Finding cost of equity capital in questions have TWO methods :
1.Dividend Growth model
2.Capital Asset Pricing model (CAPM)
The difference between the above two methods is only to know that "which method incorporates all the risks". The method which does not incorporates risk explicitly is the first model "dividend growth model" and "CAPM" incorporates the risk.
The dividend growth model has further 2 methods of incorporating the growth i.e. (a) Growth rate estimation (b) Gordon's growth estimation (it uses the formula g=br)
There are two types of risk that is unsystematic risk and the other is systematic risk. CAPM uses the systematic risk to calculate the COST OF CAPITAL. and this method seems to be the good method as compared from the first method.
NOW for the Cost of debt capital, it is of two kinds; 1- irredeemable debt 2-redeemable debt. The cost of these two debts are not the same as the name its self describes.
The answer to the question Why they produce a different estimates is described above i.e first method does not include the total risk and on the other hand, CAPM uses systematic risk (explicitly all the risk) by including a beta factor. So you should be able to know the why different estimates arises.
According to my own knowledge, COST OF CAPITAL is the cost of providing finance to business activities and Discount rate is the interest rate used to discount the future cash flows. So for example if some business needs to finance its activities in the future so they may know that what will be the cost of capital but in order to get know its present value then Discount rate is used to arrive its present value. I Think it has a relation with each other.
Cost of Capital is important to estimate or calculate as all of us know that business does not uses its wholly retained earnings to invest or re-invest in the business operations, So business may borrow funds from Banks or any other financial institutions to run the operations. But the Funds that the business is trying to borrow are free from cost. The answer is NO. So business MUST estimate how much they have to pay back the funds along with the interest to Banks and the financial institutions if they had borrowed.The other reason which can be clear your concept that is :
If you remember that in statement of Comprehensive Income **PROFIT** comes after deducting the interest. This Interest is that interest which the business has paid (the cost of capital)
before arriving at the final profit..
HOPE that you have understand now.