A simple question on bonds ?

Consider the example below (Please avoid too much jargon) :

Face value of a bond = \$1000

(1) Issuing a bond at par –

coupon rate = market rate = 10% per year or 5% semi-annually.
present value or principal = 822.7
Interest = 177.3 (because an annuity of \$50 is paid.)
TOTAL = 1000.

>>>what do i have to pay to purchase this bond ? 1000 or 822.7 ?
My textbook says that the company gets full use of the \$1000, per bond.
>>>What does this mean ?

NOW THE EVEN MORE CONFUSING PART :

(2)Issuing bond at a discount –

market rate = 12 % , coupon = 10 %
present value or principal = 792.1
Interest = 173.25
TOTAL = 965.35

The textbook says that the company RECEIVES 963.35 per bond.

>>>Why not ask people to pay 1000 instead of 965.35 and offer them interest at market rate instead of these calculations ?

NOTE : This is taken form the book “Introduction to financial accounting” by horngren, sundem and eliott.
@JKRB – you said “and the discount of 36.35 makes up for the interest the buyers are losing for purchasing the 10% bonds when the market rate is 12%”.

Can you show mathematically, that the investors are losing 36.65 ? Then it would be easier to see that a lower selling price would make the interest higher.
My book presents the concept in a rather confusing way.
I was looking for some kind of mathematical proof which will show :

money lost by investor due to low coupon = discount offered.
Hence, no money lost by investor.

1. 1. Q # 1 – You will buy at \$822.70 so that there is no liability of the company to pay half Yearly.
In their books at the time of maturity, they have the liability of \$1000 after 5 years or the case may be.

2. Q #2 same as explained above.
Company is considering Instt and pay fixed amount as known to you.
If the bank issue Bonds and pay fixed interest, these are called DEBENTURES. again they should be secured and I am sure bond can be secured and unsecured too.

2. 1) When the market rate of interest and the coupon rate of interest are equal, the bonds are issued at their par value, which is their face value. In this instance, the face value of the bond is \$1,000. So that is the amount you would pay to purchase it. By looking at the entry, you can see how they get the full use of the \$1,000 bond.
Dr Cash 1,000
Cr Bonds Payable 1,000

They received the full \$1,000 for it.

2) When the market rate of interest is more than the coupon rate, the bonds are issued at a discount (less than the face value). The reason for this is that the buyers could invest their money elsewhere at 12%, so why should they buy bonds that are paying only 10%. So the issuers of the bonds have to make the price more attractive in order to sell them. As far as your question on why the issuers don’t just offer the market rate of interest, you have to understand that it sometimes takes weeks or even months of preparations to issue bonds, there are many steps in the process. So when the bonds are being prepared to issue, the market rate of interest most likely will change and you can’t just change the coupon rate of the bonds on a moments notice. So instead, the bonds are usually sold at a discount or premium. In your example, the entry would be:
Dr Cash 963.35
Dr Discount on Bonds Payable 36.65
Cr Bonds Payable 1,000

So the company receives 963.35 and the discount of 36.35 makes up for the interest the buyers are losing for purchasing the 10% bonds when the market rate is 12%