1) Bank of Vancouver pays 7% simple interest on its savings account balances, whereas
Bank of Calgary pays 7% interest compounded annually. If you made a $6,000 deposit
in each bank, how much more money would you earn from your Bank of Calgary
account at the end of 9 years ?
Simple Interest Formula A = P (1 + rt)
Simple Interest from Bank of Vancouver = $6,000 * 7% * 9 = $3780
So Simple Interest Gain from Bank of Vancouver is $3780/Compound Interest Formula A = P( 1 + r/n) nt
Whereas
A
=
P
=
r
=
n
=
t
=
final amount
initial principal balance
interest rate
number of times interest applied per time period
number of time periods elapsed
Compound Interest from Bank of Calgary = $6,000 (1 + (0.070/1) ^1*9 = $11,030.76
So, $11.030.76 - $6,000 = $5,030.76
So, Compound Interest gained from Bank of Calgary= $5,030.76
Extra money earned through Compound Interest from Bank of Calgary is
=$5030.76 - $ 3780
Gain of = $1250.76/-
6) Calculating Interest Rates (LO3) assume the total cost of a university education will
be $300,000 when your child enters university in 18 years. You currently have $65,000
to invest. What annual rate of interest must you earn on your investment to cover the
cost of your child’s university education?
Whereas
PV = 65K
FV = 300K
N = 18,
r =?
Using the Formula of Present Value,
PV = FV/(1+r)t
65000 = 300000/(1+r)18
r = (300000/65000) 1/18-1
r = 0.887
r = 8.87%
Annual Rate of Interest is 8.87%
7) Calculating the Number of Periods (LO3) At 6.5% interest, how long does it take to
double your money? To quadruple it ?
Double your money.
We can implement the Rule of 72 to estimate how long it will take to double an
investment at a given interest rate. By dividing 72 by the interest rate to see how long
it will take to double your money on an investment.
The formula is interest rate multiplied by the number of time periods = 72:
R * t = 72
Whereas
R = interest rate per period as a percentage
t = number of periods
Commonly, periods are years so R is the interest rate per year and t is the number of
years. You can calculate the number of years to double your investment at some
known interest rate by solving for t: t = 72 ÷ R
T = 72 ÷ 6.50%
T = 11.07 Years to Double your money
Quadruple your money
The formula to calculate it is: T = ln(4) / ln(1 + r),
where T is the quadrupling time and r is the rate of growth (expressed as a
decimal).
In this case,
T = ln(4) / ln(1 + 0.065)
T = ln(4) / ln(1.065)
T = 22.013 Years to Quadruple your money
10) Calculating Present Values (LO2) Normandin Inc. has an unfunded pension liability
of $575 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 6.8%, what is the present value of this liability?
The formula to calculate the present value of an unfunded pension liability is
PV = FV / (1 + r)^t.
Where PV is the present value,
FV is the future value of the liability ($575 million),
r is the discount rate (0.068 or 6.8%,
and t is the number of years until the liability is paid (20).
PV = $575 million / (1 + 0.068)^20 = $154.26 million
Present Value is = $154.26 million
12) Calculating Future Values (LO1) Your coin collection contains fifty 1952 silver
dollars. If your grandparents purchased them for their face value when they were
new, how much will your collection be worth when you retire in 2060, assuming they
appreciate at a 4.1% annual rate?
Future value of an investment, you can use the formula: FV = PV x (1 + r)^t
Where FV is the future value,
PV is the present value (the face value of the silver dollars, $50),
r is the annual appreciation rate (0.041 or 4.1% expressed as a decimal),
and t is the number of years until you retire (2060 - 1952).
FV = $50 x (1 + 0.041)^(2060 - 1952)
FV = $50 x (1 + 0.041)^108
FV = 3 833.97
The Future Value of the coin in year 2060 will be $3833.97 considering the appreciation
rate per year is 4.1%
17) Calculating Rates of Return (LO3) The “Brasher doubloon,” which was featured in
the plot of the Raymond Chandler novel The High Window, was sold at auction in 2014
for $4,582,500. The coin had a face value of $15 when it was first issued in 1787 and had
been previously sold for $430,000 in 1979. At what annual rate did the coin appreciate
from its minting to the 1979 sale? What annual rate did the 1979 buyer earnon his
purchase? At what annual rate did the coin appreciate from its minting to the 2014
sale?
