# Problem in minimizing expression

I am trying to implement bootstrapping (https://en.wikipedia.org/wiki/Bootstrapping_(finance)) of CDS curve. To implement it I am trying to minimize a series of expression. The minimum value of expression will result from certain value of variable. I can easily do it in excel however trying to find a way in mathematica.

I am trying to find value of h1 and h2 in below code using Minimize function to minimize two expression.

'Minimize[Sum[50/10000*1/4*1/4*n*97/100*Exp[-h1*1/4*n], {n, 1, 4}] -
Sum[1/2*1/4*n*97/100*(Exp[-h1*1/4*(n - 1)] - Exp[-h1*1/4*n]), {n, 1, 4}]
&&
Sum[77/10000*1/4*1/4*n*97/100*Exp[-h1*1/4*n], {n, 1, 4}] +
Sum[77/10000*1/4*1/4*n*94/100*Exp[-h2*1/4*n], {n, 5, 8}] -
Sum[1/2*1/4*n*97/100*(Exp[-h1*1/4*(n - 1)] - Exp[-h1*1/4*n]), {n, 1, 4}] +
Sum[1/2*1/4*n*94/100*(Exp[-h1*1/4*(n - 1)] - Exp[-h2*1/4*n]), {n, 5}] +
Sum[1/2*1/4*n*94/100*(Exp[-h2*1/4*(n - 1)] - Exp[-h2*1/4*n]), {n, 6,8}],
{h1, h2}, Reals]'


I get following output but doesn't give values of h1 and h2 which will minimize the expression. I get h1 = 0.01148 and h2 = 0.01451 when I use goal seek in excel. Can anybody help in trouble shooting the issue in Minimize function

Minimize[(97 E^-h1)/80000 + (291 E^(-3 h1/4))/320000 + (97 E^(-h1/2))/
160000 + (97 E^(-h1/4))/320000 - 97/200 (-E^-h1 + E^(-3 h1/4)) -
291/800 (-E^(-3 h1/4) + E^(-h1/2)) - 97/800 (1 - E^(-h1/4)) -
97/400 (-E^(-h1/2) + E^(-h1/4)) && (7469 E^-h1)/4000000 + (
22407 E^(-3 h1/4))/16000000 + (7469 E^(-h1/2))/8000000 + (
7469 E^(-h1/4))/16000000 + (3619 E^(-2 h2))/1000000 + (
25333 E^(-7 h2/4))/8000000 + (10857 E^(-3 h2/2))/4000000 + (
3619 E^(-5 h2/4))/1600000 - 97/200 (-E^-h1 + E^(-3 h1/4)) -
291/800 (-E^(-3 h1/4) + E^(-h1/2)) - 97/800 (1 - E^(-h1/4)) -
97/400 (-E^(-h1/2) + E^(-h1/4)) + 47/50 (-E^(-2 h2) + E^(-7 h2/4)) +
329/400 (-E^(-7 h2/4) + E^(-3 h2/2)) + 47/80 (E^-h1 - E^(-5 h2/4)) +
141/200 (-E^(-3 h2/2) + E^(-5 h2/4)) + 47/100 (E^(-3 h1/4) - E^-h2) +
141/400 (E^(-h1/2) - E^(-3 h2/4)) + 47/200 (E^(-h1/4) - E^(-h2/2)) +
47/400 (1 - E^(-h2/4)), {h1, h2}, Reals]

• Are you trying to minimize 2 expressions with the && ? Commented Jun 13, 2015 at 16:46
• yes there are two expression. First expression only uses h1 but second expression uses h1 and h2 Commented Jun 13, 2015 at 16:47

expr1 = Sum[50/10000*1/4*1/4*n*97/100*Exp[-h1*1/4*n], {n, 1, 4}] -
Sum[1/2*1/4*n*97/100*(Exp[-h1*1/4*(n - 1)] - Exp[-h1*1/4*n]), {n, 1, 4}];

expr2 = Sum[77/10000*1/4*1/4*n*97/100*Exp[-h1*1/4*n], {n, 1, 4}] +
Sum[77/10000*1/4*1/4*n*94/100*Exp[-h2*1/4*n], {n, 5, 8}] -
Sum[1/2*1/4*n*97/100*(Exp[-h1*1/4*(n - 1)] - Exp[-h1*1/4*n]), {n, 1, 4}] +
Sum[1/2*1/4*n*94/100*(Exp[-h1*1/4*(n - 1)] - Exp[-h2*1/4*n]), {n, 5}] +
Sum[1/2*1/4*n*94/100*(Exp[-h2*1/4*(n - 1)] - Exp[-h2*1/4*n]), {n, 6, 8}];

Plot3D[{expr1, expr2}, {h1, -5, 5}, {h2, -5, 5}, ClippingStyle -> None,
PlotLegends -> "Expressions"]


You must minimize some function of the two expressions

NMinimize[#, {h1, h2}] & /@
{Abs[expr1*expr2], Abs[expr1 + expr2],
expr1^2 + expr2^2}


You may also wish to constrain the region of interest.

NMinimize[{expr1^2 + expr2^2, h1 >= 0, h2 >= 0}, {h1, h2}]


EDIT: Your comment below indicates that you desire both expressions to be zero

NSolve[{expr1 == 0, expr2 == 0}, {h1, h2}, Reals]


{{h1 -> 0.00998752, h2 -> -0.000767111}}

• thanks. not sure why h2 is so far away from what I get from excel goal seek. I expect h2 to be bigger than h1 in this particular case as second expression starts with 77 but 1st expression starts with 50. h is supposed to fit exponential function p= exp(-h*t), p = 1 when t =0, h is increasing function but peicewise constant. In this case constant between n = 1 to 4 which is h1. h increased and again constant between n =5 to 8 which is h2. Can you pl suggest how do I modify your code to take into these property of h? Commented Jun 13, 2015 at 18:01
• I need to minimize a system of five equation using h1 to h5. Can I just use the same logic as given in your code and extend it to 5 equation please Commented Jun 13, 2015 at 18:02
• Can you pl explain why you decided to minimize {Abs[expr1*expr2], Abs[expr1 + expr2], expr1^2 + expr2^2}? What are alternatives available please? Is that something you decide after seeing how the expression behaves after plotting in a graphical format pleas? Commented Jun 13, 2015 at 18:11
• However many expressions you have, you need to define a function of those expressions that you want to minimize. In your Excel solution, what function of the expressions were you minimizing? Commented Jun 13, 2015 at 18:11
• Your second expression goes to minus infinity as h2 approaches minus infinity. Consequently, to get a minimum other than minus infinity the range must be constrained or the function to be minimized must use the Abs, square, or similar function of expr2. Commented Jun 13, 2015 at 18:19