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Concept 8. Future Value (FV)
What is future value?
Future Value is the accumulated amount of your
investment fund.
Notations related to future value calculations:
P = principle (original invested amount)
r = interest rate for a certain period
n = number of periods
1
Simple Interest vs. Compounded
Interest
Simple interest means you only earn interest on the original invested amount.
Compounded interest rate assumes that interest earnings are automatically reinvested at the same interest rate as is paid on the original invested amount.
Example: You save $100 in a savings account with an annual r=3%
If simple interest:
End of year 3 = $100+$3+$3+$3 = $
If compounded annually: End of year 1: $100 * (1+3%) = $103. End of year 2: $103 * (1+3%) = $106. End of year 3: $106.09 * (1+3%) = $109.
2
Future Value for One-Time
Investment
To compute future value for one-time investments, one
uses Future Value Factor (FVF).
What is Future Value Factor (FVF)?
FVF= (1+r)n
What does FVF mean?
FVF is how much one dollar will generate in the future
given interest rate r and period n.
How do you use FVF to figure out the future value of
one-time investments?
FV=PFVF=P(1+r)n
3
Example
You put $10,000 in a CD account for 2 years. The
account pays a 4% annual interest rate. How much
money will you have at the end if annual compounding
is used? How about monthly compounding? How
about daily compounding?
4
Annual compounding
FV=10,000(1+4%)^2=10,0001.08160=$10,816.
Monthly compounding
Monthly interest rate: rm = 4%/12 = 0.3333%, n=2*12=
FV=10,000(1+0.3333%)^24=10,0001.
=$10,831.
Daily compounding
Daily interest rate: rd=4%/365=0.0110%, n=2*365=
FV=10,000(1+0.0110%)^730=10,0001.
= $10,836.
Note: For all FV computations please keep the decimal point to 6 digits (4 digits when % sign is used). For money amount use two digits (to cents)
You put $20,000 in a CD account for 10 years. The
account pays a 6% annual interest rate. How much
money will you have at the end if annual compounding
is used? How about monthly compounding? How
about daily compounding?
Annual compounding
FV=20,000(1+6%)^10=20,0001.790848=$35816.
Monthly compounding
Monthly interest rate: rm = 6%/12 = 0.5%, n=10*12=
FV=20,000(1+0.5%)^120=20,0001.819397=$36387.
Daily compounding
Daily interest rate: rd=6%/365=0.0164%, n=10*365=
FV=20,000(1+0.0164%)^3650=20,0001.822029= $36440.
7
FV of Periodical Investments
What is periodical investments?
Periodical investments are multiple investments that are
made at certain time intervals.
How to calculate the future value of periodical
investments? It is
probably best
illustrated
using an example.
8
Example
Suppose you have decided to save some
money to pay for a vacation. You can
afford to save $100 a month. You put the
money in a money market account
which pays an 8% annual interest rate,
compounded monthly. How much
money will you have at the end of the
th
month?
Note: we can treat this as 12 separate
$100 investments that are in the bank for
different length of time.
9
Beginning of the month calculation (deposit money on the first day of every month):
Monthly interest rate rm= 8%/12=0.6667% FV of $100 deposited on Jan. 1= $100 * (1+0.6667%)^12 = $108. FV of $100 deposited on Feb. 1= $100 * (1+0.6667%)^11 = $107. FV of $100 deposited on March 1=$100 * (1+0.6667%)^10 = $106. FV of $100 deposited on April 1 = $100 * (1+0.6667%)^9 = $106. FV of $100 deposited on May 1= $100 * (1+0.6667%)^8 = $105. FV of $100 deposited on June 1= $100 * (1+0.6667%)^7 = $104. FV of $100 deposited on July 1= $100 * (1+0.6667%)^6 = $104. FV of $100 deposited on Aug. 1= $100 * (1+0.6667%)^5 = $103. FV of $100 deposited on Sept. 1= $100 * (1+0.6667%)^4 = $102. FV of $100 deposited on Oct. 1= $100 * (1+0.6667%)^3 = $102. FV of $100 deposited on Nov. 1= $100 * (1+0.6667%)^2 = $101. FV of $100 deposited on Dec. 1= $100 * (1+0.6667%)^1 = $100. Total FV = Sum of the FVs of the 12 periodical payments = $1253.
