Prepatory Workshop

1
Prepatory Workshop Spring 2008 Time Value of
Money
2

• Time Value of Money
• When money is some how invested (and not just
  placed under a mattress or in a safe), the amount
  of money grows
• The amount of money in an account that earns a
  positive ROR will be greater in the future than
  what it is today
• Thus the money in the account has different
  values at different points in time
• This is what the term Time Value of Money
  refers to
• Future Value
• Example You deposit 100 in a savings account
  that pays 6 per year (1 compounding period per
  year). What amount of money would you have in
  this account after 1 year?
• Cash Flow Diagram

i 6
FV ?
PV 100
0
1
or this..
FV ?
i 6
0
1
PV 100

• Intuitive Solution
• The future value includes the deposit plus the
  interest payment
• How much is the interest Payment? Answer
  deposit x interest rate
• 100 x 0.06
  6
• How much is the future value? Answer 100 6
  106
• Formula Solution FV PV(1 i)n Note i k
  r
• FV 100(1 0.06)1 106
• Calculator Solution P/Y 1, N1, I/Y6, PV100
  CPT,FV FV 106
• Excel Solution
• 1) Formula
• 2) Function
• Point The amount of money in this account is
  different at different points in time

C2(1 B2)A2
3

• Present Value
• Money that is expected to be received or paid in
  the future does not have the same value as
  todays money because of TVM
• In order to determined what future money is worth
  in terms of todays dollars, we have to reverse
  the effects of TVM this is referred to as
  discounting
• Example What is the present value of 200 one
  year from now if it was invested at 4......or.
  How much would we have to deposit today into an
  account that pays 4 per year in order to have
  200 one year from now?

FV 200
PV ?
i 4
0
1
or this..
FV 200
i 4
0
1
PV ?
Formula Solution FV PV(1 i)n PV FV(1
i)n PV 200 / (1 0.04)1 192.30 Calculator
Solution P/Y 1, N1, I/Y4, FV200 CPT,PV PV
192.30 Excel Solution 1) Formula 2)
Function PV
Write down your inputs before you calculate
C2/(1 B2)A2
Reword the example above Consider a security
that yields 4 and promises to pay 200 one year
from now. What is this security theoretically
worth today? or..What is the fair market value
of this security? Answer The Present Value,
192.30
4

• Compound Interest
• When an investment is held for more than one
  interest paying period, the interest is
  compounded (interest is paid on previously
  earned interest as well as on the principle)
• Future Value (more than one compounding period)
• Example You deposit 100 in a savings account
  that pays 6 per year. What amount of money
  would you have in this account after 2 years?
  (What is the future value of 100 _at_ 6 after 2
  years?)

Calculator Solution P/Y1, N2, I/Y6, PV100
CPT, FV FV 112.36
5

• Future Value (more than one compounding
  period)(continued)
• The future value doesnt necessarily have to be
  computed at the end of the period in question
• Example You have purchased a security that
  yields 6.25 per year for 8 years. You paid
  1,000 for it, minus the sellers commission and
  other fees. How much money will accumulate in
  three years?
• Answer Find the FV at t3

Present Value (more than one compounding
period) Example Consider a security that yields
12 and promises to pay 5,000 three years from
now. What is this security theoretically worth
today?
6
Future Value (multiple uniform payments -
annuity) Example If you deposited 300 a year
(at the end of the year) into a savings account
that pays 5 APR, what would the account balance
be after 3 years? (An annuity in which the
payments occur at the end of a period is called
an Annuity in Arrears or Ordinary Annuity) Cash
Flow Diagram Formula Calculator
Solution
FV CF1(1 i)2 CF2(1 i)1 CF3
300(1 0.05)2 300(1 0.05)1 300
300(1.05)2 300(1.05)1 300
300(1.1025) 300(1.05) 300 330.7500
315.0000 300 945.75
P/Y 1, N3, I/Y5, PMT300 CPT,FV FV 945.75
Heres an explanation of what happened at each
time period
0
2
3
1
i 5
Beginning Balance 0.00
0.00 300.00
615.00 Interest Earned
0.00 0.00
15.00
30.75 Promised Payment 0.00
300.00 300.00
300.00 Ending Balance
0.00 300.00
615.00 945.75
300 x 0.05
615 x 0.05
FV
7
Future Value (multiple uniform payments -
annuity) Example If you deposited 300 a year
(at the beginning of the year) into a savings
account that pays 5 APR, what would the account
balance be after 3 years? (An annuity in which
the payments occur at the beginning of a period
is called an Annuity Due) Cash Flow Diagram
7
8
Present Value (multiple uniform payments -
annuity) Example You are considering purchasing
a security that promises 300 a year (at the end
of the year) for 3 years. It has a ROR of 5
APR. What is the fair market value of this
security? What type of annuity is this? Cash
Flow Diagram Formula PV CF1 / (1 i)1
CF2 / (1 i)2 CF3 / (1 i)3 300 / (1
0.05)1 300 / (1 0.05)2 300 / (1 0.05)3
300 / (1.05)1 300 / (1.05)2 300 /
(1.05)3 300 / (1.05) 300 / (1.1025)
300 / (1.1576) 285.7143 272.1088
259.1513 816.97
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• Other Than Annual Compounding
• Annual Compounding Not often used in
  business/finance world but it's easier to
  introduce compounding/discounting with this
  compounding arrangement
• Semiannual Compounding Used most often in bonds
• Quarterly Compounding Often used by banks for
  business loans.
• Monthly Compounding Used most often by banks
  for consumer loans and investments (CD's) also
  used in short-term bonds (with leases
• Daily Compounding Used by banks to lend/borrow
  from each other for very short terms (days
  weeks)
• Continuous Compounding Used in mathematical
  models of various more complicated financial
  concepts (i.e. duration, convexity, pricing an
  option contract, interest rate options swaps,
  etc.) (more on this later)
• Nominal Interest Rate ( isimple )
• This is often what people quote as your interest
  rate for loans, bank accounts, credit cards and
  bonds.
• It is also called the simple rate ( isimple ) or
  the quoted rate
• It must also be accompanied by a statement
  indicating the compounding frequency
• Example
• Annual inominal 8, compounded semiannually
• Semiannual inominal 8, monthly payments
• Periodic Rate
• this is the rate charged per compounding period.
• periodic Rate iperiodic inominal / m
• m is the number of payments per year
• Example

