[1-(1/1.01
)/.01]
besides investment is PV) if NPV is less than PV than its worth less than it cost, therefore not a good investment
l.
Investing in bank at insured rate:
(amount to invest)*(1+r)
PV of an investment:
(CF at Date 1)/(1+r)
NPV:
-Cost+PV
Compounding:
Loan Amount*(1+r)
2
Future Value of an Investment:
(CF to be invested today)*(1+r)
t
How much to lend today to get $1 in 2y:
PV*(1+r)
2
=$1
Present Value of an Investment:
(CF at Date T)/ (1+r)
t
PV=FV/(1+disc. Rate)
t
NPV=-C
o
+(C
1
/1+r)+(C
2
/1+r
2
)+…
Compounding Periods (semiannually):
(deposit amount)*(1+r/2)
2
where 2 is the number of compounding periods.
EAR:
(1+r/m)
m
-1, m= times compounded
APR:
r*(365/# days)
Future value with compounding:
(initial investment)*(1+r/m)
mT
Continuous Comounding:
(initial investment)*e
rT
, e=2.718
PV when Pay X at end of T year @ rate of R:
X*(1/e
rT
)=PV
PV of Perpetuity:
[C/(1+r)]+ [C/(1+r)
2
]+ [C/(1+r)3] or C/r
PV of Growing Perpetuity:
(CF/r-g)
Price of stock today:
[dividend about to pay]+[future dividend/r-g]
future dividend= dividend*1+g
Present Value of an Annuity:
C/r-C/r*[1/(1+r)
t
]
Future Value of an
Annuity:
C*[(1+r)
t
-1/r]
investment that will pay $1,000/year for 10 years.
earn a rate of 9% per year on similar investments, how much willing to pay for this
annuity?:
10 into
N
, 9 into
I/Y
, and 1000 (a cash inflow) into
PMT
. Now press
CPT
PV
to solve for the present value. The answer is -6,417.6577. Again, this is negative
because it represents the amount you would have to pay (cash outflow) today to purchase this annuity.
If borrowing $1000 each year for 10 years at a rate of 9%, and then
paying back the loan immediate after receiving the last payment.
How to repay?
All we need to do is to put a 0 into
PV
to clear it out, and then press
CPT
FV
to find that
the answer is -15,192.92972 (a cash outflow).
Present Value of 4yr college:
(expense)*[1-(1/1+r)
4
/r], take PV/(1+r)
last deposit
-> PV at Date 0. C*A
17
.14
=PV @Date 0, solve for C,
where A
17
.14
is a 17 yr annuity at 14%.
B. Present Value Annuity Problems
In a present value annuity problem
, Assume
N = 5, I/Y = 8%, PMT = $ -1, and PV = $ 3.9927.
Clear: [2nd] [CLR TVM]. 1. Present Value:
Input 5 [N], 8 [I/Y] , and 1[+/-] [PMT]. Press [CPT] [PV].
2. Payment:
Input 5 [N], 8 [I/Y] , and 3.9927 [PV]. Press [CPT] [PMT].
3. Interest Rate
: Input 5 [N], 1[+/-] [PMT], 3.9927 [PV]. Press [CPT] [I/Y]. This is the interest rate implicit in the cash flow stream and the PV. It is the Internal Rate of Return of
the annuity.
4. Number of Payments
: Input 8 [I/Y], 1[+/-] [PMT], and 3.9927 [PV]. Press [CPT] [N].
C. Future Value Annuity Problems Assume N = 5, I/Y = 8%, PMT = $ -
1, and FV = $ 5.8666. Clear: [2nd] [CLR TVM].
1. Future Value: Input 5 [N], 8 [I/Y] , and 1 [+/-] [PMT]. Press [CPT] [FV]. 2. Payment: Input 5 [N], 8 [I/Y] , and 5.8666 [FV].