Net Present Value NPV

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Lecture 4

• Net Present Value (NPV)

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NPV calculation single period case
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NPV calculation multiple periods, annual
compounding
The interest compounds when the yield on an
investment is reinvested
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NPV calculation multiple periods, annual
compounding (continued)
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NPV calculation multiple periods, within year
compounding
Compounding for m periods within the year
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Example
Consider an investment of 1,000 that is expected
to yield 500 in 1 year and 700 in 2 years. What
is the NPV of the investment at an annual
discount rate of 9 ?
What happens if the interest is compounded semi
annually?
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Effective annual interest rate (EAIR)
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Stated Vs. Effective Annual Interest Rate
SAIR is meaningful only when compounding interval
is given. On the other hand, EAIR is meaningful
itself without a compounding interval.
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Continuous compounding
Frequently continuously compounded rates are used
by financial institutions. The rate can be
obtained by compounding every infinitesimal
instant, i.e. by letting the number of
compounding periods m go to infinite.
Commonly used APR (annual percentage rate) is the
most common synonym for SAIR
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Four Classes of Cash Flow Streams
Perpetuity
A perpetuity is a constant stream of cash flows
without an end - -- Example British bonds called
consols
Consider a consol paying a coupon C each period
and forever. By convention, the first cash flow
occurs at period 1.
How can a price of a consol be determined? The
price will be the PV of all of its future coupons.
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Growing Perpetuity
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Annuity
An annuity is a finite stream T of constant
payments C. Most common kinds of financial
instruments. For example, pensions, leases,
mortgages are often annuities.
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Growing Annuity
Growing annuity is a finite number T of growing
cash flows at a rate g
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Example 1
Delayed annuity What is the PV of a 4 year
annuity 500 per year, beginning at date 6?
Interest rate is 10.
PV at date 5 500/0.1 - 500/0.1 1/(1.1)4 500
A40.1 1584.95 PV at date 0 1584.95 / (1.1)5
984.13
Annuity in advance Consider a 20 year annuity
of 50,000 per year, starting today, not a year
after. What is the PV of this annuity if r is 8 ?
PV 50,000 (payment at date 0)
50,000 A190.08 (PV of 19 year annuity)
50,000 50,000 ( 1/0.08 - 1/(0.08 (1.08)19 )
) 530,180
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Example 2
Your son just turned 5 today. You plan to start
saving for his college education by making
deposits of an equal amount semiannually in an
investment account that pays a stated annual
interest rate of 9.1, compounded semiannually.
The first deposit D is to be made 6 months hence.
You want to provide 24,000 per year for 4 years
beginning when he is 19 years old. How much
should you deposit every six months until your
son turns 18?
26 periods annuity with payment D semiannually FV
at period 26 (when he turns 18) PV (1r)26
15.0667 D (1r)26 47.911741 D PV D
( 1/r - 1/r(1r)26 ) 15.0667 D r 0.091/2
0.0455
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Example 2 (continued)
4 year annuity with payment 24000 per year Value
at his age 18 24000
77231.785 Effective annual interest rate
(10.091/2)2 - 1
0.09307025
To find D, we want to equate 47.911741 D
77231.785 D 1,611.96
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Example 3
You are 30 and wish to provide for your old age.
Suppose, beginning one year hence, you invest
5000 per year for the next 30 years at an
effective annual rate 9. Beginning at age 61,
you will tour the world for 5 years and will need
X per year at the start of each year. After you
return from the tour, you will withdraw 30,000
per year for the next 15 years.
Assuming the 9 return remains constant, what is
maximum X you can consume each year during your
world tour?
The value of 5000 annuity at age 60 5000(1.09)29
5000(1.09)28 5000(1.09) 5000
5000 ( 1 1.09 1.0929 ) 5000
( (1-1.0930 )/(1-1.09)) 681537.69
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Example 3 (continued)
The value of 30,000 annuity at age 60 1/1.095
(PV at 65) 1/1.095 (3000 A150.09 )
1/1.095 241820.85
157166.83
The value of X annuity at age 60 should be
681537.69 - 157166.83 524370.86 X A50.09
X (1/0.09 - 1/0.091.095 ) 3.8896513 X X
524370.86 / 3.8896513 134,811.79
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