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
0 1 2 3 4 5
500
200
50100
200
500
+
_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
0 1 2 3 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 1 2 3 .. .. n-1
0
A
F
0
…………..
n 1 2 3 .. .. 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
P0
Fn
n………….
Fn = 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
Fn = P (1+i)n
Fn = P (F/P, i%, n)
10
• Invest P=$1,000, n=3, i=10%
• What is the future value, F?
11
0 1 2 3
P = $1,000
F = ??
i = 10%/year
F3 = $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
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%?
14
0 1 2 3 8 9…………
F9 = $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 1 2 3 .. .. n-1
• We want an expression for the present worth P of a stream of equal, end-of-period cash flows A
0 1 2 3 n-1 n
A is given
P = ??
• Write a present-worth expression for each year individually, and add them
1 2 1
1 1 1 1..
(1 ) (1 ) (1 ) (1 )n nP A
i i i i
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
1 2 1
1 1 1 1..
(1 ) (1 ) (1 ) (1 )n nP A
i i i i
(1 ) 1 0
(1 )
n
n
iP A for i
i i
• This expression will convert an annual cash flow to an equivalent present worth amount:– (One period before the first annual cash flow)
(1 ) 1 0
(1 )
n
n
iP A for i
i i
The term in the brackets is (P/A, i%, n) Uniform series present worth factor
• Given the P/A relationship:(1 ) 1
0(1 )
n
n
iP A for i
i i
(1 )
(1 ) 1
n
n
i iA P
i
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
The future amount $F is given!
0
…………..
n 1 2 3 .. .. n-1
• Take advantage of what we know• Recall that:
and
1
(1 )nP F
i
(1 )
(1 ) 1
n
n
i iA P
i
Substitute “P” and simplify!
• First convert future to present:– Then convert the resulting P to annual
• Simplifying, we get:
1 (1 )
(1 ) (1 ) 1
n
n n
i iA F
i i
(1 ) 1nA
i
iF
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 (0.12096) = $725.76 per year
– The value 0.12096 can be computed (using the A/F formula), or looked up in a table
• Given
• Solve for F in terms of A:
(1 ) 1n
iA F
i
)=A
(1 1F
ni
i
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
Find $F, given the $A amounts
0
…………..
n 1 2 3 .. .. 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
, , (1 )nF P i n i
1, ,
(1 )nP F i n
i
(1 ) 1, ,
niF A i n
i
, ,(1 ) 1n
iA F i n
i
(1 ), ,
(1 ) 1
n
n
i iA P i n
i
(1 ) 1, ,
(1 )
n
n
iP A i n
i i