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Unformatted text preview: properly discounted by
taking the interest rate and the lease rate into account, and by ignoring any storage cost and
convenience yield (because we do not have any information on it):
F0,T = S 0 × e (r −δ l )×T
⇔ S 0 = F0,T × e − (r −δ l )×T ⇔ S 0 = 313.81 × e − (0.06−0.015 )×1 = 313.81 × 0.956 = 300.0016 46 Chapter 6 Commodity Forwards and Futures Question 6.12. a) We have the following table:
Price Quantity Revenue
1.2 Using Excel’s function STDEVP(4.5,2.4,2,1.2), we obtain a value of 1.2194 for the standard
deviation of total revenue for Scenario C.
Using any standard software’s command (or doing it by hand!) to determine the
correlation coefficient, we obtain a value of 0.7586.
Question 6.14. a)
The expected quantity of production is 0.25 × (1.5 + 0.8 + 1 + 0.6) = 0.975 million bushels
3 1.5m 3 0.8m 2 1m 2 0.6m Unhedged revenue Futures gain from shorting
− 0.5 × 0.975m
− 0.5 × 0.975m
0.5 × 0.975m
0.5 × 0.975m
= 0.4875m Total
1.6875m Using Excel’s function STDEVP(4.0125, 1.9125, 2.4875, 1.6875), we obtain a value of
0.907004 for the standard deviation of the optimally hedged revenue for Scenario C. We see that
we were able to reduce the variance of our revenues, albeit to a lesser degree than with the
optimally hedged portfolio.
Question 6.16. a)
Widgets do not deteriorate over time and are costless to store, therefore the lender does
not save anything by lending us the widget. On the other hand, there is a constant demand and
47 Part 2 Forwards, Futures, and Swaps flexible production – there is no seasonality. Therefore, we should expect that the convenience
yield is very close to the risk-free rate, merely compensating the lender for the opportunity cost.
Demand varies seasonally, but the production process is flexible. Therefore, we would
expect that producers anticipate the seasonality in demand, and adjust production accordingly.
Again, the lease rate should not be much higher than the risk-free rate.
Now we have the problem that the demand for widgets will spike up without an
appropriate adjustment of production. Let us suppose that widget demand is high in June, July
and August. Then, we will face a substantial lease rate if we were to borrow the widget in May,
to return it in September: We would deprive the merchant of the widget when he would need it
most (and could probably earn a significant amount of money on it), and we would return it
when the season is over. We most likely pay a substantial lease rate.
On the other hand, suppose we want to borrow the widget in January, and return it in June. Now
we return the widget precisely when the merchant needs it, and have it over a time where
demand is low, and he is not likely to sell it. The lease rate is likely to be very small.
However, those stylized facts are weakened by the fact that the merchant can costlessly store
widgets, so the smart merchant has a larger inventory when demand is high, offsetting the above
effects at a substantial amount.
Suppose that production is very low during June, July and August. Let us think about
borrowing the widget in May, and returning it in September. We do again deprive the merchant
of the widget when he needs it most, because with a constant demand, less production means
widgets become a comparably scarce resource and increase in price. Therefore, we pay a higher
lease rate. The opposite effects can be observed for a widget-borrowing from January to June.
Again, these stylized facts are offset by the above mentioned inventory considerations.
If widgets cannot be stored, the seasonality problems become very severe, leading to
larger swings in the lease rate due to the impossibility of managing inventory. 48 Chapter 7
Interest Rate Forwards and Futures
The coupon bond pays a coupon of $60 each year plus the principal of $1,000 after five years.
We have cash flows of [60, 60, 60, 60, 1060]. To obtain the price of the coupon bond, we
multiply each cash flow by the zero-coupon bond price of that year. This yields a bond price of
Bond Yield 1
0.02900 Zero Coupon
Par Coupon Cont. Comp. Zero
Bond Price Implied Forward
0.02859 Question 7.6.
In order to be able to solve this problem, it is best to take equation (7.6) of the main text and
solve progressively for all zero-coupon bond prices, starting with year one. This yields the series
of zero-coupon bond prices from which we can proceed as usual to determine the yields.
Bond Yield 1
0.05300 Zero Coupon
Par Coupon Cont. Comp...
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This document was uploaded on 03/11/2014 for the course FIN 402 at FPT University.
- Spring '14