The Mathematics of Financial Modelingand Investment Management

21-Bond Portfolio Man Page 670 Wednesday, February 4, 2004 1:12 PM

670 The Mathematics of Financial Modeling and Investment Management

...

n m m

∑ αiKi,^1 ∏(^1 + ρt)+ ...α+ iKim, = L^1 ∏(^1 + ρ )t + ...+ Lm

i = 1 t = 2 t = 2

ai ≥0; i ∈U

If we divide both sides of the last equation by

∏

m

( 1 + ρt)

t = 2

we see that the present value of the portfolio’s stream of cash flows must

be equal to the present value of the stream of liabilities. We can rewrite

the above expression in continuous-time notation as

n

• r t

∑[αiKi, 1 + ...α+ iKime m m

• rmtm
  , ] = L 1 + ...+ L e m
  i = 1

As in the case of CFM, if cash flows and liabilities do not occur at the

same dates, we can construct an enlarged model with more dates. At

these dates, cash flows or liabilities can be zero.

To see under what conditions this expression is insensitive to small

parallel shifts of the term structure, we perturb the term structure by a

small shift r and compute the derivative with respect to r for r = 0. In

this way, all rates are written as rt + r. If we compute the derivatives we

obtain the following equation:

n

d + m m]

∑[αiKi, 1 + ...αiKime

• (r + r)t
  [
  
  
  • (r + r)tm
    ]
  
  
  
  

,

i = 1 dL^1 + ...+ L e m m

------------------------------------------------------------------------------------------------ = -------------------------------------------------------------------

dr dr

n –(r + r)t

• ∑[αiKi, 1 + ...α+ iKimt e m m m
  
  
  • (r + r)t
    , m ] = –[L 1 + ...+ L t e m m m]
    i = 1
  
  
  
  

which tells us that the first-order conditions for portfolio immunization

are that the duration of the cash flows must be equal to the duration of