Summary So Far
Lecture 2
Four basic concepts
Future Value
Present Value
Single Cashflow
Time Value of Money and Mortgage Loans CD, Unit 2, sections 5-8
Fn = P(1 + r )
n
P=
Fn (1 + r )n
Annuity
n ⎡ (1 + r ) − 1⎤ FVn = A ⎢ ⎥ r ⎦ ⎣
−n ⎡1 − (1 + r ) ⎤ PV = A ⎢ ⎥ r ⎦ ⎣
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Quick review of Annuities
Future Value of an Annuity
You are saving for a holiday and plan to deposit $500 at the end of each month into your bank account. Your account earns 6% p.a. compounded monthly. How much will you have at the end of 12 months.
Matching cashflows with formulas
You are putting $500 at the end of each month into a retirement plan. Your plan gives you 6% p.a. compounded monthly. Suppose you retire in one year at which time you will receive the payout from your plan. How much will you receive?
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Matching cashflows with formulas
500 500 500 500 500 500 500 500 500 500 500
Matching cashflows with formulas
Remember n is the number of cashflows Future value formula gives you future value at month 11
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Solution
Present Value of an Annuity
Your university offers the following tuition payment options:
Pay $9,810 today, or Pay 10 equal payments of $1,018 at monthly intervals starting today
If you can borrow at 7% p.a. compounded monthly, which option should you choose? Answer this question by calculating the present value of the payment stream. Compare the present value to the up front payment. NOTE: The payments start today, i.e. you do not include the first payment in your annuity formula – Draw a time line.
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Other issues
When the compounding period is different than the payment interval Effective annual interest rates Deferred annuity Perpetuity
Compounding Periods and Effective Interest Rate
Formula assumes compounding at same frequency as payments
If compounding is more or less frequent than payments, use “effective” periodic interest rate For quarterly payments, compounded monthly with a nominal rate of 12% p.a.
.12 ⎞ ⎛ ⎜1 + 12 ⎟ − 1 = 3.0301% ⎝ ⎠
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Why?
⎛ 0.12 ⎞ FV3months = PV (1 + r ) = PV ⎜1 + ⎟ 12 ⎠ ⎝
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Compounding Periods and Effective Interest Rate
Suppose you deposit $100 per quarter in an account paying 12% p.a. compounded monthly.
After 8 quarters, what will your account balance be?
FV1quarter = PV(1 + reffective)1 ⎛ 0.12⎞ = FV3months = PV⎜1 + ⎟ 12 ⎠ ⎝
1 + reffective ⎛ 0.12 ⎞ = ⎜1 + ⎟ 12 ⎠ ⎝
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⎛ 0.12 ⎞ reffective = ⎜1 + ⎟ − 1 = 3.0301% 12 ⎠ ⎝
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⎡ (1.030301)8 − 1 ⎤ ⎥ FV = 100 ⎢ .030301 ⎢ ⎥ ⎣ ⎦
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Exercise
You repay a loan annually, which has an interest rate of 12% p.a. compounded monthly What is the effective interest rate you pay?
Solution
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Review Nominal Interest Rate – from Lecture 1
Most interest rates are quoted as annual nominal rates
6% p.a. compounded monthly
Compounding Frequency Makes a Difference
Which rate is better?
6% p.a. compounded monthly 6% p.a. compounded quarterly 6.1% p.a. compounded annually If you start with $100 - what does each rate give you after one year?
Convert to rate per compounding period (periodic rate)
Divide annual rate by number of compounding periods per year 0.5% per month
rper =
rann m
.005 =
.06 12
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Effective Annual Rate
EAR = (1 + r per ) − 1
m
Compounding Frequency
What happens to the future value as the number of compounding periods increases?
