### FA2 Module 5. Interest concepts of future and present value

```FA2
Module 5. Interest concepts of future and
present value
1. Time value of money
2. Basic interest concepts
3. Present and future values
a. Single payment
b. Annuity (ordinary and due)
4. Applications
1. Time value of money
Investors expect that money invested should
grow, i. e., earn a return, over time. The
future value (FV) of an investment earning
some rate of return r, after n periods, is:
FV = Investment X (1+r)n
E. g., \$100 invested at 10% will be worth:
•After one year, \$100 X (1.1)1 = \$110
•After two years, \$100 X (1.1)2 = \$121
•After three years, \$100 X (1.1)3 = \$133
1. Time value of money (cont’d)
The present value (PV) of a future cash flow
is computed using the inverse of the future
value function.
FV = Original Investment X (1+r)n
PV = FV/(1+r)n
For example, suppose I can invest at a rate of
10%, and someone promises to pay me \$100.
If the cash is received:
•Immediately, PV = \$100/(1.1)0 = \$100
•After one year, PV = \$100/(1.1)1 = \$91
•After two years, PV = \$100/(1.1)2 = \$83
1. Time value of money (basic intuition)
FV = Original Investment (PV) X (1+r)n
PV = FV/(1+r)n
• the longer the time to maturity, the lower the
PV (the greater the FV)
• The higher the interest rate (r), the lower the
PV (the greater the FV)
2. Basic interest concepts
Vocabulary
Principal: Amount borrowed
Interest: Cost of borrowing principal
(includes additional amount repaid to lender
plus any legal fees, commissions, etc.)
Interest expense: Cost of borrowing incurred
by borrower
Interest revenue: Amount in excess of
principal earned by lender
2. Basic interest concepts
Vocabulary
Interest rate: Cost of borrowing expressed as
a percentage of principal
Interest period: Period of time (e. g., month,
quarter, year) over which interest is
calculated
2. Basic interest concepts
Vocabulary
Simple interest: Original principal
outstanding for the period multiplied by
interest rate; interest is earned only on the
principal (this is very rare in business)
Compound interest: Interest is computed on
both the original principal and on past
interest that has accumulated
2. Basic interest concepts
Compound interest and interest periods
The interest on a debt is a function of the interest
rate (normally expressed as an annual rate) and the
type of compounding (number of interest periods
per year). The interest for any particular period is
the annual interest rate divided by the number of
interest periods per year, multiplied by the
principal (plus accumulated interest) outstanding
at the beginning of the interest period.
2. Basic interest concepts
Compound interest example
A8-40 (1): On Jan. 1, 20x1, \$30,000 is
deposited in a fund at 16% compound interest.
At the end of 20x5, what will the fund balance
be, assuming
a. Annual compounding?
b. Semi-annual compounding?
c. Quarterly compounding?
2. Basic interest concepts
Future value calculations in Excel
The future value of a single amount in Excel
is:
=FV(rate,nper,,pv)
Where FV is future value, rate is interest rate
per period, nper is number of periods, and
pv is the present value.
Note ,, between nper and pv. pv is entered as
a negative value.
2. Basic interest concepts
Interest rate: Nominal vs. effective
Loan agreements are typically described by
the quoted (or nominal) interest rate and the
type of compounding. The effective interest
rate is the cost of borrowing that takes into
account the effect of compounding.
Example: A8-40 (1) revisited
Example: Credit card charging 24% per
annum, compounded monthly
3. Present and future values
a. Single payment – future value
The future value (FV) of a single payment is
the value in nominal dollars of a sum of
money invested to earn some rate of return (r),
at a some future time.
FV = PV x (1+r)n
where n is the number of time periods
between the present time and the future. A840(1) is a future value exercise.
3. Present and future values
a. Single payment – present value
The present value (PV) of a single payment is
the present value of a sum of money to be
received at some future time, when money
can be invested to earn some rate of return (r).
PV = FV/(1+r)n
where n is the number of time periods
between the present time and the future.
Example: A8-40(2)
2. Basic interest concepts
Example: Present value of a single payment
A8-40 (2): On Jan. 1, 20x1, a machine is
purchased at an invoice price of \$20,000. The
full purchase price is to be paid at the end of
20x5. Assuming 12% compound interest,
what did the machine cost if compounding is
a. Annual?
b. Semi-annual?
c. Quarterly?
2. Basic interest concepts
Present value calculations in Excel
The future value of a single amount in Excel
is:
=PV(rate,nper,,fv)
Where PV is present value, rate is interest rate
per period, nper is number of periods, and
fv is the present value.
