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Homework Set #2
Question #1
What is the present value of an $11MM annuity paid out in 26 equal annual installments starting
today? Money can be invested at 9% compounded monthly.
Answer #1
Let’s begin by defining present value (PV). Present value is the amount of money today that will
have a future value (FV) in the future at a given interest rate. For example, $100 today (PV) at 10%
interested compounded annually will be worth $110 (FV) 1 year from now. After 2 years, that $100
would have value of $121 ($110 + $110*0.10). We can derive the relationship between PV, FV, n
(number of compounding terms) and R (interest rate) fairly easily through the following logic:
FV1𝑦𝑟 = PV𝑖𝑛𝑖𝑡𝑙𝑎𝑙 ∗ (1 + R)
FV2𝑦𝑟 = PV1𝑦𝑟 ∗ (1 + R)
Substituting PV1yr = FV1yr
FV2𝑦𝑟 = (PV𝑖𝑛𝑖𝑡𝑙𝑎𝑙 ∗ (1 + R)) ∗ (1 + R)
Simplifying
FV2𝑦𝑟 = PV𝑖𝑛𝑖𝑡𝑙𝑎𝑙 ∗ (1 + R)2
generalizing
𝐅𝐕𝒏 = 𝐏𝐕𝒐 ∗ (𝟏 + 𝐑)𝒏
Let’s verify that equation for our 2 above cases
FV1 = PV𝑜 ∗ (1 + R)1 = $100 ∗ (1 + 0.10)1 = $100 ∗ 1.1 = $110
FV2 = PV𝑜 ∗ (1 + R)2 = $100 ∗ (1 + 0.10)2 = $121
And all is well.
It would certainly be a trivial task to rearrange that general equation and calculate the present
value of $11,000,000 compounded monthly 25 years from now (remember that the first
payment is paid now, the second payment is made 1 year from now and so on. Meaning the
26th payment is paid 25 years from now) at an interest rate of 9% compounded monthly. The
results are as follows:
PV𝑜 =

FV𝑛
FV25×12
$11,000,000
=
=
= $1,169,166.17
r
𝑛
0.09
(1 + R)
(1 + 12)(25×12)
(1 + 12 )(25×12)

With “R” as a general rate and “r” as the specific rate given in our problem of 9% annual. In
other words, if we had $1,169,166.17 today and invested that amount of money for 25 years at
an interest rate of 9% compounded monthly, our investment would be worth $11,000,000 (after
25 years). We could also solve this in an excel spreadsheet with the following cells.

years
passed
0
0
0
0
0
0
0
0
0
0
0
0
1
1

months
passed
1
2
3
4
5
6
7
8
9
10
11
12
1
2

monthly
opening
balance
$1,169,166.17
$1,177,934.92
$1,186,769.43
$1,195,670.20
$1,204,637.73
$1,213,672.51
$1,222,775.05
$1,231,945.86
$1,241,185.46
$1,250,494.35
$1,259,873.06
$1,269,322.11
$1,278,842.02
$1,288,433.34

interest
paid
$8,768.75
$8,834.51
$8,900.77
$8,967.53
$9,034.78
$9,102.54
$9,170.81
$9,239.59
$9,308.89
$9,378.71
$9,449.05
$9,519.92
$9,591.32
$9,663.25

monthly
closing
balance
$1,177,934.92
$1,186,769.43
$1,195,67...