PV=1000, rnom=10%, time=2 years
r ⎞ ⎛ EAR = ⎜1 + nom ⎟ − 1 m ⎠ ⎝
For 6% p.a. compounded monthly:
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m
Annual Quarterly
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FV = 1000 (1 + .10 ) = 1210
2
−1 = EAR = ⎜1 + 12 ⎟ ⎝ ⎠
⎛
.06 ⎞
.10 ⎞ ⎛ FV = 1000 ⎜ 1 + 4 ⎟ ⎝ ⎠
4 ×2
= 1218.40
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Compounding Frequency
Future Value $1,222.00 $1,220.00 $1,218.00 $1,216.00 $1,214.00 $1,212.00 $1,210.00 0 100 200 300 compounding periods
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Compounding Frequency
Which rate is better?
6% p.a. compounded monthly 6% p.a. compounded quarterly 6.1% p.a. compounded annually
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Choosing a Deposit Account
A recent ANZ Bank ad offered the following nominal rates for a 3-year term deposit:
Interest paid monthly: Interest paid quarterly: Interest paid annually: 4.00% 4.05% 4.10%
Choosing a Deposit Account
Monthly interest payment at 4.00% yields an effective annual rate of:
Which is the best rate?
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Choosing a Deposit Account
Quarterly interest payment at 4.05% yields an effective annual rate of:
Choosing a Deposit Account
Of course, annual interest payment at 4.10% is equivalent to annually compounded interest of 4.10% So the best choice is quarterly interest payment with an annual yield of 4.1119%
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Deferred Annuity
HN Ltd is selling TVs on the following terms:
No payments for the first 12 months In the 13th month, begin 12 monthly payments of $75 each
Deferred Annuity
$75
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$75
$75
$75
$75
$75
$75
$75
$75
$75
$75
$75
The relevant interest rate is 6% p.a. NH Ltd sells the same TV for $750 cash
⎡ ⎛ 1 − (1.005 )−12 ⎞ ⎤ ⎟⎥ PV = ⎢75 ⎜ ⎜ ⎟⎥ .005 ⎢ ⎝ ⎠⎦ ⎣
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Deferred Annuity
Perpetuity:
an annuity with an infinite number of payments
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$75 $75 $75 $75 $75 $75 $75 $75 $75 $75 $75 $75
1 A
2 A
3 A ...
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⎡ ⎛ 1 − (1.005 )−12 ⎞ ⎤ 1 ⎟⎥ × PV = ⎢75 ⎜ ⎟ ⎥ 1.00512 .005 ⎢ ⎜ ⎠⎦ ⎣ ⎝
PV =
A A A + + + ... (1 + r )1 (1 + r )2 (1 + r )3
PV =
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A r
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Why?
1 ⎡ ⎢1 − (1 + r ) n ⎡1 − (1 + r ) ⎤ PV n = A ⎢ ⎥ = A⎢ r r ⎣ ⎦ ⎢ ⎢ ⎣
−n
Perpetuities
⎤ ⎥ ⎥ ⎥ ⎥ ⎦
How much would you pay to receive $100 per year in perpetuity if the interest rate is 5% p.a.?
What happens n increases towards infinity? Since 1+r>1
lim∞ n→ 1 = 0 (1 + r ) n
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Perpetuities
If the payment amount grows at a constant rate of g:
Where does this come from?
PV = = = = A A(1 + g ) A(1 + g ) 2 + + + ... 1 + r (1 + r ) 2 (1 + r )3 A A(1 + g ) ⎡ (1 + g ) (1 + g ) 2 ⎤ 1+ ... + + 1 + r (1 + r ) 2 ⎢ (1 + r ) (1 + r ) 2 ⎥ ⎣ ⎦ A A(1 + g ) ∞ ⎛ (1 + g ) ⎞ ⎟ + ∑⎜ 1 + r (1 + r ) 2 j =0 ⎜ (1 + r ) ⎟ ⎝ ⎠
j
PV =
A r −g
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A A(1 + g ) 1 + 1 + r (1 + r ) 2 1 − (1 + g ) (1 + r ) A A(1 + g ) (1 + r ) = + 1 + r (1 + r ) 2 ((1 + r ) − (1 + g ) ) A A(1 + g ) 1 = + 1+ r (1 + r ) (r − g ) A(r − g ) + A(1 + g ) = (1 + r )(r − g ) A(1 + r ) = (1 + r )(r − g )
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Perpetuity with inflation continued
The key here is that g