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ARCH 449 Chapter 3. Financial Mathematics. Notation. i = interest rate (per time period) n = # of time periods P = money at present F = money in future After n time periods Equivalent to P now, at interest rate i A = Equal amount at end of each time period on series E.g., annual.

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arch 449 chapter 3

ARCH 449 Chapter3

Financial Mathematics

notation
Notation
  • i = interest rate (per time period)
  • n = # of time periods
  • P = money at present
  • F = money in future
    • After n time periods
    • Equivalent to P now, at interest rate i
  • A = Equal amount at end of each time period on series
    • E.g., annual
assumptions
Assumptions

500

End of second year

+

200

200

0

1

2

3

4

5

Time

_

50

100

500

Biggining of third year

# on the cash flow means end of the period, and the starting of the next period

assumptions1
Assumptions

P

0 1 2 3 n-1 n

A

If P and A are involved the Present (P) of the given annuals is ONE YEAR BEFORE THE FİRST ANNUALS

slide5
If F and A are involved the Future (F) of the given annuals is AT THE SAME TIME OF THE LAST ANNUAL

:

F

…………..

1 2 3 .. .. n-1

n

0

Assumptions

0

A

slide6

F

…………..

1 2 3 .. .. n-1

n

0

Assumptions

P

0

A

overview
Overview
  • Converting from P to F, and from F to P
  • Converting from A to P, and from P to A
  • Converting from F to A, and from A to F
present to future and future to present
Present to Future,

and Future to Present

converting from present to future
Converting from Present to Future

Fn

………….

n

P0

Fn = P (F/P, i%, n)

To find F given P:

derive by recursion
Derive by Recursion
  • Invest an amount P at rate i:
    • Amount at time 1 = P (1+i)
    • Amount at time 2 = P (1+i)2
    • Amount at time n = P (1+i)n
  • So we know that F = P(1+i)n
    • (F/P, i%, n) = (1+i)n
    • Single payment compound amount factor

Fn = P (1+i)n

Fn = P (F/P, i%, n)

example present to future
Example—Present to Future

F = ??

P = $1,000

i = 10%/year

0 1 2 3

F3 = $1,000 (F/P, 10%, 3) = $1,000 (1.10)3

= $1,000 (1.3310) = $1,331.00

Invest P=$1,000, n=3, i=10%

What is the future value, F?

converting from future to present
Converting from Future to Present

Fn

………….

n

P

(P/F, i%, n) = 1/(1+i)n

  • To find P given F:
    • Discount back from the future
converting from future to present1
Converting from Future to Present
  • Amount F at time n:
    • Amount at time n-1 = F/(1+i)
    • Amount at time n-2 = F/(1+i)2
    • Amount at time 0 = F/(1+i)n
  • So we know that P = F/(1+i)n
    • (P/F, i%, n) = 1/(1+i)n
    • Single payment present worth factor
example future to present
Example—Future to Present

F9 = $100,000

…………

0 1 2 3 8 9

P= ??

i = 15%/yr

P = $100,000 (P/F, 15%, 9) = $100,000 [1/(1.15)9]

= $100,000 (0.1111) = $11,110 at time t = 0

Assume we want F = $100,000 in 9 years.

How much do we need to invest now, if the interest rate i = 15%?

annual to present and present to annual
Annual to Present,

and Present to Annual

converting from annual to present
Converting from Annual to Present

Fixed annuity—constant cash flow

P = ??

…………..

1 2 3 .. .. n-1

n

0

$A per period

converting from annual to present1
Converting from Annual to Present

We want an expression for the present worth P of a stream of equal, end-of-period cash flows A

P = ??

0 1 2 3 n-1 n

A is given

converting from annual to present2
Converting from Annual to Present

Write a present-worth expression for each year individually, and add them

The term inside the brackets is a geometric progression.

This sum has a closed-form expression!

converting from annual to present3
Converting from Annual to Present

Write a present-worth expression for each year individually, and add them

converting from annual to present4
Converting from Annual to Present

This expression will convert an annual cash flow to an equivalent present worth amount:

(One period before the first annual cash flow)

  • The term in the brackets is (P/A, i%, n)
  • Uniform series present worth factor
converting from present to annual
Converting from Present to Annual

Given the P/A relationship:

We can just solve for A in terms of P, yielding:

Remember:The present is always one period before the first annual amount!

  • The term in the brackets is (A/P, i%, n)
  • Capital recovery factor
future to annual and annual to future
Future to Annual,

and Annual to Future

converting from future to annual
Converting from Future to Annual

Find the annual cash flow that is equivalent to a future amount F

$F

…………..

1 2 3 .. .. n-1

n

0

0

The future amount $F is given!

$A per period??

converting from future to annual1
Converting from Future to Annual

Take advantage of what we know

Recall that:

and

Substitute “P” and simplify!

converting from future to annual2
Converting from Future to Annual

First convert future to present:

Then convert the resulting P to annual

Simplifying, we get:

  • The term in the brackets is (A/F, i%, n)
  • Sinking fund factor (from the year 1724!)
example
Example

How much money must you save each year (starting 1 year from now) at 5.5%/year:

In order to have $6000 in 7 years?

example1
Example

Solution:

The cash flow diagram fits the A/F factor (future amount given, annual amount??)

A= $6000 (A/F, 5.5%, 7) = 6000 (0.12096) = $725.76 per year

The value 0.12096 can be computed (using the A/F formula), or looked up in a table

converting from annual to future
Converting from Annual to Future

Given

Solve for F in terms of A:

  • The term in the brackets is (F/A, i%, n)
  • Uniform series compound amount factor
converting from annual to future1
Converting from Annual to Future

Given an annual cash flow:

$F

…………..

1 2 3 .. .. n-1

n

0

0

Find $F, given the $A amounts

$A per period