Simple yield to maturity (SYTM) is the approximate annual interest rate at which a bond yields the same return, provided the investor holds the bond until maturity and receives all of the coupon payments. You cannot compute the interest rate by hand using the exact equation for yield to maturity (YTM), as that equation is too complex. Online bond calculators have algorithms for solving the exact equation. However, you can calculate a close approximation manually using a simplified equation.
Call the face value of the bond F, the purchase price B, the annual coupon payment C and the number of years N. For example, suppose you buy a £650 bond for £585, and the bond pays £45 every year for five years, then F = 1,000, B = 900, C = 70 and N = 5.
Compute C + (F-B)/N. This is the average annual return of the bond. Using the example above, the average annual return is £58 since 70 + (1,000 - 900)/5 = 90.
Compute (F+B)/2. This is the average price of the bond. Using the example above, the average price is £617 since (1,000 + 900)/2 = 950.
Divide the average annual return by the average price to obtain the simple yield to maturity. Since 90/950 = 0.0947, the approximate yield to maturity is 9.47 per cent.
The exact equation for yield to maturity is C(1+R)^(-1) + ... + C(1+R)^(-N) + F(1+R)^(-N) = B, where R is the YTM. Even if you know the values for C, F, B and N, this equation cannot be solved for R. However, the simple yield to maturity formula gives a good approximation for R.