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FIRST PRINCIPLES OF VALUATION
TIME VALUE OF MONEY (TVM)
MORE TVM
INVESTMENT RETURNS: IRR and NPV
TYPES OF LOANS
TIME VALUE OF MONEY:Six Functions of a Dollar
• Future Value of a One Time Deposit• Present Value of a Single Future Cash Flow• Future Value of an Annuity• Present Value of an Annuity• Sinking Fund Payment• Payment to Amortize Debt
TVM: Future Value of a One Time Deposit with Annual Compounding
What is the value in two years of $1,000 deposited today in a savingsaccount bearing 6% annual interest compounded annually?
Value at the end of year 1:
FV = $1,000 (1 + 0.06) = $1,060
Value at the end of year 2:
FV = $1,000 (1 + 0.06) (1 + 0.06)
= $1,000 + $60 + $60 + $3.60
= $1,123.60
TVM: Future Value of a One Time Deposit with Annual Compounding
HP 10B Keystrokes
CLEAR ALL
1 P/YR
1000 PV
6 I/YR
2 N
FV
Clear registers
Annual compounding
$1,000 present value
6% annual interest
Two year term
Compute future value
TVM: Future Value of a One Time Deposit with Annual Compounding
F V P V i n ( )1
w h e r e
P V = t h e a m o u n t o f t o d a y ’ s d e p o s i t
i = t h e a n n u a l i n t e r e s t r a t e
n = t h e n u m b e r o f y e a r s i n t h e i n v e s t m e n t t e r m
F V = t h e f u t u r e v a l u e
TVM: Future Value of a One Time Deposit with Periodic Compounding
Suppose the deposit in the previous example was compoundedmonthly instead of annually. What is the future value?
Monthly interest rate = 0.06/12 = 0.005Number of compounding periods = 2 x 12 = 24
End of month Future Value
1 $1000.00 (1.005) = $1005.00 2 $1005.00 (1.005) = $1010.25 .12 $1056.40 (1.005) = $1061.68 .24 $1121.55 (1.005) = $1127.16
TVM: Future Value of a One Time Deposit with Periodic Compounding
HP 10B Keystrokes
CLEAR ALL
12 P/YR
1000 PV
6 I/YR
2
FV
x P/YR
Clear registers
Monthly compounding
$1,000 present value
6% annual interest
24 months
Compute future value
TVM: Future Value of a One Time Deposit with Periodic Compounding
F V P Vi
kn k ( )1
w h e r eP V = t h e a m o u n t o f t o d a y ’ s d e p o s i t
i = t h e a n n u a l i n t e r e s t r a t e
n = t h e n u m b e r o f y e a r s i n t h e t e r m
k = t h e n u m b e r o f c o m p o u n d i n g p e r i o d s p e r y e a r
F V = t h e f u t u r e v a l u e
TVM: Quoted Rates vs Effective Annual Rates
Quoted interest rate (QR): the interest rate expressed interms of the interest paymentmade each period.
Effective Annual Rate (EAR): the interest rate expressed asif it were compoundedONCE per year.
Compounding can produce significant differences betweenquoted and effective annual rates.
TVM: Quoted Rates vs Effective Annual Rates
6% annual interest, compounded annually, produced a futurevalue of $1060 at the end of year 1-- the quoted and effectiveinterest rates are the same.
6% annual interest, compounded monthly, produced a futurevalue of $1061.68 at the end of year 1 -- the effective interestrate is 6.168%.
TVM: Quoted Rates vs Effective Annual Rates
I n g e n e r a l , t h e e f f e c t i v e a n n u a l r a t e ( E A R ) i s :
w h e r e Q R = t h e q u o t e d ( a n n u a l ) i n t e r e s t r a t e , a n d
k = t h e n u m b e r o f c o m p o u n d i n g p e r i o d sp e r y e a r .
