Chap5 6(Edited)

8/2/2019 Chap5 6(Edited)

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NOTESCHAPTER- 5

Future value It measures the nominal future sum of money that a given sum of money is"worth" at a specified time in the future assuming a certain interest rate, or more generally, rate

of return; it is the present value multiplied by the accumulation function. The value does not

include corrections for inflation or other factors that affect the true value of money in the future.

This is used in time value of money calculations.

To determine future value using compound interest:

Present value, also known as present discounted value, is the value on a given date of a

payment or series of payments made at other times.

If the payments are in the future, they are discounted to reflect the time value of money and other

factors such as investment risk. If they are in the past, their value is correspondingly enhanced toreflect that those payments have been (or could have been) earning interest in the intervening

time. Present value calculations are widely used in business and economics to provide a means to

compare cash flows at different times on a meaningful "like to like" basis.

The most commonly applied model of the time value of money is compound interest. To

someone who can lend or borrow for years at an interest rate per year (where interest of

"5 percent" is expressed fully as 0.05), the present value of the receiving monetary unitsyears in the future are:

This is also found from the formula for the future value with negative time.

The purchasing power in today's money of an amount of money, years into the future,

can be computed with the same formula, where in this case is an assumed future inflationrate.

The expression enters almost all calculations of present value. Where the interest

rate is expected to be different over the term of the investment, different values for may be

included; an investment over a two year period would then have PV of:


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Spreadsheets commonly offer functions to compute present value. In Microsoft Excel,

there are present value functions for single payments (=NPV) and series of equal,

periodic payments (=PV). Programs will calculate present value flexibly for any cashflow and interest rate or for a schedule of different interest rates at different times.

Compound interest arises when interest is added to the principal, so that, from that moment on,

the interest that has been added also earns interest. This addition of interest to the principal is

called compounding. A bank account, for example, may have its interest compounded every

year: in this case, an account with 1000 initial principal and 20% interest per year would have a

balance of 1200 at the end of the first year, 1440 at the end of the second year, and so on.

Present Value Interest FactorPVIF

A factor is one that can be used to simplify the calculation, for finding the present value of aseries of values. PVIFs can be presented in the form of a table with PVIF values separated byrespective period and interest rate combinations.

Definition of 'Discounted Cash Flow - DCF

A valuation method used to estimate the attractiveness of an investment opportunity. Discountedcash flow (DCF) analysis uses future free cash flow projections and discounts them (most often

using the weighted average cost of capital) to arrive at a present value, which is used to evaluatethe potential for investment. If the value arrived at through DCF analysis is higher than thecurrent cost of the investment, the opportunity may be a good one.


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Calculated as:

Future Value of Single / Multiple Cash Flows

To find out the future value of cash flows, we have to apply the compounding technique.Compounding may be yearly, half-yearly, quarterly, monthly etc.

Future Value of Single Cash Flow

Future value can be computed by the following formula:

FVn = PV (1 + r)n

Where

FV = Future value

PV = Present value

r = Rate of Interest

n = Number of periods

Example: FV of single cash flow compounded annually

Let us calculate the future value of an investment of $ 2,000 compounded annually at the rate of12%, after 4 years period.

FV = $ 2,000 x (1 + 0.12) 4

= $ 3,147.04

Frequent Compounding:

Interest is compounded often more than once a year. In such cases, the formula for FV becomes:In this case, the formula for FV becomes:

FVn = PV (1 + (r / m))n x m


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Where:m = Number of total compounding periods in a yearIf compounded semi-annually, m=2If compounded quarterly, m = 4 and so on. The more frequent compounding occurs in a year, the morewould be the future value as illustrated below.

Example: FV of single cash flow compounded semi-annually

In the above example, let us assume that the interest of 12% is compounded semi-annually; rest of detailsbeing the same, the future value after 4 years would be:

Here:

m = 2

FV = $ 2,000 x (1 + (0.12 / 2)) 4 x 2

FV = $ 3,187.70

Thus, when interest is compounded yearly once, the FV is only $3,147.04 whereas if it iscompounded twice a year, the FV is $3,187.70. Similarly, if interest is compounded quarterly ormonthly, FV would accordingly be greater.

Future value using Simple InterestIf no interest is earned on the interest on the investment, it is called as simple interest. The futurevalue of an investment in such cases would be calculated by the following formula:

FVn = PV (1 + [n x r])

Where

n = Number of years

r = Interest Rate

The future value using a simple interest would obviously be lower than the future value usingcompound interest as there is no interest earned on the interest portion of the investment.


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Example:

An investment of $10,000, if invested at 13% simple interest rate will in 6 years be:

FV = $ 10,000 x (1 + [6 X 0.13])

FV = $ 17,800

Future Value of Multiple Cash Flows

In many instances, we may be interested in the future value of series of payments of differentamounts at different time periods. In such cases, we can find the FV as illustrated below:

Example:

A person deposits $1000, $2000, $3000, $4000 and $5000 at the end of each of the 5 respective

years. The interest rate is 10%, compounded annually. Find the future value.

