Financial Mathematics 1

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 2

# on the cash flow means end of the period, and the starting of the next period _ Time End of second year Biggining of third year

If P and A are involved the Present (P) of the given annuals is ONE YEAR BEFORE THE FİRST ANNUALS n-1 n A P

If F and A are involved the Future (F) of the given annuals is AT THE SAME TIME OF THE LAST ANNUAL : 0 A F 0 …… …….. n n-1

0 A F 0 …… …….. n n-1 P

7 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 8

To find F given P: 9 P0P0 FnFn n …… ……. F n = P (F/P, i%, n)

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 F n = P (1+i) n F n = P (F/P, i%, n) 10

Invest P=$1,000, n=3, i=10% What is the future value, F? P = $1,000 F = ?? i = 10%/year F 3 = $1,000 (F/P, 10%, 3) = $1,000 (1.10) 3 = $1,000 (1.3310) = $1,331.00

To find P given F: –Discount back from the future 12 P Fn Fn n …… ……. (P/F, i%, n) = 1/(1+i) n

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 13

Assume we want F = $100,000 in 9 years. How much do we need to invest now, if the interest rate i = 15%? ………… F 9 = $100,000 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

Annual to Present, and Present to Annual

Fixed annuity—constant cash flow $A per period P = ?? 0 …… …….. n n-1

We want an expression for the present worth P of a stream of equal, end-of- period cash flows A n-1 n A is given P = ??

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!

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

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

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

Find the annual cash flow that is equivalent to a future amount F 0 $A per period?? $F$F The future amount $F is given! 0 …… …….. n n-1

Take advantage of what we know Recall that: and Substitute “P” and simplify!

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!)

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?

Solution: –The cash flow diagram fits the A/F factor (future amount given, annual amount??) –A= $6000 (A/F, 5.5%, 7) = 6000 ( ) = $ per year –The value can be computed (using the A/F formula), or looked up in a table

Given Solve for F in terms of A:  The term in the brackets is (F/A, i%, n)  Uniform series compound amount factor

Given an annual cash flow: 0 $A per period $F$F Find $F, given the $A amounts 0 …… …….. n n-1

Single-Payment Compound-Amount Factor Single-Payment Present-Worth Factor Equal-Payment-Series Compound-Amount Factor Equal-Payment-Series Sinking-Fund Factor Equal-Payment-Series Capital-Recovery Factor Equal-Payment-Series Present-Worth Factor