Download PV of an ordinary annuity (PVOA)

Mod 8 Present Values and
Long-Term Liabilities
PRESENT VALUES:
• A dollar received today is worth much more than a
dollar to be received in 20 years. Why?
– Inflation, risk of not receiving the dollar, and immediate
versus deferred spending
• Two components determine the “time value” of
money:
– interest (discount) rate
– number of periods of discounting
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Present Value Concepts
• To record activities in the general ledger
dealing with future cash flows, we should
calculate the present value of the future cash
flows using present value formulas or
techniques.
• Accounts that often require PV calculations:
– bonds payable and bond investments
– long-term notes payable
– Long-term leases
– Employee pensions
2
Types of Present Value Calculations
• PV of a single sum (PV1): discounting a future value of
a single amount that is to be paid in the future (ex:
face value of bonds payable).
• PV of an ordinary annuity (PVOA): discounting a set
of payments, equal in amount over equal periods of
time, where the first payment is made at the end of
each period (ex: interest on bonds payable).
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Present Value of a Single Sum
All present value calculations presume a discount
rate (i) and a number of periods of discounting
(n).There are 4 different ways you can calculate the
PV1:
n
1. Formula: PV1 = FV1 [1/(1+i) ]
2. Tables: Table 1
PV1 Table
PV1 = FV1(
)
i, n
3.Calculator (with time value functions)
4.Spreadsheet
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Illustration 1: L.T. Notes Payable
• Long-term, usually issued to financial institutions.
• May be interest bearing or non-interest bearing (we
will look at non-interest bearing).
• Illustration 1: On January, 2, 2009, Pearson Company
purchases a section of land for its new plant site.
Pearson issues a 5 year non-interest bearing note, and
promises to pay $50,000 at the end of the 5 year
period. What is the cash equivalent price of the land
today, if a 6 percent discount rate is assumed?
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Illustration 1 Solution
See Table 1
PV1 Table 1
PV1 = FV1(
)
i, n
PV1 Table1
i=6%, n=5
Journal entry Jan. 2, 2009:
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2. Present Value of an Ordinary Annuity (PVOA)
An annuity is defined as equal payments over equal
periods of time. An ordinary annuity assumes that each
payment occurs at the end of each period.
PVOA calculations presume a discount rate (i), where (A) =
the amount of each annuity, and (n) = the number of
annuities (or rents), which is the same as the number of
periods of discounting. There are 4 different ways you can
calculate PVOA:
n
1. Formula: PVOA = A [1-(1/(1+i) )] / i
2. Tables: see page A-2 (back of text), Table 2
PVOA Table
PVOA = A(
)
i, n
3.Calculator (with time value functions)
4.Spreadsheet
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Illustration 2: Long Term Notes Payable
• Illustration 2: On January, 2, 2008, Pearson Company
purchases a section of land for its new plant site.
Pearson issues a 5 year non-interest bearing note, and
promises to pay $10,000 per year at the end of each of
the next 5 years. What is the cash equivalent price of
the land, if a 6 percent discount rate is assumed?
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Illustration 2 Solution
See Appendix A, Table 2
PVOA Table
PVOA = A (
)
i, n
PVOA Table
=
i=6%, n=5
Journal entry Jan. 2, 2008:
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Bonds Payable
• Bonds payable are issued by a company (usually
to the marketplace) to generate cash flow.
• The bonds represent a promise by the company
to pay a stated interest each period (yearly,
semiannually, quarterly), and pay the face
amount of the bond at maturity.
• The marketplace values bonds by discounting the
cash flows using the market rate of interest. This
is also called the yield rate, discount rate, or
effective rate.
• There are two types of cash flows with bonds:
PVOA (for interest payments) and PV1(for
payment of maturity value).
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Illustration 3: Bonds Payable (Discount)
• On January 1, 2006, Corvette Corporation issues
$100,000 of its 5 year bonds which have an annual
stated rate of 5%, and pay interest annually each
December 31, starting December 31, 2006. The
bonds were issued to yield 6% annually.
• Calculate the issue price of the bond:
What are the cash flows and factors?
(1) Face value at maturity =
(2) Stated Interest =
Face value x stated rate x time period
Number of periods = n = 5 yrs
Discount rate = 6% per year
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Illustration 3 : Present Value Calculations
PV of interest annuity:
PVOA Table
PVOA = A(
PVOA Table
)=
i, n
i = 6%, n=5
PV of face value:
PV1 Table
PV =FV1(
PV1 Table
)=
i, n
I = 6%, n=5
Total issue price =
Issued at a discount of $4,212 because the company was
offering an interest rate less than the market rate, and
investors were not willing to pay as much for the lower
interest rate.
