# Annuities and loans #2 ```Introduction to Agribusiness Finance– Annuities
and Loans Part II
Objectives:
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Practice more complicated annuity problems
In-class work (more loan amortization, complex annuity problems)
Advanced Time Value of Money Problems
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Deferred Annuities
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When the annuity doesn’t begin in year 0 or 1.
Example would be our social security payments, which won’t start
until we are 62
Combining Multiple Annuities
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When an annuity is followed by a different annuity
Example would be annual savings followed by mortgage payment or
social security income.
Advanced Time Value of Money Problems
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We already know how to solve these problems – they are just
cases of unequal cash flows!
The annuity equations just give us additional ways to solve
them.
Tips for Solving Advanced TVM Problems
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Review the information given in the problem and jot down
what is known
Draw a timeline
Select a focal date
Determine whether cash flows are lump-sums or annuities
Determine the compounding frequency of the problem
and decide whether the cash flows are beginning or end
of period
Timeout! Let’s review the “timing” of =PV()
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=PV(rate, nper, payment)
𝑃𝑃𝑃𝑃 = 𝑃𝑃𝑃𝑃𝑃𝑃 &times;
1−
1
1+𝑟𝑟 𝑛𝑛
𝑟𝑟
These assume that the first payment is received 1 year from
today.
Today = year 0
Deferred Annuities

Example – A contract agreement requires the payment of
\$15,000 at the end of the 3rd, 4th, and 5th years but nothing in
the 1st and 2nd years. Find PV when discount rate = 0.09.
0
Cash Flow
PV
1
0
2
0
=0/(1.09)^0 =0/(1.09)^1
0
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3
0
=0/(1.09)^2
0
Total PV = \$31,958.10
4
15000
=15000/(1.09)^3
0
11,582.75
5
15000
=15000/(1.09)^4
10,626.38
15000
=15000/(1.09)^5
9,748.97
Deferred Annuities
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We can solve this problem another way using the PV of
annuity formula.
1. Find PV of annuity at start of annuity (end of year 2)
𝑃𝑃𝑃𝑃 = 15000 ∗
1
1.093
1−
0.09
= \$37,969.42
2. Discount this amount back to “today” (i.e. end of year zero).
\$37,969.42
𝑃𝑃𝑃𝑃 =
= \$31,958.10
2
1.09
Combined Annuities
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This is just an annuity with other payments mixed in.
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Two ways to solve:
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You can either write out on timeline and adjust each cash flow to
present value individually and sum them, OR
You can use the annuity formulas for the annuities and the PV and
FV formulas for individual cash flows and sum them.
Imbedded and Combined Annuities
Example:
You turned 30 years old today and don’t have a penny to your
name. You want to retire at age 60 with \$2,200,000 in the
bank. Realistically, you know that the most that you can save
year​ (you only save on your​ birthdays!). How much do you have
to save each year from your 51st to your 60th birthday in order
to achieve your retirement goal if you can earn 5.5​% on
your​ savings?
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In-Class Activity