Good day, I have been tasked with this question and I am unsure if my understanding and work so far is correct.
On January 1st 2018 AAA shares were trading at $30USD$ per share.
The shares pay a quarterly dividend of $0.60c$ per share if the shareholder has held the share for at least 10 days before the dividend payment.
Calculate the fair strike of the one year forward on AAA given that the $USD$ interest rate is 1%.
What I have done so far is find the value of the forward in 1 year.
$Fwd=30*(1.01)-0.6*(1.0025)^3-0.6*(1.0025)^2-0.6*(1.0025)^1-0.6=27.89098098 USD$
I am not sure if I understood the concept of a fair strike of a forward as asked in the question
Let be $f(S, t,T)$ the value of forward, $F(S, t, T)$ the forward price, $T$ the time to expiration, $S_t$ the spot price of the underlying asset at $t$, $B(t,T)$ the value of an unit par discount bond with time to maturity $T$, $D$ the present value of all dividends received from holding the asset during the life of the forward contract.
If the dividends $d_k$ are discrete, $D=\sum_{k=1}^n d_k v^k$, with $v=\frac{1}{1+i}$ the discount factor at the interest $i$.
If the interest rate $r$ is constant and interests are compounded continuously, then $B(t,T)=\mathrm{e}^{-r(T-t)}$.
Then $$ f=S_t-[D+KB(t,T)] $$
By definition the forward price $F(S,t,T)$ is the "fair strike" $K$ set at $t$ so that the value at $T$ of the forward contract is zero. that is the fair strike is $$ f=0\qquad\Longrightarrow\qquad K=F(S,t,T)=\frac{S-D}{B(t,T)} $$
So in your case, $i=\frac{1\%}{3}=0.33\%$, $v=\frac{1}{1.0033}=0.996678$, $d_k=0.6$, and $D\approx 1.78$; $S_t=30$, $r=1\%$, $B=\mathrm{e}^{-0.01\times 1}\approx 0.99$. So we have $$ K=F(S,t,T)=\frac{S-D}{B(t,T)}=\frac{30-1.78}{0.99}\approx 28.50 $$