E A RQ R
kk [ ]1 1
TVM: Quoted Rates vs Effective Annual Rates
E x a m p l e : t h e e f f e c t i v e a n n u a l r a t e f o r a d e p o s i t t h a t r e c e i v e s6 % a n n u a l i n t e r e s t , c o m p o u n d e d m o n t h l y , i s :
= 1 . 0 0 5 1 2 - 1
= 1 . 0 6 1 6 7 8 - 1
= 0 . 0 6 1 6 7 8 o r 6 . 1 6 7 8 %
E A R [.
]10 0 6
1 211 2
TVM: Quoted Rates vs Effective Annual Rates
HP 10B Keystrokes
Clears registersMonthly compounding6% annual interestCompute effective rateDisplay 4 decimal
places
CLEAR ALL
12 P/YR
I/YR
EFF %
DISP
6
4
TVM: Quoted Rates vs Effective Annual Rates
Continuous compounding occurs when deposits receiveinstantaneous interest compounded throughout the year.
The effective interest rate with continuous compounding is:
EAR = eQR - 1, where QR is the quoted rate.
Example: the EAR for a deposit that receives 6% annualinterest, compounded continuously, is
EAR = e0.06 - 1 = 0.061837 or 6.1837%
TVM: Quoted Rates vs Effective Annual Rates
HP 10B Keystrokes
Clears registers
Compute e to 0.06 power
Subtract 1
Multiply by 100 to get
the percent.
CLEAR ALL
0.06 e X
- 1 =* 100 =
TVM: Quoted Rates vs Effective Annual Rates
The following table provides the effective annual rates (EARs)for a deposit that receives 6% annual interest for variouscompounding frequencies:
Compounding Frequency EARAnnual compounding (k = 1) 6.0000%Semi-annual compounding (k = 2) 6.0900%Quarterly compounding (k = 4) 6.1364%Monthly compounding (k = 12) 6.1678%Weekly compounding (k = 52) 6.1800%Daily compounding (k = 365) 6.1831%Continuous compounding (k = large) 6.1837%
TVM: Present Value of a Future CFAnnual Discounting
An investment promises $1000 cash in two years. A potentialinvestor believes the investment should return 8% annually.How much would this investor be willing to pay today for thisopportunity?
$1000 = PV (1 + 0.08)2
= PV (1.1664), so
PV = $1000/1.1664
= $857.34At a required return of 10%, the present value is $826.45.
TVM: Present Value of a Future CF Annual Discounting: HP 10B Keystrokes
CLEAR ALL
P/YR
FV
N
I/YR
PV
1
1000
2
8
Clears registers
One payment per year
$1,000 future value
CF in two years
8% annual interest
Compute present value
TVM: Present Value of a Future CFAnnual Discounting
T h e p re s e n t v a lu e (P V ) o f a s in g le fu tu re c a s h f lo w (C F ) i s :
w h e re F V = th e s in g le fu tu re c a s h f lo w ,
d = th e a n n u a l d is c o u n t r a te , a n d
n = th e n u m b e r o f y e a r s in th e in v e s tm e n t te rm .
P V F Vd n
1
1( )
TVM: Annuities
Annuity: a finite sequence of equal paymentsmade across equally spaced timeintervals.
Ordinary Annuity: the payments occur at the endof the period.
Examples?
Annuity Due: the payments occur at the beginning of the period.
Examples?
TVM: Future Value of an Annual Ordinary Annuity
How much will $1,200, deposited at the end of each year,accumulate to in three years if the deposits earn 6% annualinterest (compounded annually).
Year Future Value + Contribution = Total FV
1 $0 + $1,200 = $1,200.00
2 $1,200x1.06 + $1,200 = $2,472.00
3 $2,472x1.06 + $1,200 = $3,820.32
TVM: Future Value of an Annual Ordinary Annuity
HP 10B Keystrokes
Clear registers
One payment per year
$1,200 payment/year
6% annual interest
For 3 years
Compute future value
CLEAR ALL
P/YR
PMT
I/YR
N
FV
1
1200
6
3
TVM: Future Value of an Annual Ordinary Annuity
F V = P M T ( 1 + i ) n - 1 + P M T ( 1 + i ) n - 2 + . . . . . . . . +
P M T ( 1 + i ) 2 + P M T ( 1 + i ) + P M T
= P M T [ ( 1 + i ) n - 1 + . . . . + ( 1 + i ) 2 + ( 1 + i ) + 1 ]
= P M T i P M Ti
it
n n t n
( ){ ( ) }
11 1
1
TVM: Future Value of a Periodic Ordinary Annuity
Instead of depositing $1,200 at the end of the year for three years, computethe future value of the annuity if the deposits are $100 at the end of eachmonth for 36 months and earn 6% annual interest, compounded monthly.
