An investor owns a bond that is redeemable for 250 in 6 years from now. The investor has just received a coupon of $c$ and each subsequent semiannual coupon will be 2% larger than the preceding coupon. The present value of this bond immediately after the payment of the coupon is 582.53 assuming an annual effective yield rate of 4%. Calculate $c$.
My Solution: First, I converted the effective rate of interest $i$ to nominal rate compounded semianually, which is $$1.04 = (1+j)^2 \Rightarrow j=1.09804$$I really think that the first coupon payment was $c$, followed by $c(1.02)$, $c(1.02)^2,...,c(1.02)^{11}$, hence having the equation $$582.53 = cv + c(1.02)v^2 +...+c(1.02)^{11}v^{12} + 250v^{12}$$ $$\Rightarrow 582.53 = cv(1+1.02v +(1.02v)^2+...+(1.02v)^{11}) + 250v^{12}$$ $$\Rightarrow 582.53 = cv\left[\dfrac{1-(1.02v)^{12}}{1-1.02v}\right] + 250v^{12}$$ $$\Rightarrow c = 32.68$$
SOA solution: It appears that the FIRST coupon payment was $c(1.02)$, hence the equation $$582.53 = c(1.02)v + c(1.02v)^2 +...+c(1.02v)^{12} + 250v^{12}$$ $$\Rightarrow c=32.04$$
By the way, both answers are in the multiple choice. My question is, why is it that the first payment is $c(1.02)$? As far as I know, "the investor received a coupon of c, and each subsequent semiannual coupon will be $2\%$ larger", hence I really thought my equation was correct. Thank you for your help.