# When does funding cost of a portfolio enter into the portfolio's present value?

This question comes from some confusion when reading Hull's book and from the general concept of no-arbitrage/self-financing portfolios in stochastic finance books. I am not fully seeing the distinction made between the present value of a portfolio and the cash flows made in setting up the portfolio, and when to use which.

In some cases when doing a no-arbitrage proof, I have seen that the funding cost of a portfolio doesn't enter into the portfolio present value. For example from https://maths.ucd.ie/~vlasenko/MST30030/fm5_0.pdf is a typical proof for European put-call parity:

So here, in Portfolio A, for example, the trader buys a European call and a zero coupon bond worth K at maturity, but the Portfolio's value considers them to be 'positive'. This seems to me 'as if the call and zero-coupon bond were free'. So here it seems like the cash flows themselves are not taken into consideration.

From Hull's book, Chapter 5.14:

We suppose that the speculator puts the present value of the futures price into a risk-free investment while simultaneously taking a long futures position. The proceeds of the risk-free investment are used to buy the asset on the delivery date. The asset is then immediately sold for its market price.

The present value of this portfolio/investment is written as (according to Hull): $$-F_0e^{-rT} + E(S_T)e^{-kT}$$

So here the trader is putting money into a zero coupon bond that will give $$F_0$$ at maturity and also entering into a futures contract. But the present value of the bond investment is negative, rather than positive, unlike the earlier example. And now, even though the futures contract has 0 value at initiation the present value is written as $$E(S_T)e^{-kT}$$.

I could see this portfolio value arising from calculating the present value of a portfolio which in the future will consist of 'exchanging an amount $$F_0$$ for $$S_T$$', but then I am still confused why the portfolio takes into account the funding cost ($$F_0$$) of buying the stock, whereas the put-call parity example ignores funding costs.

Another issue that adds to the confusion is when showing that an arbitrage exists, the present value of the portfolio includes the proceeds of the transaction. For example, when a european call is mispriced and the present call value is above the present stock price, an arbitrage can be done by shorting the call and longing the stock. So the present value (at time $$t$$) of the portfolio would be $$-c_t + S_t + proceeds$$, where $$proceeds = c_t - S_t$$. So the present value of the portfolio is 0. So here the portfolio consists of both the assets and the cash flow. I am unsure why the funding costs are relevant here but not in other examples.

All the examples I gave above make sense to me in their own way, but I am unsure in which situations I would include the funding costs or not, and what the rationale is for (not) doing so.

Any clarifications/help would be greatly appreciated! Thanks!

• The two ways are equivalent if you think about it. In the "portfolio centric" view you write down the initial value of the portfolio $c+K e^{-rT}$. In the "person centric" view you write down what you do to set up the arbitrage: you initially borrow $F_0 e^{-rT}$ from someone or from yourself (which is why the sign is negative) and you add the expected present value of what you receive later, setting the whole thing to zero. (Formally, if you want you can move the $-F_0 e^{-rT}$ to the other side of the equal sign and make it positive just like in the earlier example). – Alex C Jan 25 at 14:45
• I am still unsure what the 'actual' present value of the portfolio would be though, and in which cases to use which approach. Hopefully someone can answer and clarify. Thanks! – Slade Jan 25 at 14:48
• When you think present value think "present value of flows with delivery dates >= today". So whatever you just paid for a portfolio has no impact on its present value. If you wish to keep track of what you just paid then consider a different portfolio consisting of the original one plus the funding of the amount you just paid. – Antoine Conze Jan 25 at 15:29
• Okay, I think your answer helps clarify a lot. I didn't realize we could think of it as two separate portfolios. Does this mean that the present value for a portfolio that includes the asset and funding is always 0? For example, a portfolio of a stock plus the funding is $S_t - S_t$. I think this would make sense with the idea of 'investments being priced to have 0 net present value' – Slade Jan 25 at 15:42
• A portfolio of a stock is worth $S_t$. A portfolio of "a stock and the cash spent at $t=0$ to buy the stock" is worth $S_t - S_0 e^{rt}$, assuming the rate of interest on cash is $r$. In this example the stock position is "funded at the rate $r$". – Antoine Conze Jan 25 at 15:57