Time Value 2

7/28/2019 Time Value 2

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The Time Value of Money

Money NOW

is worth more thanmoney LATER!


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To simplify this material as much

as possible, you should

understand that there are only afew basic types of problems,

though each has severalvariations.

Future value or present value

Future value of an annuity

Present value of an annuity

Perpetuities and growing perpetuities


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Required Rate of Return Would an investor want Rs. 100 today or after one year?

Cash flows occurring in different time periods are notcomparable.

It is necessary to adjust cash flows for their differences in timing

and risk.

Example : If preference rate =10 percent

An investor can invest if Rs. 100 if he is offered Rs 110 after

one year.

Rs 110 is the future value of Rs 100 today at 10% interest rate.

Also, Rs 100 today is the present value of Rs 110 after a year at

10% interest rate.

If the investor gets less than Rs. 110 then he will not invest.

Anything above Rs. 110 is favorable.


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Time Value Adjustment

Two most common methods of adjusting cash

flows for time value of money:

Compoundingthe process of calculating

future values of cash flows and

Discountingthe process of calculating

present values of cash flows.


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A sum of money today is called a

present value.

We designate it mathematically with asubscript, as occurring in time period 0

For example: P0 = 1,000 refers to $1,000today


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A sum of money at a future time

is termed a future value

We designate it mathematically with a

subscript showing that it occurs in time

period n.

For example: Sn = 2,000 refers to $2,000

after n periods from now.


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As already noted, the number of

time periods in a time valueproblem is designated by n.

n may be a number of years

n may be a number of months

n may be a number of quarters

n may be a number of any defined time

periods


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The interest rate or growth rate in

a time value problem isdesignated by i

i must be expressed as the interest rate perperiod.

For example if n is a number of years, i

must be the interest rate per year. If n is a number of months, i must be the

interest rate per month.


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The first of the general type of

time value problems is calledfuture value and present value

problems. The formula for these

problems is:

Sn = P0(1+i)n


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An example problem:

If you invest $1,000 today at an interest rate of 10

percent, how much will it grow to be after 5 years?

Sn = P0(1+i)n

Sn = 1,000(1.10)

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Sn = $1,610.51 Table Sn = P0(CVFn,i)Compounded value factor


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Future Value Example

If you deposited Rs 55,650 in a bank whichwas paying 15 percent rate of interest on a ten

year time deposit how much the deposit grow

at the end of ten years

Sn = P0(CVFn,I= 55650x 4.046 = 225,159.90

TABLE A Pg 796


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Present Value of a Single Cash Flow

Assume you will receive an inheritance of$100,000, six years from now. How much couldyou borrow from a bank today and spend now,such that the inheritance money will be exactlyenough to pay off the loan plus interest when it isreceived? Assume the bank charges an interestrate of 12 percent?

Sn = P0(1+i)n so, P0 = Sn/(1+i)

n

P0 = 100,000/(1.12)

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P0 = $50,663 Table (Pg798)

P0 = Sn X PVFn,i


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Example TABLE C Pg 798

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Another example problem:

How long will it take for $10,000 to grow to

$20,000 at an interest rate of 15% per year?

Sn = P0(1+i)n

20,000 = 10,000(1.15)n

n = 4.96 years (or, about 5 years)


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One more example problem:

If you invest $11,000 in a mutual fund today, andit grows to be $50,000 after 8 years, whatcompounded, annualized rate of return did you

earn?

Sn = P0(1+i)n

50,000 = 11,000(1+i)8

i = 20.84 percent per year


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The next two general types of

time value problems involveannuities

An annuity is an amount of money thatoccurs (received or paid) in equal amounts

at equally spaced time intervals.

These occur so frequently in business thatspecial calculation methods are generally

used.


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For example:

If you make payments of $2,000 per year

into a retirement fund, it is an annuity. If you receive pension checks of $1,500 per

month, it is an annuity.

If an investment provides you with a returnof $20,000 per year, it is an annuity.


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For the future value of an

annuity:

FV = PMT[(1+i)n - 1]/i

TABLE

The term within bracket is the

compounded value factor for anannuity of Re 1 which shall be

FV = PMT CVFAn,i


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An example problem:

If you save $5000 per year at 6percent perannum, how much will you have at the endof 4 years?

Note that since time periods are months, i =6% per period, for 4 periods.

FV = PMT[(1+i)n - 1]/i

FV = 5000[(1.06)4 - 1]/.06

FV = $21,873

TABLE =5000(CVFA4,0.06 = 5000x4.3746 = 21873 Table B Pg 800


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Present Value of an Annuity

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ap tal Reco er and Loan


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ap tal Recovery and LoanAmortisation

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Example Table D Pg 802

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Another example problem:

If you want to save $10,000 for retirement

after 3 years, and you earn 9 percent per

annum, how much must you save eachyear?

FV = PMT[(1+i)n - 1]/i

10,000 = PMT[(1.09)3 - 1]/.9

PMT = $3,951 per year


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An example problem:

If you borrow $100,000 today at 9 percent

interest per annum, and repay it in equal

annual payments over 10 years, how muchare the payments?

PV = PMT[(1+i)

n

-1]/[i(1+i)n

] 100,000 = PMT[(1+.09)10 -1]/[.09(1.09)10]

PMT = $15,582 per year


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Loan Amortisation Schedule

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End Payment Interest Principal Outstanding

of Year Repayment Balance0 10,000

1 3,951 900 3,051 6,949

2 3,951 625 3,326 3,623

3 3,951 326 3,625* 0

Present Value o an Uneven


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Present Value o an UnevenPeriodic Sum

In most instances the firm receives a stream of

uneven cash flows. Thus the present value

factors for an annuity cannot be used.

The procedure is to calculate the present value

of each cash flow and aggregate all present

values.

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PV o Uneven ash Flows:


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PV o Uneven ash Flows:Example

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A last type of time value problem

involves what are called,perpetuities

A perpetuity is simply an annuity that

continues forever (perpetually).

The formula for finding the present value ofa perpetuity is:

PV = PMT/i


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Present Value of Perpetuity

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A variation to perpetuity

problems is the case ofgrowing

perpetuities

If an annuity continues forever, and grows

in amount each period at a rate g, then

PV = PMT1/(i - g)


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An example problem:

If you invest in a stock that will pay adividend of $10 next year and grow at 5percent per year, and you require a 14percent rate of return, how much is thestock worth to you today?

PV = PMT1/(i - g)

PV = 10/(.14-.05) PV = $111.11


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Sinking Fund

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Example


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Value of an Annuity Due

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Future Value o n nnu ty:


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Future Value o n nnu ty:Example

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Future Value of an annuity due

Future Value of an annuity due =

1x4.375x1.06

= Rs 4637


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Example

The present value of Re 1 paid at the

beginning of each year for 4 years is

1 3.170 1.10 = Rs 3.487

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Time Value 2