## Costs of Carry

The preceding analysis assumed that there are no storage costs associated with holding the underlying asset This is not always the case Holding a physical asset such as gold entails storage costs, such as vault rental and insurance fees Holding a security may, alternatively, entail negative costs, representing dividend or coupon payments These costs (or incomes) affect the theoretical forward price.

We shall use a discrete-time (multiperiod) model to describe this situation The delivery date T is M periods (say, months) in the future, We assume that storage is paid periodically, and we measure time according to these periods, The carrying cost is c(k) per unit for holding the asset in the period from k to k 4- I (payable at the beginning of the period) The forward price of the asset is then determined by the structure of the forward interest rates applied to the holding costs and the asset itself

For ward price formula with carrying costs Suppose an asset has a holding cost of c(k) per unit in period k, and the asset can he sold short Suppose the initial spot price is S Then the theoretical forward price is

For ward price formula with carrying costs Suppose an asset has a holding cost of c(k) per unit in period k, and the asset can he sold short Suppose the initial spot price is S Then the theoretical forward price is

where d(k, M) is the discount factor from k to M Equivalently,

where d(k, M) is the discount factor from k to M Equivalently,

Proof: The simple version of the proof is this: Buy one unit of the commodity on the spot market and enter a forward contract to deliver one unit at time T. The cash flow stream associated with this is (—S—c(0), — t(l), ~c(2),, . ,

~c(M — 1), F). The present value of this stream must be zero, and this gives the stated formula for F We shall also give a detailed proof based on the no-arbitrage condition .

Suppose that F is greater than that given by (10.2). We can set up an arbitrage as follows. At the initial time, short one unit of a forward contract with forward price F and buy one unit of the asset for price S Simultaneously, borrow an amount of cash S and arrange to borrow amounts c(k) each time k — 0, 1,.. , , M — 1. All of these loans are to be repaid at the final time M, so each is governed by the corresponding forward interest rate between k and M. The initial cash flow associated with this plan is zero, since we immediately borrow enough to pay for the asset. Furthermore, the cash flow during each period is also zero, because we borrow enough to cover the carrying charge. Hence there is no net cash flow until the final period,

At the final period we deliver the asset as required, receive F, and repay ail loans, which now total Sjd(0, lo* M) Under our inequality assumption this will represent an arbitrage profit, so our original assumption of inequality must be false The details are shown in Table 10 .3 Assuming short selling is possible, we may reverse this argument to prove that the opposite inequality is likewise not possible. (See Exercise 5.) I

The alternative formula (103) is obtained from (10.2) by multiplying through by d(0, M) and using the fact that d(0, M) = d(0, k)d(k, M) for any k This alternative formula is probably the simplest to understand, since it is a standard present value equation. We recognize that we can buy the commodity at price S and deliver it according to a forward contract at time M in a completely deterministic fashion. The cash flow incurred while holding the commodity will be the carrying charges and the delivery price. The present value of this stream must equal the price S

Time 0 action |
Time 0 cost |
Time k cost |
Receipt at time M |

Short 1 forward |
0 |
0 |
F |

Borrow $5 |
-5 |
0 |
- S'(0, M) f/(0, M) |

Buy 1 unit spot |
■S |
0 |
0 |

Borrow c(AO's forward |
—c(0) |
~c(k) |
yp c(k) |

Pay storage |
c(0) |
c(k) |
0 |

Tota! |
0 |
0 |
5 y^ c(k) <r/(0, M) fa d(k.M) |

10.3 FORWARD PRICES

Example 10,4 (Sugar with storage cost) The current price of sugar is 12 cents per pound We wish to find the forward price of sugar to be delivered in 5 months The carrying cost of sugar is 1 cent per pound per month, to be paid at the beginning of the month, and the interest rate is constant at 9% per annum,

The interest rate is .09/12 = 0075 per month The reciprocal of the 1-month discount rate (for any month) is 1.0075. Therefore we find

F = (1,0075)5(. 12) + [(1.0075)5 + (1 0075)4 + (1.0075)3 + (1 0075)2 -f 1 0075]( 001) = 1295 =12.95 cents

Example 10.5 (A bond forward) Consider a Treasury bond with a face value of $10,000, a coupon of 8%, and several years to maturity. Currently this bond is selling for $9,260, and the previous coupon has just been paid. What is the forward price for delivery of this bond in 1 year? Assume that interest rates for I year out are flat at 9%

We recognize that there will be two coupons before delivery: one in 6 months and one just prior to delivery. Hence using the present value form (10,3) and a 6-month compounding convention, we have immediately

S9.260

(1.045)2 1.045 This can be solved [or turned around to the form (10.2)] to give

F = $9,260(1.045)2 - $400 - $400(1.045) = $9,294.15 (in decimal form, not 32nd1s).

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