10
Modify FV and PV Formulas to Account for Other
Than Annual Compounding
iperiodic
FV PV(1 inominal/m)n PV FV / (1
inominal/m)n
Future Value (Other Than Annual
Compounding) Example If today you deposit 1,000
in to an account that pays 7.2000 per annum with
monthly compounding, how much will you have in
the account three years from now?
1) Determine the number of payments / compounding
periods m12, T3 n m x T 12 x 3 36
Formula FV PV (1 i/m)n 1,000(1
0.072/12)36 1,240.30
Calculator Solution 1) Compute periodic rate
iperiodic inominal/m 7.2/12 0.6 2) P/Y1,
N36, I/Y0.6, PV1000 CPT, FV FV 1,240.30
or P/Y12, N36, I/Y7.2, PV1000 CPT, FV
FV 1,240.30
11
Present Value of an Annuity (Other Than Annual
Compounding) Example An ordinary annuity pays
50 semiannually for two years. If the current
market interest rate for this annuity is 4, what
is it worth today?
Formula PV CF1/(1 i/m)1 CF2/(1 i/m)2
CF3/(1 i/m)3 CF4(1 i/m)4 50/(1
0.04/2)1 50/(1 0.04/2)2 50/(1 0.04/2)3
50/(1 0.04/2)4 50/(1.02)1 50/(1.02)2
50/(1.02)3 50(1.02)4 50/1.02
50/1.0404 50/1.0612 50/1.0824 49.0196
48.0584 47.1161 46.1923 190.39
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Future Value of an Annuity (Other Than Annual
Compounding) Example Today you invested 1,200
in a mutual fund account that pays 10.7800 APR.
You plan to deposit 1,200 at the beginning of
every 3 months thereafter. How much money would
you have in this account after 2.75 years?
13
Computing Payments of an Annuity (Other Than
Annual Compounding) Example You are considering
financing a new car which cost 48,999 with an
amortized loan. Your bank offers a nominal rate
of 7.200 per annum for a 6 year loan with
monthly payments. How much will each payment be?

14
Present Value of Uneven Cash Flows (Other Than
Annual Compounding) Example You are tasked with
estimating the fair market value of a security
that promises uneven future payments. The table
below shows the quarterly payment schedule (each
cash flow occurs at the end of the quarter). You
consider 7.2000 APR to be the appropriate
opportunity cost. What is the theoretical value
of this security?
Formula PV CF1/(1 i/m)1 CF2/(1 i/m)2
CF3/(1 i/m)3 CF4(1 i/m)4 300/(1
0.072/4)1 400/(1 0.072/4)2 500/(1
0.072/4)3 700/(1 0.072/4)4
300/(1.018)1 400/(1.018)2 500/(1.018)3
700(1.018)4 300/1.018 0 400/1.03632
500/1.05498 700/1.07397 294.6955
385.97972 473.9426 651.7889 1806.41
Calculator Solution 1) Compute periodic rate
iperiodic isimple/m 7.2000/4 1.8000 CF,
2nd, CLR WORK (Clears Cash Flow Registers) 0,
ENTER, ?, 300, ENTER ?, ?, 400, ENTER ?, ?, 500,
ENTER ?, ?, 700, ENTER NPV, 1.8, ENTER ?, CPT
1,806.41
Excel Solution Formula
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• Continuous Compounding
• Used in mathematical models of various more
  complicated financial concepts (i.e. duration,
  convexity, pricing an option contract, interest
  rate options swaps, etc.)
• Formula FV PVeiT or PVert where I (r) is an
  annual rate and T (t) is time in years
• Example If today you deposit 1,000 in to an
  account that pays 7.2000 per annum with
  continuous compounding, how much will you have in
  the account three years from now?
• FV PVeiT 1,000e(0.072)(3) 1,000e(0.216)
  1,000(1.2411) 1,241.10
• Compare this answer to that of the example from
  p. 12
• Example (from p.12) If today you deposit 1,000
  in to an account that pays 7.2000 per annum with
  monthly compounding, how much will you have in
  the account three years from now?
• P/Y12, N36, I/Y7.2, PV1000 CPT, FV FV
  1,240.30
• Perpetuities
• A type of annuity
• The uniform payments go on forever

PMT
0
8
1
2
3
4
5
PV
PV


PMT (1 i/m)n

PMT (i/m)

Example A corporation wishes to establish an
endowment fund that provides 5,000 per month.
The fund pays 6.0000 per annum. How much
should the corporation deposit into the
account? PV PMT / (i/m) 5,000 / (0.06/12)
5,000 / 0.005 1,000,000
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• Perpetuities (continued)
• Stock Valuation
• Example What is the theoretical value of a share
  of stock that pays a constant 0.25 dividend
  every quarter? ks 12
• PV PMT / (ks/m) P0 Div / (ks/m) 0.25/
  (0.12/4) 0.25 / 0.03 8.33
• Capitalize (Capitalization)
• Example What is the value of a firm that earns
  100m per year and its cost of debt is 10? (This
  firm is totally financed by debt)
• VFirm 100m / 0.10 1 billion