Note: With beginning of the month (BOM) calculation the last deposit, deposited on Dec. 1, earns one month of interest.
10
End of the month calculation (deposit money on the last day of every month): Monthly interest rate rm= 8%/12=0.6667% FV of $100 deposited on Jan. 31= $100 * (1+0.6667%)^11 = $107. FV of $100 deposited on Feb. 28= $100 * (1+0.6667%)^10 = $106. FV of $100 deposited on March 31=$100 * (1+0.6667%)^9 =$106. FV of $100 deposited on April 30 = $100 * (1+0.6667%)^8 = $105. FV of $100 deposited on May 31= $100 * (1+0.6667%)^7 = $104. FV of $100 deposited on June 30= $100 * (1+0.6667%)^6 = $104. FV of $100 deposited on July 31= $100 * (1+0.6667%)^5 = $103. FV of $100 deposited on Aug. 31= $100 * (1+0.6667%)^4 = $102. FV of $100 deposited on Sept. 30= $100 * (1+0.6667%)^3 = $102. FV of $100 deposited on Oct. 31= $100 * (1+0.6667%)^2 = $101. FV of $100 deposited on Nov. 30= $100 * (1+0.6667%)^1 = $100. FV of $100 deposited on Dec. 31= $100 * (1+0.6667%)^0 = $100. Total FV = Sum of the FV of the 12 periodical payments = $1244.
With end of the month calculation, the last deposit, which is deposited on Dec. 31, does not earn any interest. In fact, every deposit earns one month less of interest compared to the beginning of month situation.
Are there simpler ways of calculating FV for periodic
investments?
If the monthly payments are equal, then we can simplify
the problem by using Future Value Factor Sum (FVFS)
200 6. 213535 1242. 71
1 ) 1 %
( 1 1 %) 1 200 (
1 )
( 1 ) 1 (
( 1 %, 6 , )
61
1
= ร =
โ
โ
= ร = =
r
r P
FV P FVFSr n BOM
n
p
p
n=6 months, monthly r=12%/12=1%=0.01,
beginning of the month calculation
19
Compute the FV of saving $200 on the last day of each
month for 6 months (withdraw at the end of the sixth
month) at 12% annual interest rate, monthly
compounding.
20
6
= ร =
= ร
= ร
= ร = =
r
r
P
FV P FVFSr n EOM
n
p
p
n=6 months, monthly r=12%/12=1%=0.01, end
of the month calculation
21
Suppose you save $500 on the first day of each month
for 6 months at 12% annual interest rate, compounded
monthly, and then only put $200 on the first day of
each month starting from the 7th month for another 6
months at the same interest rate. How much money
will you have at the end of the 12th month?
22
126
61
12
1
12 1
= ร ร =
โ ร +
= ร
โ ร +
= ร
= ร = = ร +
โ
โ
โ
n
n
p
n p
r
r
r
P
FV P FVFSr n BOM r
This is a more complicated scenario. You have to treat this as two investments. Investment one is a periodical investment of $500 per month for 6 month. After 6 months whatever amount there is will be treated as a one-time investment for another six months. Investment two is a periodical investment of $200 each month for 6 months. The total is the sum of these two investments.
Investment 1:
61
1
2
= ร =
= ร
= ร
= ร = =
r
r P
FV P FVFSr n BOM
n
p
p
Investment 2
Next 6 months: Pp=200, monthly r=12%/12=1%=0.01,
n=
Total FV
FV=FV1+FV2=3297.90+1242.71=4540.
Saving for Vacation
Mary is planning on saving some money for her next
vacation. Her goal is to have $2000 saved after one
year. If she decides to put an equal amount of money
in a bank savings account every month on the first day,
and the interest rate is 6% annually, how much should
she save every month, if interest is compounded
monthly?
121
ร = = =
FVFSr n BOM
M
M FVFSr n BOM
This is an application of FVFS because this is
related to periodical payments
Denote the monthly saving amount as M
Monthly interest rate r = 6%/12 = 0.5%
