The compound interest formula is
(1.0)
where
EXAMPLE #1
Given PV0 = $1000, i = 12%, and n = 20 years. Find the ending value if interest is compounded (a) annually; (b) monthly; and (c) daily.
SOLUTION
(a) $1000(1 + .12)20 = $9,646.29
(b) $1000(1 + .12/12)240 = $10,892.55
(c) $1000(1 + .12/365)7300 = $11,018.83
If you solve equation (1.0) for PV0, you get the present value formula:
(2.0)
Present value is the amount to invest today in order to have TV n years hence, if the annual interest rate is i and the compounding frequency per year is m.
EXAMPLE #2
Find the present value of receiving $10,000 20 years hence if i = 10% and the compounding frequency is monthly.
SOLUTION
PV0 = $10,000/(1 + .10/12)(12)(20)
= $10,000/7.328073627
= $1364.62
Present value is less than terminal value, not because of inflation, but because of the time value of money. Using equation (1.0), we have
($1364.62)(1 + .10/12)240 = $10,000
Given the interest rate, compounding frequency, and the time period, present value is the amount to invest now in order to have $10,000 20 years later.
Given: annual compounding, TV30 = $100,000 and PV0 = $20,000 and all cash flows or interest earned are reinvested in the same investment (are kept "internal"). Find the internal rate of return, IRR.
SOLUTION
$20,000(1 + IRR)(30) = $100,000
(1 + IRR)30 = 5
I + IRR = 51/30
IRR = 5.5113064%
CHECK: 20,000(1 + .055113064)30 = 100,000
Comment:
On a calculator, 51/30 can be obtained as follows:
5[xy]30[1/x] = (gives 1.055113064)
where [xy] is the power key and [1/x] is the reciprocal key (on some calculators, it is called x-1 which is the same as 1/x).
EXAMPLE #4
Given: monthly compounding, TV20 = $50,000 and PV0 = $5,000. Find the annual internal rate of return, IRR.
SOLUTION
$5,000(1 + IRR/12)(12)(20) = $50,000
(1 + IRR/12)240 = 10
1 + IRR/12 = 101/240 = 1.009640276
IRR = 11.5683306%
Check: 5,000(1 + .115683306/12)240 = 50,000
EXAMPLE #5
Assume: annual compounding and that the value of your investment ten-tuples every 15 years. Your initial investment is $2,000.
(a) How much will you have after
(b) What was your annual IRR?
SOLUTION:
(a)
(b) $2,000(1 + IRR)15 = $20,000
1 + IRR = (10)1/15
IRR = 16.5914401%
EXAMPLE #6
Given: initial investment of $100,000 and annual compounding. Your returns varied widely from year to year and some were losses and others were gains. After 10 years, however, your investment was worth $250,000. Find your annualized internal rate of return.
SOLUTION
$100,000(1 + IRR)10 = $250,000
(1 + IRR)10 = 2.5
IRR = 9.5958226%
EXAMPLE # 7
Given: initial investment = $10,000 and annual compounding. The annual rate of growth in the value of your investment was 5% (year 1), 14% (year 2), and 12% (year 3).
(a) How much was your investment worth after 3 years? (b) What was your annualized ROR (rate of return)?
SOLUTION
(a) $10,000(1.05)(1.14)(1.12) = $13,406.40
(b) $10,000(1 + ROR)3 = $13,406.40
ROR = 10.2649262%
Check: 10,000(1 + .102649262)3 = 13,406.40
PROBLEM #1
Given: PV0 = $5,000 and i = 8%. Find TV15 if interest is compounded (a) annually; (b) monthly; (c) daily.
Answer:
(a) $15,860.85
(b) $16,534.61
(c) $ 16,598.40
PROBLEM #2
Find PV0 if i = 5%, m = 12, and TV30 = $25,000
Answer:
$5,595.66
PROBLEM #3
How much should you invest today if you want $50,000 after 15 years if i = 12% and m =12?
Answer:
$8,339.17
PROBLEM #4
Given: Daily compounding. TV30 = $100,000 and PV0 = $1,000. Find the annualized internal rate of return, IRR.
Answer
15.3537954%
PROBLEM #5
The value of your investment triples every 8 years. Assume monthly compounding. Find the annualized IRR.
Answer:
13.8115316%