ARCH 449 Chapter 3

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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 Chapter3

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
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

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

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

F

…………..

1 2 3 .. .. n-1

n

0

Assumptions

P

0

A

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

Converting from Present to Future

Fn

………….

n

P0

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

To find F given P:

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

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

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 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

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

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 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 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 Present

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

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

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

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 Annual

Take advantage of what we know

Recall that:

and

Substitute “P” and simplify!

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

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?

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

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 Future

Given an annual cash flow:

\$F

…………..

1 2 3 .. .. n-1

n

0

0

Find \$F, given the \$A amounts

\$A per period