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 and we established that at 1150 which is essentially the spot price plus the borrowing rate to borrow $1,000 plus the carrying cost of the gold what I want to do in this video is to think of a reasonable lower bound and 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 well let's just say it is 1050 dollars so it's still higher than the spot price but let's think about it from the point of view who of someone who wants to hold gold in a year and let's say that you are that someone and right now you have right now you have $1,000 that you want to use to give you gold so that you have and you want to hold the gold for the long term so you have two options here you could just buy gold now buy one ounce with your $1000 right now and then when you go when you go forward a year from then you're going to have that one ounce of gold one ounce plus you're going to have to pay the carrying cost the the safety deposit box or and the insurance on the goal so you're going to have one ounce of gold - $50 that's what you're going to have to pay the other option you say hey look instead of putting my thousand dollars into gold right now let me put my thousand dollars into a risk-free bond into a risk-free bond so 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 so I'm agreed to be the buyer at what at ten fifty so if you go forward a year your thousand dollars risk-free bond is going to give you five percent Interest a year so you're going to have you're going to have one thousand and fifty dollars that's just the interest on your bond and then you can use that you can use that to go and buy the gold which you already locked in the price by agreeing to be the buyer on the forward contract so then you buy by the gold and in this in both situations you end up with an ounce of gold a year from now starting with your thousand dollars but in this situation you just have an ounce of gold and you didn't have to pay any carrying costs in this situation you have the ounce of gold and you did have to pay carrying costs so clearly any rational person assuming they're buying the gold for the long term would want to do this situation they save $50 over the course of the year so really the rational price if we didn't so if this were to happen everyone would want to be the buyers over here so now we're talking about 1050 price that's right over here this was the price in the last example so everyone want to be the buyer on the futures contract or the forward contract so that would increase the price over there and no one and you'd have less buyers on the spot contract so that would decrease the price over there so in general this price seems too low and the price at which people would be neutral is if this price was $50 higher because then because then it becomes completely equivalent so a rational lower bound given all of this would have been a price of $1,100 $1,100 and in general just going back to the last video a rational upper bound will be the spot price plus the carrying cost and the rate at which someone could borrow money and a rational lower bound a rational lower bound will be the spot price will be the spot price plus the risk-free interest rate and the carrying cost so in general the only difference between those upper and lower bound is this rate and this rate this rate and this rate and so if these were the same the rational price would be the same so if these were both 10% then the rational price would be 1150 if these were both 5% the rational price the upper and the lower bound would have been $1,100