Why would I not buy a bond for less than face value?

As well as credit risk there's also interest risk. If a bond has a face value of $100, pays 1% and matures in 20 years' time then you expect to receive a total of $120 from buying it now -- $1 per year for 20 years and $100 at the end.

But if you can get a 3% return elsewhere, then if you invest your $80 there instead you will get $2.40 per year for 20 years and then $80 at the end, making a total of $128 (and you also get more of the money sooner). So even $80 for the $100 bond is a bad buy, and you should invest elsewhere.

A (very) simplified bond-pricing equation goes thus:

Fair_Price:
{Face_Value * (1 + Interest - Expected_Market_Return) ^ (Years_To_Maturity)}
* P(Company_Will_Default_Before_Maturity)

To reiterate, that is a very simplified model. But it allows us to demonstrate the 3 key factors that drive "Fair" Value:

1. The interest relative to the current market rate. If your AAA bond yields 1%, but an equally-good AAA bond currently sells at 3% in the market, then the "Equivalent" value is the face value minus 2% (1% - 3%) for every year to maturity.

2. Years to maturity. Because 1) is multiplied for every year to maturity, longer-dated bonds are more sensitive to changes in market rates. If your bond yields 2% less than market but matures in a year, then it's worth $98, but if it matures in 56 years, then it's only worth 0.98^56 = $32. Conversely, if your bond yields more than the market rate, then its' price will be greater than face value.

3. The company might default on the debt. If a Bond has a "Fair" Value of $100, but you think there's a 50% chance that the company will default, then it's only worth $50. In fact, it can be worth even less because getting paid on a defaulted bond can often take time and/or money and/or lawyers.

In your case, because your bond matures in 56 years but yields ~5% (well above the current market rate), for it to be below Face value implies a strong probability of default, or a strong belief that market returns will be above 5% over the next 56 years.

The time value of money is very important in understanding this issue. Money today is worth more than money next year, two years from now, etc. It's a well understood economics concept, and well worth reading about if you have some, well, time.

Not only is money literally worth more now than later due to inflation, but there is the simple fact that, assuming you have money for the purpose of doing something, being able to do that thing today is better than doing that same thing tomorrow. "A bird in the hand is worth two in the bush" gets to this rather directly; having it now is better than probably having it later. Would you rather have a nice meal tonight, or eat beans and rice tonight and then have the same nice meal next year?

That's why interest exists, in part: you're offered some money now, for more money later; or in the case of buying a bond, you're offered more money later for some money now. The fact that people have different discount rates for money later is why the loan market can exist: people with more money than they can use now have a lower discount for future money than people who really need money right now (to buy a house, to pay their rent, whatever).

So when choosing to buy a bond, you look at the money you're going to get, both over the short term (the coupon rate) and the long term (the face value), and you consider whether $80 now is worth $100 in 20 years, plus $2 per year. For some people it is - for some people it isn't, and that's why the price is as it is ($80). Odds are if you have a few thousand USD, you're probably not going to be interested in this - or if you have a very long term outlook; there are better ways to make money over that long term. But, if you're a bank needing a secure investment that won't lose value, or a trust that needs high stability, you might be willing to take that deal.