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# Lower bound on forward settlement price

Video transcript

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.