Chapter 5 Flashcards
time value of money
money can be invested today to earn interest and grow to a larger dollar amt in the future
has nothing to do with the worth or buying power of the dollars
ex:
invested in bank $100
annual yield is 6%
FV is $106
time value of money is useful for
valuing a variety of assets/liabilities and rev/exp
interest
amount of money paid/received in excess of the amt of money borrowed or lent
simple interest
initial investment x annual interest rate x period of time
can also do: [ I x R x T ]/100
compound interest
includes interest not only on the initial investment, but also on the accumulated interest in previous periods
when money remains invested for multiple periods
how else can you calculate compound interest
future value of a single amount
effective rate
actual rate at which money grows per year
interest rate per compounding period for effective rate (what must be done to %)
semiannually –>
12% / 2 = 6%
quarterly –>
12% / 4 = 3%
monthly –>
12% / 12 = 1%
future value (of a single amount)
is the amt of money that a dollar will grow to at some point in the future
also principle + interest
formula for future value (of a single amount)
FV = I (1+i)^n
FV = FV of invested amt
I = amt invested at the beginning period
i = interest rate per compounding period
n = number of compounding periods
what must change to the FV formula when interest is compounded semiannually
if its 1,000 @ 10%, for three years
semi annual is 2 times a year
must multiply n x 2
divide r / 2
so:
1000 [1+ (.01/2)] ^ 3 x 2
NOTE: if using table, your table amount would be multiplied by 1,000
BUT what is the formula for future value of a single amt when using the table
FV = I x FV Factor
example: assume invested $1,000 in an investment account for three years paying 10% interest compounded annually
FV = $1,000 x FV Factor
FV = $1,000 x 1.331
FV = $1,331
present value (of a single amount)
PV = [FV]/[(1 + i)^n]
example: present value of $1,331 received at the end of three years when the interest rate is 10%
PV = FV x PV Factor
PV = $1,331 x .75131
PV = $1,000
What is a result of PV
requires the removal of compound interest
higher the interest rate –> lower the present value
further into the future –> lower the present value
determining an unknown interest rate
PV = 500
FV = 605
n = 2
FV/PV = 605/500 = 1.21 (in table)
look under n=2
so i = 10%
Determining an unknown number of periods
PV = 10,000
FV = 16,000
i = 10%
n =?
10,000/16,000 = .625
under 10% –> .625 –> n = 5
valuing a note: one payment, explicit interest
company sold shoes to Sporting Goods Inc. for $50,000.
Shoe Company agrees to accept a note in payment for the shoes requiring payment for $50,000 in one year plus interest at 10%
FV is 55,000
because 50,000 –> then 10% of 50,000
what is PV today?
55,000 x .90909 = 50,000
.90909 comes from table (n=1, i = 10)
valuing a note: one payment, no interest stated
Shoe Co sells shoes to Sporting Goods Inc.
Terms of sale require Sporting Goods Inc to sign a non-interest bearing note of 60,500 with payment due in 2yrs
know that the FV is 60,500
n = 2
i = 10%
what is PV?
to find PV (price of shoes) have to know either the cash price of the shoes? or the interest rate? (we were given interest rate)
60,500 (fv) x .82645 = 50,000 (pv)
.82645 comes from table ( n= 2, i = 10%)
annuity
series of cash flows of the same amount received or paid each period
examples of when need annuities
loan where periodic interest is paid in equal amounts
a lease paid in equal installments during a specified period
ordinary annuity
cash payments occur at the END of each period
if have to make payments “due at the end of each year” –>
today (NO)
1: end of year one
2: end of year two
3: end of year three
annuity due
cash payments occur at the beginning of each period
so if have to make three payments “immediately” –> 1: today,
2: end of year 1,
3: end of year two
future value of ordinary annuity
“accumulate” ,
rather than investing single amount today,
investing 10,000 over the next three years,
10% interest compounded annually,
first payment is made one year from today,
how much will accumulate in the acct by the end of year 3?
SOLVE:
- formula
- PMT x table value
Future value of annuity due
not investing single amount today,
growing to future value, decide to invest 10,000 a year over the next three years, 10% interest compounded annually,
making the first payment immediately
how much will you have in acct at end of three years?
SOLVE:
- formula
- PMT x table
present value of ordinary annuity
want to determine the cost today,
of a three yr graduate program,
program cost is 10,00 each year,
payments made at end of each year
10%
SOLVE:
- formula
- PMT x table
NOTE!!!!!!!!!
for example, when looking at FVAD:
you know how to solve for the first payment (what was done above)
we take n = 3 and i = 10% and find our corresponding table value to multiply by 10,000
but this can be broken down into each payment
so for first payment, 10,000 x table value of (n = 1 and i = 10%) = FV at end of year 1
this can be done for
payment n =1,
payment n = 2,
payment n =3
add up payment table values and the corresponding PV from each year
present value or ordinary annuity
want to determine the cost today of three year grad program,
cost is 10,000 each year,
payments are made at the end of year,
10%
SOLVE:
- formula
- 10,000 x table value
present value of an annuity due
want to determine the cost today,
three year grad program,
10,000 each year,
payments made at beginning of year,
10% interest rate
what is the cost today of the program?
