Bonds ... whats is it and what is a discount? I am studying about bonds and I am a bit puzzled.  So far my understanding is that bonds are loans issued by the government or a company and it works very similarly to loans.
However, I have a question regarding a premium bond and a discount bond.  I am not quite understanding who is making money and who is losing. Can someone explain to me using the following example?

Smith purchases a bond of face value $100,000$ at a rate of $10\%$ for $10$ years. The yield rate is $12\%$.  

 A: Bonds have an inverse relationship with yield. So if Smith purchases a $10\%$ bond and now the interest rate is $12\%$, the face value of the bound has gone down. Therefore, the bond is trading at a discount at present.
If at some time later, the interest rate was $8\%$, the bond would be worth more and trading at a premium. Assuming there is still some time value left on the bond. 
As $t\to 10$, the bond converges to its par value regardless if it was trading at a discount or premium and the entity that issued it isn't going bankrupt. 

Suppose Smith buys this bond when it was issued new. At the time of the bond inception, the interest rate was determined to be $10\%$. At the time the bond came to the market (let's say today), the interest is $12\%$.
The present value of an ordinary annuity is
$$
PVOA = C\Bigl(\frac{1 - (1 + i)^{-n}}{i}\Bigr)
$$
where $C$ is cash flow, $i$ is interest rate, and $n$ is the number of payments. We will assume semi-annual payments so the cash flow is
$$
C = 1000*0.1*100 = 10000
$$
or $5000$ semi-annually.
$$
PVOA = 5000\Bigl(\frac{1 - (1 + .06)^{-20}}{.06}\Bigr) = 57349.6
$$
The present value of the bond is
$$
PV_{\text{bond}}= \frac{F_v}{(1+i)^{-n}}
$$
where $F_v$ is the face value at maturity.
$$
PV_{\text{bond}}= \frac{100000}{(1+.06)^{-20}} = 31180.5
$$
The bonds total value is $PVOA + PV_{\text{bond}}$.
The bond is trading at the discounted rate of $\$ 88530.1$
Since we assumed semi-annually payments, the interest rate was divided by $2$ and $n$ was multipled $2$.


Is buying a premium bond ever worth it? Yes.

Consider two $10$ year bonds where one is trading at a premium and the other a discount.
Let the coupon on the premium bond be $10\%$ and the coupon on the discount bond be $4\%$.
Premium Bond:
Initial cost $\$120,000$, annual cash flow $\$10,000$, and 10 year cash flow $\$100,000$.
Premium Net Cash Flow: $100000 - (120000 - 100000) = \$80000$
Discount Bond:
Initial cost $\$90,000$, annual cash flow $\$4,000$, and 10 year cash flow $\$40,000$.
Discount Net Cash Flow: $40000 - (90000 - 100000) = \$50000$
Therefore, we can't say that buying a premium bond is a bad thing. It depends on the other bonds available to the investor at the time. A premium could be a better bet. In the toy problem above, we could either raise the interest rate of the discounted bond by another $3\%$ or increase the discounted price by $\$30,000$ before they two bonds break even or we could do a combination of change the interest rate and the discount. That is, a discounted bond can be structured to look more attractive then a premium which may conjure a negative connotation since we are paying more than par but unless the situations are analyzed we don't know what the better deal is. 