Month Future Value + Contribution = Total FV
1 $ 0 + $100 = $ 100.00
2 $100 x (1.005) + $100 = $ 200.50
3 $200.5 x (1.005) + $100 = $ 301.50 . .36 $3,814.54 x (1.005) + $100 = $3,933.61
TVM: Future Value of a Periodic Ordinary Annuity
HP 10B Keystrokes
CLEAR ALL
P/YR
PMT
I/YR
12
100
3
6
FV
x P/YR
Clears registers
Monthly payments
$100 payment/mo.
36 months (3 years)
6% annual interest
Compute future value
TVM: Future Value of a Periodic Ordinary Annuity
The total future value is simply the sum of 36 future values:
FV = $100(1.005)35 + $100(1.005)34 + .... + $100
= $100[(1.005)35 + (1.005)34 + .... + 1]
= $100[1.1907 + 1.1848 + .... + 1]
= $100[39.33610]
= $3,933.61
TVM: Future Value of a Periodic Ordinary Annuity
In general,
FV = PMT
where PMT = the equal periodic payment made at theend of the period;
i/k = the periodic interest rate;nk = the number of periods in the annuity;FV = the future value.
( )11
i
kt
nknk t
TVM: Future Value of a Periodic Annuity Due
What is the future value of the three year, 6% annual interestrate, $100 monthly annuity if the cash flows occur at thebeginning, rather than the end, of the month? Interest iscompounded monthly.
FV = $100(1.005)36 + $100(1.005)35 +...+ $100(1.005)
= $100[1.1967 + 1.1907 + .... + 1.005]
= $100[39.5328]
= $3,953.28
TVM: Future Value of a Periodic Annuity Due
HP 10B Keystrokes
CLEAR ALL
P/YR
I/YR
PMT
BEG/END
FV
12
6
100
3 x P/YR
Clear registers
12 payments per year
6% annual interest
$100 payment/mo.
Annuity due (BEGIN)
36 monthly payments
Compute future value
TVM: Future Value of a Periodic Annuity Due
In general,
FV = PM T
w here
PM T = the equal paym ent that occurs atthe beginning of the period;
i/k = the periodic in terest rate; and
nk = the num ber of periods in the annuity.
( )11
1
i
kt
nknk t
TVM: Present Value of an Annual Ordinary Annuity
What is the present value of receiving $1,200 at the end the
year for three years if future cash flows are discounted at 8%
annually?
Year Present Value of Receipt Total PV
1 $1,200 x (1/1.08) = $1,111.11 $1,111.11
2 $1,200 x (1/1.082) = $1,028.81 $2,139.92
3 $1,200 x (1/1.083) = $ 952.60 $3,092.52
TVM: Present Value of an Annual Ordinary Annuity
HP 10B Keystrokes
CLEAR ALL
P/YR
PMT
N
I/YR
PV
1
1200
3
8
Clears registers
1 payment per year
$1,200 payment/year
3 year term
8% annual interest
Compute present value
TVM: Present Value of an Annual Ordinary Annuity
The total present value is the sum of the present values:
PV = $1,200 (1/1.08) + $1,200 (1/1.082) + $1,200 (1/1.083)
= $1,200 [(1/1.08) + (1/1.082) + (1/1.083)]
= $1,200[0.9259 + 0.8573 + 0.7938]
= $1,200[2.5771]
= $3,092.52
TVM: Present Value of an Annual Ordinary Annuity
The PV of an annual ordinary annuity is:
PV = PMT + ..... + PMT
= PMT
=PMT
dd
n11
1
{
( )}
where d is the annual discount rate.
