(This is a tough one, for readers who actually studied the appendices to Chapter 9 on lognormal option pricing.) The Johannesburg branch of Shanghai Chartered Bank (SCB) is considering a three-month loan to Bechuana Coffee Plantations (BCP), to be backed by BCP's export receipts. The expected harvest is about 100 tons, and the expected world coffee price is about 7,000 crowns/ton.
(a) SCB must decide how much it can lend if it can use BCP's entire export revenue as security. What precautions could SCB take to make sure that the export revenue is actually used to pay back the loan?
(b) One of SCB's analysts is asked to estimate the worst-case export revenue. Unfortunately, both BCP and the coffee market have changed quite a lot since the company's founding 20 years ago, so that the analyst cannot simply use the history of BCP's export revenue to assess the risk.
The analyst assumes that the actual output (O) and the price (P) are lognormally distributed, because this distribution is more consistent with the non-negativity of outputs and prices than a normal distribution and because then the revenue, (O × P), is also conveniently lognormal. On the basis of commodity option prices and output data from similar plantations, the analyst then estimates the parameters of output and prices separately. The plan is to compute the confidence intervals for the normally distributed variable ln(O × P) = ln(O) + ln(P), which has mean and variance equal to [μo + μp] and [σo2 + 2 covo,p + σp2], respectively. From the lower bound on ln(O ×P) the analyst can then infer the lower bound on (O × P). From traded commodity option prices, SCB's analyst infers that the standard deviation of the log price is 10 percent over three months (20 percent p.a.). From past data on planted acreage and output for similar plantations, the standard deviation of BCP's output is estimated to be 15 percent over three months. Using the output and price expectations given above, what are μo and μp-the expected values of ln(O) and ln(P), rather than O and the price P?
(c) The analyst argues that, since Botswana has only a small share in the coffee market, the variance of the export revenue can be computed as if the covariance between local output and the world price is zero. Is this a conservative assumption or not?
(d) How would SCB compute a 90 percent confidence interval for BCP's entire export revenue?
e) It turns out that BCP needs far less than 500,000 crowns. BCP signs a contract with CEH Jouy-en-Josas, a well-known and solid French coffee trader, to deliver 40 tons at the forward price of 6,900 crowns/ton. When computing the maximum amount it can lend on the strength of this forward contract, should SCB take a similar safety margin relative to the expected revenue from this transaction as the one computed in part (e)?
(f) Suppose instead that CEH agrees to buy 50 tons at the (as yet unknown) future spot price for coffee. How should the analyst assess the risk in this case?