much I would be willing to pay (or invest) today such that I earn
a necessary rate of return (or interest)
•
The “opportunity cost” represents the next best alternative
rate of return for an investment of similar risk and time duration
•
The present value represents what I am willing to sacrifice today
in order to invest in an asset (stock, bond, property, business,
etc.), such that I (at the least) earn my opportunity cost
FIN 300 - TVM Lump Sum
18

Lump-Sum Discounting
•
Let’s say that you buy a “piece of paper” (some
sort of a financial security or asset) that promises
to pay (a single lump-sum of) exactly $10k in
exactly 6 years
•
Also, let’s say that you require a 6% annual rate of
return for investment in assets of a similar risk
•
What would you be willing to pay today for this
piece of paper – in other words, what is its
“value”
FIN 300 - TVM Lump Sum
19

Lump-Sum Discounting
•
Do the work for the previous…
𝑃𝑃
=
$10,000
1.06
6
= $7,049.61
•
It would be valued at $7, 049.61
–
Note that you would be thrilled to get it cheaper than
that
–
But, as long as you don’t pay any more than
$7,049.61, you are earning at least as good a return as
you could get elsewhere for the same risk
FIN 300 - TVM Lump Sum
20

Lump-Sum Discounting
•
If we were to diagram the previous example
FIN 300 - TVM Lump Sum
21
t=0
t =6
Invest $7,049.61
(negative cash-flow)
Cash Out $10,000
(positive cash-flow)
Discounting
Backward in Time

Compounding Periods
•
Assume that you will invest $1
–
You are told that you will earn 12% “APR” (annual
percentage rate) with annual compounding
–
How much will you have in 1 year?
–
Pretty simple answer…..$1.12!
•
What if it said that the $1 investment earns
12% APR with “semi-annual” compounding?
–
How much will you have in 1 year?
FIN 300 - TVM Lump Sum
22

APR
•
APR (annual percentage rate) is a “quoted
rate”
–
Must be paired with the frequency of
compounding
–
The APR is not a true, honest-to-goodness interest
rate – the APR is the number of compounding
periods per year multiplied by the effective rate
per period
𝐴𝑃𝐴
=
𝑚 ∗ 𝑖
(where m = # of compounding periods/yr. and the i = rate of
return per period (aka. period rate)
FIN 300 - TVM Lump Sum
23

APR
•
Back to investing $1 for year at a 12% APR,
with semi-annual compounding
–
This doesn’t mean that you earn 12%/year
–
This does mean that the actual (or, effective) rate
is 6% (1/2 of 12%) per semi-annual period
–
Go back to the compound-interest formula
$1(1.06)(1.06) = $1.1236
–
Note that your actual (or, “effective”) rate of
return for the year is 12.36%
–
This 12.36% is termed the EAR (or, effective
annual rate)
FIN 300 - TVM Lump Sum
24

APR vs. EAR
Continuing the previous with more frequent
compounding and 12% APR, investing $1 for a year
–
Quarterly: really earning 3% per 3-month period
•
$1(1.03)
4
=$1.125509
•
So, the EAR (effective annual rate) = 12.5509%
–
Monthly : earning 12%/12=1% per month
•
$1(1.01)
12
=$1.126825
•
EAR = 12.6825%
–
Weekly: earning a period rate of 12%/52 per week
•
$1(1 + 0.12/52)
52
=$1.127341
•
EAR = 12.7341%
FIN 300 - TVM Lump Sum
25

APR vs. EAR
Continuing the previous with more frequent

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- Fall '08
- Olander
- Time Value Of Money, Corporate Finance, Interest, Interest Rate, Net Present Value, TVM Lump Sum