Convert between spot & forward rate

Can anyone please help me understand & make a rational or think of consistent formula that I can use to convert between spot & forward rates. I thought I mastered the calculations, then bumped into this question and couldn’t solve it!

An investor wants to take advantage of the 5-year spot rate, currently at a level of 4.0%. Unfortunately, the investor just invested all of his funds in a 2-year bond with a yield of 3.2%. The investor contacts his broker, who tells him that in two years he can purchase a 3-year bond and end up with the same return currently offered on the 5-year bond. What 3-year forward rate beginning two years from now will allow the investor to earn a return equivalent to the 5-year spot rate? Answer is 4.5%

For this type of problem, you want to equate the ending amount of a 5 year buy and hold with 2 year investment and reinvest for a further 3 years (assuming bond equivalent yields):

(1+ 0.04/2)(25) = [(1+ 0.032/2)(22)] * [(1 + i/2)(2*3)]

1.0210 = [1.0164] * [(1+i/2)6]

1.218994 = 1.065552 *[(1+i/2)6]

(1.218994/1.065552)(1/6) = 1 + i/2

1.022675 = 1 + i/2

i= 0.045351 = 4.5351%

Baidar - I look at the calculation like the below. It helped me early on w synthetic FRA and Spot/Forward problems to draw things out…IE draw a line for the full spot (5 Year) and then a line below broken into the known (2 year bond) and the unkonwn (3 year @ x rate).

(1.04)5 = (1.032)2 * (1.X)3

(1.04)5 / (1.032)2 = (1.x)3

1.1424 = (1.X)3

1.14241/3 = 1.x

1.X = 1.045

1.045 - 1 = 4.5%