Endof

Year

AmountDeposited

No. of yearscompounded

CompoundedInterestfactor

FutureValue(B x D)

A B C D E

1 $ 1,000 4 1.4641 $ 1,464.10

2 $ 2,000 3 1.331 $ 2,662.00

3 $ 3,000 2 1.21 $ 3,630.00

4 $ 4,000 1 1.1 $ 4,400.00

5 $ 5,000 0 1.0 $ 5,000.00TOTAL $ 17,156.10

The compounded interest factor is calculated as given below:

For 4 years = (1 + 0.10)4 = 1.4641

For 3 years = (1 + 0.10)3 =1.331 and so on...


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CHAPTER6

Annuity The term annuity is used in finance theory to refer to any terminating stream of fixedpayments over a specified period of time.This usage is most commonly seen in discussions offinance, usually in connection with the valuation of the stream of payments, taking intoaccount time value of money, concepts such as interest rate and future value.Examples of annuities are regular deposits to a savings account, monthly home mortgagepayments and monthly insurance payments. Annuities are classified by the frequency of paymentdates. The payments (deposits) may be made weekly, monthly, quarterly, yearly, or at any otherinterval of time.

Present ValueAnnuity

An annuity is a series of equal annual cash flows. In general terms, the present value of an

annuity may be expressed as follows:

PVAn

= A A A

+ + ........(1 + r)1 (1 + r)2 (1 + r)n

OR

PV An = A { n 1 / (1 +r)t }

t = 1

Where

PV An = Present value of an annuity

A = annuity amount (even cash inflows)

r = discount rate and

n = number of years of annuity (the last year)

Example: Let us find out the present value of an annuity of $10,000 to be received in the next 4years time, the discount rate being 10%.

Years Cash inflows

PV factor at

10%

PV of cash

inflows1 $10,000 0.9091 $9,091

2 $10,000 0.8264 $8,264

3 $10,000 0.7513 $7,513

4 $10,000 0.6830 $6,830

Total $31,698.00


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Graphic Presentation of Time Line of Present value of Annuity

0 1 2 3 4

$ 10,000$10,000

$10,000

$10,000

$10,000

$ 9,091

$ 8,264

$ 7,513

$ 6,830

$31,698

(Total PV)

Alternatively, the PV of an annuity can also be calculated by the formula:

PV An =A [ { 1 - (1/ (1 +r))n } / r ]

Future ValueAnnuity

An annuity is a stream of equal annual cash flows occurring at regular intervals of time. Whenthe cash flows occur at the end of each period, the annuity is called as an ordinary annuity or a

deferred annuity. When the cash flows occur at the beginning of each period, the annuity iscalled as an annuity due. The Future Value of an Annuity is given by the following formula:

FV An = A (1 + r)n-1 + A (1 + r)n-2 + ........ A

= A [(1 + r)n -1] / r

Where

FV An = Future value of an annuity at time n

A = Annuity periodic amount

r = Interest rate

n = Number of periods


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Example: A person deposits $10,000 annually in a bank for 5 years which earns an interest of9%, compounded annually. Let us calculate the future value of his series of deposits at the end of5 years.

FV

An

= A [(1 + r)n - 1] / r

A = $10,000

r = 9% or .09

n = 5

Future value ofannuity

= $10,000 [(1 + 0.09)5 - 1] / 0.09

= $59,847.11

Time Line for an Annuity

1 2 3 4 5

$ 10,000 $ 10,000 $ 10,000 $ 10,000 $ 10,000

+

$ 10,900 +

$ 11,881 +

$ 12,950.29 +

$ 14,115.82

TOTAL $ 59,847.11

Example:

To find out the required annual deposits to accumulate a sum in future:

A company needs to accumulate $100 million in the next 6 years to retire its long-term debts.Calculate the amount the company needs to deposit annually in a fund which earns 14% interestper annum.

FVAn

= A [(1 + r)n - 1] / r


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r = .14 or 14%

n = 6

FVAn

= $100 million

Substituting the values:

$100million

= A [(1 + 0.14)6 - 1] / 0.14

A =$100 million / 8.5355187 = $11.716million

Present Value of a Perpetuity

Perpetuity is an annuity of infinite years or period. The present value of a perpetuity may beexpressed as follows:

P = A

r

Where

P = Present value of a perpetuity

A = Perpetual cash flow or annuity amount

r = Discount rate (in decimal form)

Example: Suppose if a person would receive $10,000 in perpetuity and the interest rate is 10%,the present value would be:

PV =$

10,000=

$100,000

0.10


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Effective Annual Interest Rate

An investment's annual rate of interest when compounding occurs more often than once a year.Calculated as the following:

Effective Annual Interest Rate: Consider a stated annual rate of 10%. Compounded yearly, thisrate will turn $1000 into $1100. However, if compounding occurs monthly, $1000 would growto $1104.70 by the end of the year, rendering an effective annual interest rate of10.47%. Basically the effective annual rate is the annual rate of interest that accounts for theeffect of compounding.

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Chap5 6(Edited)