SOLVE:
- formula
- table
NOTE: when breaking up annuity due
because making payment on first day for three years:
n = 0,
n = 1,
n = 2
deferred annuity
when the first cash flow occurs more than one period after the date the agreement begins
present value of a deferred annuity
buying investment,
provide three equal payments of 10,000,
received at the end of three consecutive years,
the first payment is not expected until three years from now,
10%
how much would you be willing to invest?
n = 0 (today)
n = 1 (end of year one)
n = 2 (end of year two)
n = 3 (end of year three)
n = 4 (end of year four)
n = 5 (end of year five)
BREAK DOWN
BREAK DOWN: PV of deferred annuity
step one: calculate the PV of the annuity as the beginning of the annuity period (using PV of ordinary annuity)
step two: reduce the single amount calculated in (1) to its present value as of today (using PV of $1)
Step 1:
PVA = 10,000 x table value (10%, n = 3) –> 24,869
Step 2:
PV = 24,869 x (10%, n =2)
24,869 is a FV that need to be converted to PV
determining the annuity amount when other variables are known (PMT)
assume borrow $700 from friend,
intend to repay in 4 equal annual installments beginning one year from today,
friend wants to be reimbursed for time value of money at 8% annual rate,
what is the required annual payment that must be made, to repay the loan in four years??
basically, what is the pv of ordinary annuity amount??
700 is PV
n =4
i = 8%
solving for PMT
determining the annuity amount when other variables are known (years)
PV of ordinary annuity
700 is pv,
100 is PMT (annuity amount)
i = 7%
Divide 700/100
so n = 10yrs (when looking at table value
determining interest rate when other variables are known
a friend asked to borrow 331 from you (331 is PV),
promised to repay you 100 (annuity amt) at the end of each of the next four years.
What is the annual interest rate?
Looking at PV of ordinary annuity
divide 331/100
n = 4
i = ?
(this interest rate is known as the effective interest rate)
Determining interest when other variables are known - unequal cash flows
borrowed $400 from a friend and promised to repay the loan,
three annual payments of $100 at the end of each of the next three years,
plus a final payment of $200 at the end of year four?
What is the interest rate implicit in this agreement?
$100 single annuity, n = 3, i = ?
$200 single payment has n = 4, i = ?
valuation of long term bonds
from previous semester
valuation of long term bonds - calculating interest expense
(what sold for or calculated price of bond) x 12% x 1/2
the 12% is the MR
Debit: Interest expense
Credit: Interest Payable
valuation of long-term leases
25yr lease
10,000 PMT
beginning of each year (which means PV of ordinary annuity)
i = 10%
trying to solve for the face value of the note
so…
PVAD = 10,000 x table value = x
PVAD represents face value of note
10,000 is PMT
journal entry for long term lease
debit: right of use asset
credit: lease payable
valuation of installment notes
35,000 machine,
5,000 down payment, 5yr installment note, remaining 30,000 payed on a note
note calls for annual installment payments, 1st payment is Jan 1 2024
i = 4%
paying loan over 5yrs
we are solving for PMT
have to use PV of annuity due table because given PVAD of 30,000
PVAD = PMT x table value
30,000 = PMT x table value
PMT = 6,480
journal entry for valuation of installment notes for acquisition of machinery
Debit: machine 35,000
Credit: cash 5,000
Credit: NP 30,000
journal entry for first installment payment on machinery
Notes Payable 6,480
Cash 6,480
when valuing installment notes is there a need for first installment payment on interest?
no because no interest has accrued
on the 2nd payment will need to pay interest and there will be a reduction on the loan
Debit: Interest Expense 941
Debit: Notes Payable 5539
Credit: cash 6,480
how is interest expense calculated for valuation of installment notes?
Interest expense:
4% x (30,000 - 6480) = 941
interest x (face value of note - cash payed to reduce note on first payment)
valuation of pension obligation
Jan 1, 2024
hired new manager,
manger must work 25yrs before retirement on Dec 31, 2048
annual retirement payments will be paid at the end of each year during his retirement period (expected to be 20yrs)
first payment will be on Dec 31, 2049
during 2024, manager earned annual retirement benefit estimated to be 2,000 per year
company plans to contribute cash to pension fund that will accumulate amount sufficient to pay this benefit
6% on all funds in pension plan
How much would the company have to contribute at the end of 2024 to pay for the pension benefits earned in 2024?
BREAK DOWN of valuation of pension obligation
Deferred annuity
must begin with PV of ordinary annuity and convert to PV of $1
PVA = 2,000 x table value
PVA = 2,000 (PMT) x 11.46992 = $22,940
Table value uses n = 20 and i = 6%
then convert PVA, which is FV to PV
PV = 22,940 (future amount) x .24698 = 5,666
(n = 24 and i = 6%)
2,000 is annual payments for 20 yrs
but for PV of $1, n = 24 because 2048-2024 = 24yrs