1
1 d
1
1( ) d n
1
11 ( ) d tt
n
TVM: Present Value of a Periodic Ordinary Annuity
Compute the present value of an annuity that pays $100 at theend of each month for three years if future cash flows arediscounted monthly at an annual rate of 8%.
PV = $100 + ..... + $100
= $100[0.9934 + 0.9868 + ..... + 0.7872]
= $100[31.9118]
= $3,191.18
1
10812
1(.
)
1
10812
36(.
)
TVM: Present Value of a Periodic Ordinary Annuity
Hp 10B Keystrokes
CLEAR ALL
P/YR
PMT
I/YR
PV
12
100
3
8
x P/YR
Clear registers
Monthly payments
$100 payment/mo.
For 36 months
8% annual interest
Compute present value
TVM: Present Value of a Periodic Ordinary Annuity
In general, the PV of a periodic ordinary annuity is:
PV = PMT
where PMT = the periodic payment;
d/k = the periodic discount rate;
nk = the number of discounting periods inthe annuity term.
1
11 ( ) d
ktt
nk
TVM: Present Value of a Periodic Annuity Due
Recompute the present value of the $100 monthly annuity if thecash flows are received at the beginning of the month and arediscounted monthly at an annual rate of 8% .
PV = $100 + $100 + ... + $100
= $100 [ 1 + 0.9934 + ..... + 0.7925 ]
= $100 [32.1246]
= $3,212.46
1
108
121(
.)
1
10812
35(.
)
TVM: Present Value of a Periodic Annuity Due
HP 10B Keystrokes
CLEAR ALL
12 P/YR
100 PMT
8 I/YR
3
BEG/END
PV
x P/YR
Clear registers
Monthly payments
$100 monthly payment
8% annual interest36 months
Annuity due (BEGIN)
Compute present value
TVM: Present Value of a Periodic Annuity Due
In general, the present value of a periodic annuity due is:
PV = PMT
where d/k = the periodic discount rate,
nk = the number of discounting periods, and
PMT = the equal periodic payment.
1
1 11 ( ) d
ktt
nk
TVM: Annual Sinking Fund Payment for Ordinary Annuities
How much has be set aside at the end of each year in anaccount paying 6% annual interest (compounded annually) toaccumulate to a future value of $5,000 in 3 years?
$5,000 = PMT (1.06)2 + PMT (1.06) + PMT
= PMT [ 1.1236 + 1.06 + 1]
= PMT [ 3.1836 ]
Solve for the annual sinking fund payment:
PMT = $5,000/3.1836 = $1,570.55
TVM: Annual Sinking Fund Payment for Ordinary Annuities
HP 10B Keystrokes
CLEAR ALL
1 P/YR
6 I/YR
5000 FV
3 N
PMT
Clears registers
One payment per year
6% annual interest
$5,000 future value
In three years
Compute the annual
payment.
TVM: Periodic Sinking Fund Payment for Ordinary Annuities
In general, the periodic sinking fund payment (PMT) forordinary annuities is:
PMT = FV
where i/k = the periodic interest rate,
nk = the number of periodic sinking fund payments,
FV= the desired future value.
1
11
( )
i
kt
nk nk t
TVM: Periodic Sinking Fund Payment for Ordinary Annuities
Example: compute the quarterly payment, made at the end ofeach quarter, necessary to accumulate to $25,000 in six years ifthe payments earn 8% annual interest compounded quarterly.
PMT = $25,000
= $821.78
1
1008
41
24 24
(.
)
t
t
TVM: Periodic Sinking Fund Payment for Ordinary Annuities
HP 10B Keystrokes
4
CLEAR ALL
P/YR
25000 FV
6
8 I/YR
PMT
x P/YR
Clear registers
Quarterly payments
$25,000 future value
24 payments (6 years)
8% annual interest
Compute quarterly
payment
TVM: Periodic Sinking Fund Payment for an Annuity Due
Example: compute the quarterly payment, made at thebeginning of the period, necessary to accumulate to $25,000 insix years if the payments earn 8% annual interest compoundedquarterly.
