Among many submissions to the NIST post-quantum cryptography (PQC) project, NewHope is a promising key encapsulation mechanism (KEM) based on the Ring-Learning with errors (Ring-LWE) problem. Since NewHope is an indistinguishability (IND)-chosen ciphertext attack secure KEM by applying the Fujisaki-Okamoto transform to an IND-chosen plaintext attack secure public key encryption, accurate calculation of decryption failure rate (DFR) is required to guarantee resilience against attacks that exploit decryption failures. However, the current upper bound of DFR on NewHope is rather loose because the compression noise, the effect of encoding/decoding of NewHope, and the approximation effect of centered binomial distribution are not fully considered. Furthermore, since NewHope is a Ring-LWE based cryptosystem, there is a problem of error dependency among error coefficients, which makes accurate DFR calculation difficult. In this paper, we derive much tighter upper bound on DFR than the current upper bound using constraint relaxation and union bound. Especially, the above-mentioned factors are all considered in derivation of new upper bound and the centered binomial distribution is not approximated to subgaussian distribution. In addition, since the error dependency is considered, the new upper bound is much closer to the real DFR than the previous upper bound. Furthermore, the new upper bound is parameterized by using Chernoff-Cramer bound in order to facilitate calculation of new upper bound for the parameters of NewHope. Since the new upper bound is much lower than the DFR requirement of PQC, this DFR margin is used to improve the security and bandwidth efficiency of NewHope. As a result, the security level of NewHope is improved by 7.2 % or bandwidth efficiency is improved by 5.9 %.