Lower bound on forward settlement price

In the last video we established a reasonable upper bound on the 1-year forward settlement price of gold. We established that at $1150 which is essentially the sport price plus the borrowing rate to borrow $1000, plus the carrying cost of the gold. What I want to do in this video is to think of a reasonable lower bound. To do that let's imagine a world where instead of the forward settlement price being $1150, let's imagine a world where the forward settlement price is $1050. It's still higher that the spot price. Let's think of it from the the point of view of someone who wants to hold gold in a year. Let's say that you are that someone and right now you have $1000 that you want to use to give you gold. You want to hold the gold for the long term. You have 2 options here. You could just buy gold now. Buy 1 ounce with your $1000 right now. When you go forward a year from then your going to have that 1 once of gold plus you're going have to pay the carrying cost. The safety deposit box and the insurance on the gold. You're going to have 1 ounce of gold minus $50. That's what you're going have to pay. The other option, you say hey look instead of putting my $1000 into gold right now, let me put my $1000 into a risk-free bond. Into a risk-free bond. Risk-free bond, and at the same time agree to be the buyer on the 1-year forward settlement. On the 1-year forward contract. I'm agreed to to be the buyer at $1050. If you go forward a year, your $1000 risk-free bond is going to give you 5% interest a year. You're going to have $1050. This is just the interest on your bond. Then you can use that to go and buy the gold. Which you all ready locked in the price by agreeing to be the buyer on the forward contract. Then you buy the gold. In both situations you end up with an ounce of gold a year from now starting with your $1000. In this situation you just have an ounce of gold and you didn't have to pay any carrying cost. In this situation you have the ounce of gold and you did have to pay carrying cost. So, clearly any rational person, assuming their buying the gold for the long term, would want to do this situation. They save $50 over the course of the year. Really, the rational price, if this were to happen everyone would want to be the buyers over here. We're talking about $1050 price. That's right over here. This was the price in the last example. Everyone would want to be the buyer on the futures contract or the forward contract. That would increase the price over there. Knowing you'd have less buyers on the spot contract. That would decrease the price over there. In general this price seems too low. The price at which people would be neutral is if this price was $50 higher. Then it becomes completely equivalent. A rational lower bound given all of this, would have been a price of $1100. $1100. In general just going back to the last video, a ration upper bound will be the spot price plus the carrying cost and the rate at which someone could borrow money. A rational lower bound will be the spot price plus the risk-free interest rate and the carrying cost. In general, the only difference between those upper and lower bound is this rate and this rate. This rate and this rate. If these were the same the rational price would be the same. If these both were 10%, then the rational price would $1150. These were both 5% the rational price, the upper and the lower bound would have been $1100.