the time value of money chapter 9. chapter 9 - outline time value of money perpetuity future value...
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Chapter 9 - OutlineTime Value of MoneyPerpetuityFuture Value and Present ValueEffective Annual Rate (EAR)Annuity
Time Value of MoneyThe basic idea behind the concept of time value
of money is:– $1 received today is worth more than $1 in the
future OR – $1 received in the future is worth less than $1
today
Why?– because interest can be earned on the money
The connecting piece or link between present (today) and future is the interest rate
2 Questions to Ask in Time Value of Money Problems
Future Value or Present Value?Future Value: Present (Now) FuturePresent Value: Future Present (Now)
Single amount or Annuity?Single amount: one-time (or lump) sumAnnuity: same amount per year for a number
of years
Perpetuity: Constant Payment Forever
PV = PMT/iThis is the present value
of receiving a constant payment forever.
C
r
EXAMPLE:
• Suppose you wish to endow a chair at your old university. The aim is to provide $100,000 forever and the interest rate is 10%.
$100,000 PV = = $1,000,000 .10
A donation of $1,000,000 will provide an annual income of .10 x $1,000,000 = $100,000 forever.
PV =
Valuing perpetuities
Future Value and Present ValueFuture Value (FV) is what money today will
be worth at some point in the futureFV = PV x FVIFFVIF is the future value interest factor
Present Value (PV) is what money at some point in the future is worth todayPV = FV x PVIFPVIF is the present value interest factor
Future Value of a Lump Sum
FV = PV * (1+i)n
Why is this formula correct?
This is the amount that will be accumulated by investing a given amount today for n periods at a given interest rate.
Simple & compound interest
Simple interest rates) are calculated by multiplying the rate per period by the number of periods. Compound interest rates recognize the opportunity to earn interest on interest.
i ii iii iv vPeriods Interest Value Annuallyper per APR after compoundedyear period (i x ii) one year interest rate
1 6% 6% 1.06 6.000%
2 3 6 1.032 = 1.0609 6.090
4 1.5 6 1.0154 = 1.06136 6.136
12 .5 6 1.00512 = 1.06168 6.168
52 .1154 6 1.00115452 = 1.06180 6.180
365 .0164 6 1.000164365 = 1.06183 6.183
Effective Annual Rate (EAR) or Yield (EAY)
EAR or EAY = (1+inom/m)m-1This is used to calculate the compounded
yearly rate. It considers interest being earned on interest.
Adjusting for Non-Annual Compounding
Interest is often compounded quarterly, monthly, or semiannually in the real world
Since the time value of money tables and calculators often assume annual compounding, an adjustment must be made in those cases:– the number of years is multiplied by the
number of compounding periods– the annual interest rate is divided by the
number of compounding periods
FV With Compounding IntervalsFV of lump sum for various compounding
intervals:FV = PV * (1+i/m)n*m
where m=number of compounding periods per year
At an extreme there could be continuous compounding, then FV can be calculated as follows: FV = PV (ein) where e=2.7183...
PV of a Lump Sum
PV=FV/(1+i)n
This is the value today of a future lump sum to be received in the future after n periods of time at a given discount rate.
Present valuesexample: saving for a new computer
Suppose: - you need $3000 next year to buy a computer - the interest rate = 8% per yearHow much do you need to set aside now?
3000PV of $3000 = = 3000 x .926 = $2777.77 1.08 1- year discount factor
By end of 1 year $2777.77 grows to $2777.77 x 1.08 = $3000
Suppose you can postpone purchase until Year 2.
3000PV = = 3000 x .857 = $2572.02 1.082
2-year discount factor
PV With Compounding IntervalsPV of a lump sum for various compounding
intervals is calculated as:PV=FV/(1+i/m)n*m
where m=number of compounding periods per year
At an extreme there could be continuous discounting, then PV=FV/(ein) where e=2.7183...
PV of an Annuity
PV=A/(1+i)n = A*{(1/i) - (1/i) [1/(1+i)n]}This is the value today of a series of equal
payments to be received at the end of each period for n periods at a given interest rate.
Asset Year of payment PV
1 2 . . t t+1 . .
Perpetuity (first payment year 1)
Perpetuity (first payment year t + 1)
Annuity from year 1 to year t (1+r)
1t)
Cr(-
Cr
(1+r))rC 1
( t
Cr
An annuity is equal to the difference between two perpetuities
Using the annuity formulaExample: valuing an 'easy payment' scheme
Suppose: a car purchase involves 3 annual payments of $4000 the interest rate is 10% a year
1 1PV = $4000 x - .10 .10(1.10)3
= $4000 x 2.487 = $9947.41
ANNUITY TABLE
Number Interest Rate of years 5% 8% 10%
1 .952 .926 .909 2 1.859 1.783 1.736 3 2.723 2.577 2.487 5 4.329 3.993 3.791 10 7.722 6.710 6.145
FV of an Annuity
FV=A* (1+i)n = A*{[(1+i)n -1]/i} This is the accumulated value of equal
payments for n years at a given interest rate.
Annuity DueAnnuity due: Payments received at the
beginning of each period. Will be worth more (higher PV) since it gets
payments sooner.Will have higher FV since it has one extra
period to earn interest.Calculations are the same as before
except now we multiply by (1+i).
Solving for Annuity Payments (Present Value)
Recall thatPV=A*{(1/i) - (1/i) [1/(1+i)n]}, then
A=PV/{(1/i) - (1/i) [1/(1+i)n]}
A is the payment necessary for n years at given interest rate to amortize a present (loan) amount.
Solving for Annuity Payments (Future Value)
Recall thatFV=A*{[(1+i)n-1]/i}, thenA=FV/{[(1+i)n-1]/i}
A is the amount needed to be invested each period at a given interest rate to accumulate a desired future amount at the end of n years.
Solving for Rate of Return (i)For Lump Sum Case:
Since PV=FV/(1+i)n, then(1+i)n=FV/PV, and it follows that(1+i) = (FV/PV)1/n, and thereforei= (FV/PV)1/n-1
Solving for Rate of Return (i)Annuities:
In the annuity case, you could also solve for i using annuity relationship once you know the annuity.
You do not need a cash flow register.
Solving for Rate of Return (r) with uneven cash flows
0 = C0 + + + . . .
•Spreadsheets (use financial function =IRR)
•Financial calculators (IRR using cash flow register)
•Manual (Trial and error until PV of all cash flows equal zero)
C1 C2
(1 + r)1 (1 + r)2
Solving for Number of Periods (n)
Since PV=FV/(1+i)n, then(1+i)n=FV/PV, and it follows thatnLN(1+i)=LN(FV/PV), and thereforen=LN(FV/PV)/LN(1+i)
Net Present Value in the General discounted cash flow formula
NPV = C0 + + + . . .
Note: It is today’s cost of capital that matters
C1 C2
(1 + r)1 (1 + r)2
Example
If C0 = -500, C1 = +400, C2 = +400
r1 = r2 = .12
NPV = -500 + +
= -500 + 400 (.893) + 400 (.794)
= -500 + 357.20 + 318.80 = +176
400 400
1.12 (1.12)2