a) The time line from minting to the first sale is:
1979-1787 = 192
-$15 to $430,000 in 192 Years
To answer this question, we can use either the FV or the PV formula. Both will give
the same answer since they are the inverse of each other. We will use the FV formula,
that is:
FV = PV(1 + r)t
Solving for r, we get:
r = (FV/PV)1/t - 1
r = ($430,000/$15)1/192 - 1
r = 0.0549, or 5.49%
The rate of return for the coin from its minting in 1787 to the 1979 sale is 5.49%
b) The time line from the first sale to the second sale is:
2014-1979 = 35
-$430,000 to $4,582,500
We will use the FV formula, that is:
FV = PV(1 + r)t
Solving for r, we get:
r = (FV/PV)1/t - 1
r = ($4,582,500/$430,000)1/35 - 1
r = 0.0699, or 6.99%
The rate of return from the 1979 sale to the 2014 sale is 6.99%
c) The time line from minting to the second sale is:
2014-1787 = 227
-$15 to $4,582,500
We will use the FV formula, that is:
FV = PV(1 + r)t
Solving for r, we get:
r = (FV/PV)1/t - 1
r = ($4,582,500/$15)1/227 - 1
r = .0572, or 5.72%
The Rate of return from its minting to the 2014 sale is 5.72%.
Note:
It is important to note that these calculations only consider the price appreciation of the coin
and do not take into account any costs associated with buying or selling the coin, such as
commission fees or taxes, which can affect the actual rate of return.
18) Calculating Present Values (LO2) Suppose you are still committed to owning a
$190,000 BMW (see Question 9). If you believe your mutual fund can achieve a 12%
annual rate of return and you want to buy the car in nine years on the day you turn 30,
how much must you invest today?
We know the following values:
BMW
$190,000
Investment time (years)
9
Mutual fund rate of return 11.20%
To calculate how much we need to invest today we are going to use the Present Value (PV)
formula.
PV = FV / (1 + r)^t
Where:
FV is the future value
r is the applied rate
t are the periods of time
Know we just need to use the values we know to obtain the present value of our investment.
PV = $190,000 / (1 + 0.12)^9 = $68,515.90
Our investment today should be $68,515.90 if we want to afford the BMW in 9 years.
20) Calculating Future Values (LO1) You are scheduled to receive $15,000 in two years.
When you receive it, you will invest it for six more years at 7.1% per year. How much
will you have in eight years?
Givens:
Investment
$20,000
Rate (r)
7.3%
Period (n)
6
For the period (n), it is six years instead of eight since the amount to be invested will be
received two years later.
To calculate the future value of an investment, you can use the formula:
FV = PV x (1 + r)^t
FV = $15,000 x (1 + 0.071)^6
FV = $22,637.48
Plugging in the given values, in eight years you will have approximately $22,637.48
21) Calculating the Number of Periods (LO3) You expect to receive $10,000 at
graduation in two years. You plan on investing it at 11% until you have $75,000. How
long will you wait from now?
To calculate this we need to use the formula of Present Value (PV) and rearrange it to obtain
the value of time for each case.
PV = FV / (1 + r)^t
So
t=ln(FV/PV)/ln(1+r)
Whereas
FV is the future value
t are the times the rate is applied
r is the rate applied
t = ln(75000/10000) / ln (1 + 0.11)
t = 19.30
We need to wait 2 years for the amount to be received hence 2 years should be added
in the final calculation as the investment will be done after 2years.
T = t+2
T = 19.30+2
T = 21.30
You will need to invest 21.30 years considering 11% constant growth rate to
reach the target of $75,000.00.
CHAPTER 6
2) Present Value and Multiple Cash Flows (LO1) Investment X offers to pay you $6,000
per year for nine years, whereas Investment Y offers to pay you $8,000 per year for six
years. Which of these cash flow streams has the higher present value if the discount rate
is 5%? If the discount rate is 22%?
6). Calculating Annuity Values (LO1) your company will generate $68,000 in annual
revenue each year for the next seven years from a new information database. If the
appropriate interest rate is 8.5%, what is the present value of the savings?
PV = CF / r x (1 - (1 + r)^-n)
P = PMT * [1 – [ (1 / 1+r)^n] / r]