Multivariate reward processes with reward functions of constant rates, defined on a semi-Markov process, first were studied by Masuda and Sumita, 1991. Reward processes with nonlinear reward functions were introduced in Soltani, 1996. In this work we study a multivariate process , , where are reward processes with nonlinear reward functions respectively. The Laplace transform of the covariance matrix, ?(t), is specified for given , and if they are real analytic functions, then the covariance matrix is fully specified. This result in particular provides an explicit formula for the variances of univariate reward processes. We also view ?(t) as a solution of a renewal equation.