The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 123 Friday, March 12, 2004 12:39 PM

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Principles of Calculus 123

Vx()= C C CM+

------------------+------------------+...+--------------------

( 1 +i)

1

( 1 +i)

2

( 1 +i)

N

• x C C NC M( + )
  ------------------+------------------+...+---------------------------
  ( 1 +i)
  2
  ( 1 +i)
  3
  ( 1 +i)
  N+ 1

(^12 2) C 32 ⋅⋅C (NN( + 1 ))(CM+ )
+---x ------------------+--------------------+...+---------------------------------------------------
(^2) ( 1 +i)^3 ( 1 +i)^4 ( 1 +i)N+^2
1 3 32 ⋅⋅C 432 ⋅⋅⋅C

• -----------x ----------------------------+----------------------------+...
  32 ⋅ ( 1 ++ i ξ)^4 ( 1 ++i ξ)^5
  NN( + 1 )(N+ 2 )(CM+ )
  +------------------------------------------------------------------
  ( 1 ++ i ξ)N+^3

Asset managers, however, are primarily interested in percentage price

change. We can now compute the percentage price change as follows:

∆V= Vx()– V() 0

V V() 0

= –x ------------------C +------------------C +...+NC M--------------------------( + )-

( 1 +i)

2

( 1 +i)

3

( 1 +i)

N+ 1

1

×------------------------------------------------------------------------------------

C C CM+

------------------+------------------+...+--------------------

( 1 +i)

1

( 1 +i)

2

( 1 +i)

N

(^12 2) ⋅C 32 ⋅⋅C NN( + 1 )(CM+ )
+---x ------------------+--------------------+...+----------------------------------------------
(^2) ( 1 +i)^3 ( 1 +i)^4 ( 1 +i)N+^2
1
×-----------------------------------------------------------------------------------------
C C CM+
------------------+------------------+...+--------------------
( 1 +i)^1 ( 1 +i)^2 ( 1 +i)N