:
Future Value (FV) = $100
Interest Rate (r) = 4% or 0.04
Present Value (PV) =?

Present value of future amount made six months: (Time (t) = 6 months or 6/12 = 0.5 years)
PV = FV / (1+r) t
PV = 100 / (1+0.04) 0.5
PV = 100 / (1.04) 0.5
PV = 100 / 1.0198
PV = $98.06

Excel Hint:
=PV (rate, nper, pmt, [fv], [type])
=PV (4%, 0.5, 0, -100, 0) OR =PV (4%, 6/12, 0, -100, 0)

(Additional information: Power of 0.5 on a variable x is equal to the square root of x. e.g. x0.5 = √x.)

Present value of future amount made five years: (Time (t) = 5 years)
PV = FV / (1+r) t
PV = 100 / (1+0.04) 5
PV = 100 / (1.04) 5
PV = 100 / 1.2167
PV = $82.19

Excel Hint:
=PV (rate, nper, pmt, [fv], [type])
=PV (4%, 5, 0, -100, 0)

Present value of future amount made five years: (Time (t) = 10 years)
PV = FV / (1+r) t
PV = 100 / (1+0.04) 10
PV = 100 / (1.04) 10
PV = 100 / 1.4802
PV = $67.56

Excel Hint:
=PV (rate, nper, pmt, [fv], [type])
=PV (4%, 10, 0, -100, 0)

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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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Problem 1:
Compute the future value of $100 at an 8 percent interest rate five, ten, and fifteen years into the future.

Solution:
Present Value (PV) = $100
Interest Rate (r) = 8% or 0.08 per year

FV5 =? When Time (t) = 5 years
FV10 =? When Time (t) = 10 years
FV15 =? When Time (t) = 15 years

FV5 = PV * (1 + r) t
FV5 = 100 * (1 + 0.08) 5
FV5 = 100 * (1.08) 5
FV5 = 100 * 1.4693280768
FV5 = $146.93

FV10 = PV * (1 + r) t
FV10 = 100 * (1 + 0.08) 10
FV10 = 100 * (1.08) 10
FV10 = 100 * 2.15892499727278669824
FV10 = $215.89

FV15 = PV * (1 + r) t
FV15 = 100 * (1 + 0.08) 15
FV15 = 100 * (1.08) 15
FV15 = 100 * 3.172169114198268924301601144832
FV15 = $317.22

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Money, Banking, and Financial Markets written by Stephen G. Cecchetti.
We are going to solve, all the questions of this book for you!!!
Enjoy!
About the Author:
Stephen G. Cecchetti joined the Brandeis University faculty in 2003 as a Professor of International Economics and Finance at the International Business School. He is also the Director of Research at the Rosenberg Institute for Global Finance at Brandeis. Previously, Professor Cecchetti taught at the New York University Stern School of Business and, for approximately 15 years, was a member of the Department of Economics at The Ohio State University. He has been a Visiting Professor of Economics at Princeton University, Oxford University, the University of Melbourne, and Boston College.