PMT = $25,000
= $805.66 ($16.11 less than the ordinary annuity pmt)
1
1008
41
24 24 1
(.
)
t
t
TVM: Periodic Sinking Fund Payment for an Annuity Due
HP 10B Keystrokes
CLEAR ALL
4 P/YR
25,000 FV
6
8 I/YR
BEG/END
PMT
x P/YR
Clear registersQuarterly payments$25,000 future value24 quarterly payments8% annual interestAnnuity due (BEGIN)Compute quarterly
payment
TVM: Annual Payment to Amortize Debt for Ordinary Annuities
Compute the annual payment necessary to fully amortize(completely repay) an 8%, 3 year, $5000 loan if the paymentsare made at the end of the year.
$5,000 = PMT/1.08 + PMT/1.082 + PMT/1.083
= PMT [ 1/1.08 + 1/1.082 + 1/1.083 ]
= PMT [ 0.9259 + 0.8573 + 0.7938 ]
= PMT [ 2.5771 ]
So, PMT = $5,000/2.5771 = $1,940.17
TVM: Annual Payment to Amortize Debt for Ordinary Annuities
HP 10B Keystrokes
CLEAR ALL
1 P/YR
5,000 PV
3 N
8 I/YR
PMT
Clear registers
Annual payments
$5,000 present value
3 year term
8% annual interest
Compute annual payment
TVM: Periodic Payment to Amortize Debt for Ordinary Annuities
In general, the periodic payment (PMT) made at the end of theperiod necessary to amortize (completely repay) a loan is:
PMT = PV
wherePV = the loan amount, or present value,
d/k = the periodic discount (or interest) rate, and
nk = the number of periods in the loan term.
11
11( ) d
ktt
nk
TVM: Periodic Payment to Amortize Debt for Ordinary Annuities
Example: compute the constant monthly payment necessary tofully amortize a $100,000, 7.5% annual interest rate, 30 yearmortgage.
PMT = $100,000
= $ 699.21
11
10075
121
360
(.
)
tt
TVM: Periodic Payment to Amortize Debt for Ordinary Annuities
HP 10B Keystrokes
CLEAR ALL
P/YR
PV
I/YR
PMT
12
100,000
7.5
30 x P/YR
Clear registers
Monthly payments
$100,000 loan (PV)
7.5% annual interest
360 monthly pmts
Compute monthly payment
MORE TIME VALUE OF MONEY
• Perpetuities
• Growing Perpetuities
• The PV of Uneven Cash Flows
• The PV of Grouped Cash Flows
MORE TIME VALUE OF MONEY:Perpetuities
PERPETUITY: an infinite sequence of equal periodic cash flows (also called CONSOLS).
The present value of a perpetuity with periodic cash flow CFand periodic discount rate r is:
PV =
=
CF
r
CF
r
CF
r( ) ( ) ( ).......
1 1 12 3
CF
r
MORE TIME VALUE OF MONEY:Perpetuities
T h e p re s e n t v a lu e o f a p e rp e tu i ty th a t p ro v id e s $ 5 0 ,0 0 0 p e ry e a r ( fo re v e r ) d is c o u n te d a t 1 0 % a n n u a l ly is :
P V = $ 5 0 ,
.$ 5 0 0 ,
0 0 0
0 1 00 0 0
MORE TIME VALUE OF MONEY:Growing Perpetuities
The present value of a periodic perpetuity with end of periodcash flows
discounted at rate r (for r > g) is:
PV =
=
CF CF gjj ( )1
CF g
r
CF g
r
CF g
r
( )
( )
( )
( )
( )
( ).....