Note ,, between nper and fv. fv is entered as a
negative value.
3. Present and future values
b. Annuities – ordinary and due
An annuity is a series of uniform payments
occurring at uniform intervals over a specified
investment time frame, with all amounts
earning compound interest at the same rate.
An ordinary annuity is one in which the
payments occur at the end of each period (e.
g., loan payment). An annuity due is one in
which the payments occur at the beginning of
each period (e. g., rent payment).
3. Present and future values
b. Annuities – future value
The future value of a stream of cash flows can
be viewed as the sum of the future values of a
series of individual cash payments occurring
at different points in time.
3. Present and future values
b. Annuities – future value
Example: A8-40 (5). On Jan. 1, 20x1, a
company decided to establish a fund by
making 10 equal annual deposits of \$6,000,
starting on Dec. 31. The fund will be
increased by 9% compounded interest. What
will be the fund balance at the end of 20x10
(i. e., immediately after the last deposit)?
3. Present and future values
b. Annuities – future value in Excel
The future value function of an annuity in
Excel is
=FV(rate, nper,pmt,,type)
rate is the interest per period, nper is the
number of periods, pmt is the amount of each
payment, and type indicates type of annuity (0
= ordinary annuity; 1 = annuity due). pmt is
usually entered as a negative number.
3. Present and future values
b. Annuities – present value
The present value of a stream of cash flows
can be viewed as the sum of a series of
individual cash payments occurring at
different points in time, all discounted to their
present values. The present value calculation
involves calculating the present value of each
of the cash flows.
3. Present and future values
b. Annuities – present value
Example: A8-40 (8). Ace Company is
considering the purchase of a unique asset on
Jan. 1, 20x1. The asset will earn \$8,000 net
cash inflow each Jan. 1 for five years, starting
Jan. 1, 20x1. At the end of 20x5, the asset
will have no value. Assuming a 14%
compound interest rate, what should Ace be
willing to pay for this unique asset on Jan. 1,
20x1?
3. Present and future values
b. Annuities – present value in Excel
The future value function of an annuity due in
Excel is
=PV(rate, nper,pmt,,type)
rate is the interest per period, nper is the
number of periods, pmt is the amount of each
payment, and type indicates type of annuity (0
= ordinary annuity; 1 = annuity due). pmt is
usually entered as a negative number.
4. Applications
a. Loans
Example: A8-37. On Jan. 1, 20x4, Terry
Corporation borrowed \$100,000 from the
Canadian Bank. The loan will be repaid in five
equal annual instalments, including both principal
and compound interest at 10%; interest is
compounded annually.
Required. Compute the annual loan payment that
would be made if (1) first payment is made Jan. 1,
20x4 or (2) first payment is made on Dec. 31,
20x4. Prepare a debt amortization schedule for
each alternative.
4. Applications
Solving for annuity payment in Excel
We can solve for the annuity payment in Excel
with the PMT function:
=PMT(rate, nper,pv,,type)
rate is the interest per period, nper is the number
of periods, pv is the present value of the annuity,
and type indicates the type of annuity (0 =
ordinary annuity; 1 = annuity due). pv is usually
entered as a negative number.
4. Applications
b. Retirement planning
My wife plans to retire in 15 years. She would
like to make sure that we have sufficient
retirement savings in place to generate retirement
income of \$40,000 per year for 25 years. The
interest rate is 8%. How much would we need if:
a. We want to set aside a sufficient lump sum of
cash right now?
b. We decide to make equal deposits over the next
15 years, at the beginning of each year?
4. Applications
c. Buy now, pay later
Jack and Jill bought themselves a “shabby chic”
living room suite for \$7,500 on January 1, 20x7.
The Block Furniture Warehouse has agreed that J
and J need not “pay a dime until Jan. 1, 20x9.”
The relevant interest rate is 7%.
Required
1. How much should the Block record as revenue
at the date of sale (assume no collectibility
problems)?
2. How much interest income should the Block
record in 20x7, 20x8 and 20x9?
4. Applications
d. Equipment rental
On Jan. 1, 20x6, Lessee Ltd signed an agreement to
rent a piece of equipment from Lessor for five
years. Lessee will pay rent in the amount of
\$12,000 per year, payable at the beginning of each
year. At the end of the five-year term, legal title to
the equipment passes to Lessee. The equipment
has a useful life of five years. Lessee can borrow
money at a rate of 9%.
Required: From an accounting point of view, is this
transaction really a rental? How should Lessee
record this transaction?
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