 # Kaplans' Derivatives Question - seems wrong

Hi All, Could some of you look at these question from quiz 57.1. To me the answers seem wrong.

1. Derivatives pricing models use the risk-free rate to discount future cash flows because these models: A. are based on portfolios with certain payoffs. B. assume that derivatives investors are risk-neutral. C. assume that risk can be eliminated by diversification.

2. The price of a forward or futures contract: A. is typically zero at initiation. B. is equal to the spot price at expiration. C. remains the same over the term of the contract

3. For a forward contract on an asset that has no costs or benefits from holding it to have zero value at initiation, the arbitrage-free forward price must equal: A. the expected future spot price. B. the future value of the current spot price. C. the present value of the expected future spot price.

1.A 2.C 3.B

They look good to me.

Why do you think that they’re incorrect?

Something wrong the way I understood them, never mind.

Not “never mind”.

You do need to understand them.

Let me know your thinking and I’ll help fix it.

Hello nikks99:

“1. Derivatives pricing models use the risk-free rate to discount future cash flows because these models: A. are based on portfolios with certain payoffs” (CORRECT)

“B. assume that derivatives investors are risk-neutral.”

This answer is incorrect because these models do not assume investors are risk neutral. Most models assume that investors are risk averse.

“C. assume that risk can be eliminated by diversification.”

This answer is incorrect because although nonsystematic risk can (theoretically) be eliminated by diversification, systematic risk cannot be eliminated by diversification. So all portfolios will have some amount of systematic risk. It follows that a fair return on a fully diversified portfolio should be above the risk-free rate of return, by virtue of the fact that the portfolio is not entirely risk-free.

“2. The price of a forward or futures contract: A. is typically zero at initiation.”

One should exercise care when using the terminology of futures and forwards. In this context, the “price” refers to the futures price, i.e., the price paid for the underlying asset when the contract matures. If you agree to pay \$100 for a share of stock one year from now, then the share of stock is the underlying asset and the futures price is \$100. The error someone might make is to conflate the idea of futures price with value of the futures contract. They are not the same thing.

Answer A is wrong since the value of a futures contract, meaning the present value of the futures price, minus the current spot price, must be zero at initiation. However, the futures price would not be zero at initiation of the contract.

“B. is equal to the spot price at expiration.”

Answer B is incorrect since once the futures price is agreed to at initiation of the contract, it does not change. However, the value of the futures contract will certainly change throughout the life of the futures contract, since the value of the underlying will change. The value of the futures contract should be zero at contract initiation.

“C. remains the same over the term of the contract” (CORRECT)

“3. For a forward contract on an asset that has no costs or benefits from holding it to have zero value at initiation, the arbitrage-free forward price must equal: A. the expected future spot price. B. the future value of the current spot price. C. the present value of the expected future spot price.”

The spot price can be thought of as a random variable. If you were able to predict the mean of the spot price in the future, without reference to the risk free rate, the expected value of the spot price in the future would simply be the same as the spot price now. Thus choice A is incorrect. The present value of that expected future spot price would be something less than the current spot price because the term “present value” implies you are discounting at the risk-free rate back to the present. Thus choice C would be incorrect.

If the no-arbitrage condition holds, (which in theory it does, in practice it might) the relationship between the current spot price and the forward price must take into account the risk free rate: _forward price = (spot price)(_1 + r)^T.

I hope this is helpful to you.

1 Like

hi @S2000magician @Analyst0718
I also have some confusions for the first question “1. Derivatives pricing models use the risk-free rate to discount future cash flows because these models: A. are based on portfolios with certain payoffs” (CORRECT)”

the explanation on notes saying “Derivatives pricing models use the risk-free rate to discount future cash flows
(risk-neutral pricing) because they are based on constructing arbitrage relationships that
are theoretically riskless. (LOS 49.a)”

why it’s “arbitrage relationships”? i thought the whole forward price= future value of spot price is because of no-arbitrage.
Thanks!!

You construct an arbitrage relationship (i.e., a model for an arbitrage profit), then price the derivative so that the arbitrage profit is zero.

so if i take an example of selling an apple. spot price is 5 euro, one year risk free rate is 6%. and i can get a forward contract where i sell it after one year at 5.30. Is it an “arbitrage relationship”? (because then i expect to profit from fwd contract if the spot price drops? but when both parties enter the fwd contract, there should be no arbitrage right? otherwise the other party wouldnt agree)

The arbitrage relationship is that you take the EUR 5 you would spend on the apple today and invest it risk-free for one year, whereupon you buy the apple. The forward price of the apple should eliminate any arbitrage profit, so it has to be the amount you receive on your risk-free investment: EUR 5.30. If it were less, you would earn an arbitrage profit (you’d have an apple and money left over); if it were more, the apple monger would earn an arbitrage profit.

Note a couple of important details: EUR 5 for an apple is awfully expensive, and nobody in their right mind will buy a 1-year-old apple.

1 Like

then this derivative is non-arbitrage right? because the forward price=spot*(1+risk free rate) (1year).
then why the answer to the questions says " constructing arbitrage relationships that
are theoretically riskless."? and "are based on portfolios with certain payoffs? what’s the payoff if there’s no arbitrage profit?

The payoff for the short is the guaranteed forward price; the payoff for the long is one (incredibly stale) apple. ok got it. thanks for helping out!