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Illustration 3: Journal Entry
The journal entry to record the initial issue of
the bond would be:
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Illustration 4: Bonds Payable(Premium)
• On July 1, 2005, Mustang Corporation issues
$100,000 of its 5 year bonds which have an annual
stated rate of 7%, and pay interest semiannually each
June 30 and December 31, starting December 31,
2005. The bonds were issued to yield 6% annually.
• Calculate the issue price of the bond:
What are the cash flows and factors?
(1) Face value at maturity = $100,000
(2) Stated Interest =
Face value x stated rate x time period
100,000 x .07 x 1/2 = $3,500
Number of periods = n = 5 yrs x 2 = 10
Discount rate = 6% / 2 = 3% per period
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Illustration 4 - Solution
PV of interest annuity:
PVOA Table
PVOA = A(
PVOA Table
) = 3,500 (8.53020) = $29,856
i, n
i = 3%, n = 10
PV of face value:
PV1 Table
PV =FV1(
PV1 Table
) = 100,000(0.74409)=$74,409
i, n
Total issue price =
i=3%, n=10
$104,265
Issued at a premium of $4,265 because the
company was offering an interest rate greater than
the market rate, and investors were willing to pay
more for the higher interest rate.
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Illustration 4: Journal Entry
The journal entry to record the initial issue of
the bond would be:
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Illustration 5: Annuity Income
• Other present value applications include financial
decisions. For example:
• Illustration 4: On January, 2, 2008, Donna Smith won
the lottery. She was offered an annuity of $100,000
per year for the next 20 years, or $1,000,000 today as
an alternative settlement. Which option should Donna
choose? Assume that she can earn an average 4
percent return on her investments for the next 20
years.
• Solution: calculate the present value of the annuity at a
discount rate of 4%, then compare to the $1,000,000
settlement.
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Illustration 5 Solution
PVOA Table
PVOA = A (
)
i, n
PVOA Table
=
i=4%, n=20
Which should she choose?
At approximately what interest (discount) rate would
she choose differently? (based on whole
percentage rate)
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Types of Future Value Calculations
• FV of a single sum (FV1): compounding a future
value of a single amount that is to be
accumulated in the future. Example:
– projected future value of a savings bond.
• FV of an ordinary annuity (FVOA):
compounding the future value of a set of
payments, equal in amount over equal periods
of time, where the first payment is made at the
end of the first period. Examples:
– projected balance in a retirement account.
– amount of payments into retirement fund.
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3. Future Value of a Single Sum (FV1)
There are 4 different ways you can calculate the FV1:
n
1. Formula: FV1 = PV1 [(1+i) ]
2. Table 3
FV1 Table
FV1 = PV1(
)
i, n
3.Calculator (with time value functions).
4.Excel spreadsheet.
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Illustration 6: Investment
Holliman Company wants to invest $200,000 cash it
received from the sale of land. What amount will it
accumulate at the end of 10 years, assuming a 6%
interest rate, compounded annually?
FV1 Table
FV1 = PV1
(
)
i=6%, n=10
FV1 Table
FV1 =
=
i=6%, n=10
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4. Future Value of an Ordinary Annuity (FVOA)
FVOA calculations presume a compound rate (i), where (A)
= the amount of each annuity, and (n) = the number of
annuities (or rents), which is the same as the number of
periods of compounding. There are 4 different ways you
can calculate FVOA:
1. Formula: FVOA = A [(1+i)n - 1] / i
2. Table 4
FVOA Table
FVOA = A(
)
i, n
3.Calculator (with time value functions).
4.Excel spreadsheets.
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Illustration 7: Future Value of Investment
Jane Smith wants to invest $10,000 at the end of each year
for the next 20 years, for her retirement. What balance will
she have at the end of 20 years (after the last deposit),
assuming a 6% interest rate, compounded annually?
FVOA Table 4
FVOA =
A
(
) =
i=6%, n=20
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Illustration 8: Future Value of Investment
James Holliman wants to accumulate $200,000 at the end
of 10 years, for his son’s education fund. What equal
amount must he invest annually to achieve that balance,
assuming a 6% interest rate, compounded annually?
FVOA Table
FVOA =
A
(
) =
i=6%, n=10
FVOA Table
=
i=6%, n=10
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