1
1
1
1
1
1
2 3
32
CF g
r g
( )1
MORE TIME VALUE OF MONEY:Growing Perpetuities
T h e p r e s e n t v a l u e o f a $ 5 0 , 0 0 0 a n n u a l p e r p e t u i t y t h a t i n c r e a s e s4 % p e r y e a r a n d i s d i s c o u n t e d a t 1 0 % a n n u a l l y i s :
P V $ 5 0 , ( . )
.
$ 5 0 , ( . )
.. . . .
0 0 0 1 0 4
1 1
0 0 0 1 0 4
1 1
2
2
P V $ 5 2 ,
.
$ 5 4 ,
.. . . . .
0 0 0
1 1
0 8 0
1 1 2
P V
$ 5 2 ,
. .
$ 5 2 ,
.$ 8 6 6 , .
0 0 0
0 1 0 0 4
0 0 0
0 0 66 6 6 6 7
MORE TIME VALUE OF MONEY:The PV of Uneven Cash Flows
P V =
T h e p r e s e n t v a l u e o f a s e q u e n c e o f d i f f e r e n t p e r i o d i c c a s hf l o w s ( C F 1 , C F 2 , C F 3 , . . . . . ) , d i s c o u n t e d a t t h e p e r i o d i c d i s c o u n tr a t e r , i s :
C F
r
C F
r
C F
r1 2
23
31 1 1( ) ( ) ( ). . . . . .
MORE TIME VALUE OF MONEY:The PV of Uneven Cash Flows
W h a t i s t h e p r e s e n t v a l u e o f r e c e i v i n g $ 5 0 0 o n e y e a r f r o mt o d a y , $ 1 , 5 0 0 t h r e e y e a r s f r o m t o d a y , a n d $ 2 , 5 0 0 f o u r y e a r sf r o m t o d a y i f f u t u r e r e c i e p t s a r e d i s c o u n t e d a n n u a l l y a t 1 0 % ?
P V =
= $ 4 5 4 . 5 5 + $ 1 , 1 2 6 . 9 7 + $ 1 , 7 0 7 . 5 3
= $ 3 , 2 8 9 . 0 5
$ 5 0 0
.
$ 1 ,
.
$ 2 ,
.1 1
5 0 0
1 1
5 0 0
1 13 4
MORE TVMThe PV of Uneven Cash Flows
HP 10B Keystrokes
CLEAR ALL
P/YR
CFj
I/YR
NPV
CFj
CFj
CFj
CFj
1
0
500
0
1500
2500
10
Clears registers
One payment per year
Initial CF = 0
1st CF = $500
2nd CF = $0
3rd CF = $1,500
4th CF = $2,500
Discount rate = 10%
Compute (net) present value
MORE TIME VALUE OF MONEY:The PV of Grouped Cash Flows
W h a t i s t h e p r e s e n t v a l u e o f a n i n v e s t m e n t t h a t i s e x p e c t e d t or e t u r n $ 1 0 , 0 0 0 p e r y e a r a t t h e e n d o f t h e n e x t t h r e e y e a r s ,$ 1 5 , 0 0 0 a t t h e e n d o f y e a r s 4 a n d 5 , a n d t h e n $ 1 0 0 , 0 0 0 a t t h ee n d o f y e a r 6 i f e x p e c t e d c a s h f l o w s a r e d i s c o u n t e d a n n u a l l y a t1 2 % ?
P V =
= $ 9 2 , 7 2 5 . 6 0 .
$ 1 0 ,.
$ 1 5 ,.
$ 1 0 0 ,
.0 0 0
1
1 1 20 0 0
1
1 1 2
0 0 0
1 1 21
3
4
5
6tt
tt
MORE TIME VALUE OF MONEY:The PV of Grouped Cash Flows
HP 10B Keystrokes
CLEAR ALL
P/YR1
0
10000
3
15000
2
100000
12
CFj
CFj
CFj
CFj
I/YR
NPV
N j
N j
Clear registers
One payment per year
Initial CF = $0
1st grouped CF = $10,000
CF occurs 3 times
2nd grouped CF = $15,000
CF occurs 2 times
3rd CF = $100,000 (1 time)
Annual discount rate = 12%
Compute (net) present value