b. Buy the relatively cheap futures, sell the relatively expensive stock and lend the
proceeds of the short sale:
CF Now CF in 6 months
Buy futures 0 ST 1,914
c. If you do not receive interest on the proceeds of the short sales, then the $1,600 you
receive will not be invested but will simply be returned to you. The proceeds from the
strategy in part (b) are now negative: an arbitrage opportunity no longer exists.
CF Now CF in 6 months
Buy futures 0 ST 1,914
d. If we call the original futures price F0, then the proceeds from the long-futures,
short-stock strategy are
CF Now CF in 6 months
Buy futures 0 ST F0
F0 can be as low as 1,860 without giving rise to an arbitrage opportunity.
On the other hand, if F0 is higher than the parity value (1,917), then an arbitrage
opportunity (buy stocks, sell futures) will exist.
CF Now CF in 6 months
Sell futures 0 F0 ST
Therefore, the no-arbitrage range is
26. a. Call p the fraction of proceeds from the short sale to which we have access. Ignoring
transaction costs, the lower bound on the futures price that precludes arbitrage is the
following usual parity value (except for the factor p):
The factor p arises because only this fraction of the proceeds from the short sale can
be invested in the risk-free asset. We can solve for p as follows:
b. With p = 0.9, the no-arbitrage lower bound on the futures price is
The actual futures price is 1,951. The departure from the bound is therefore 5.83. This
CF Now CF in 1 Year
Buy futures 0 ST 1,951
CFA PROBLEMS
1. a. By spot-futures parity:
b. The lower bound is based on the reverse cash-and-carry strategy.
Action Now CF in $ Action at period-end CF in $
Buy one TOBEC index
futures contract 0Sell one TOBEC index
futures contract $100 × (F1 − F0)
(Note that F1 = S1 at expiration.)
The lower bound for F0 is: 19,040/100 = 190.40
2. a. The strategy would be to sell Japanese stock index futures to hedge the market risk of
Japanese stocks, and to sell yen futures to hedge the currency exposure.
b. Some possible practical difficulties with this strategy include
• Contract size on futures may not match size of portfolio.
3. a. The hedged investment involves converting the $1 million to foreign currency,
investing in that country, and selling forward the foreign currency in order to lock in
Japanese government Swiss government
Convert $1 million
$1,000,000 × 133.05 =
$1,000,000 × 1.5260 =
b. The results in the two currencies are nearly identical. This near-equality reflects the interest
c. The 90-day return in Japan is 1.5793%, which represents a bond-equivalent yield of
4. The investor can buy X amount of pesos at the (indirect) spot exchange rate and invest the pesos
in the Mexican bond market. Then, in one year, the investor will have
These pesos can then be converted back into dollars using the (indirect) forward exchange
rate. Interest rate parity asserts that the two holding period returns must be equal, which can
be represented by the formula:
The left side of the equation represents the holding-period return for a U.S.
dollar-denominated bond. If interest rate parity holds, then this term also corresponds to the
U.S. dollar holding-period return for the currency-hedged Mexican one-year bond. The right
side of the equation is the holding-period return, in dollar terms, for a currency-hedged
peso-denominated bond. Solving for r US:
5. a. From parity:
0.5 0.5
0 0
11.0010
124.30 122.06453
1 1.0380
Japan
US
r
F E r
+
æ ö æ ö
= ´ = ´ =
ç ÷ ç ÷
+è ø
è ø
b.
Action Now
CF in $
Action at
CF in ¥
Borrow $1,000,000 in U.S. $1,000,000 Repay loan −($1,000,000 × 1.0350.25 ) =
−$1,008,637.446
Sell forward $1,008,637.446
Convert borrowed dollars to yen;
¥124,300,000 × 1.0050.25 =
7. Situation A. The market value of the portfolio to be hedged is $20 million. The market value
of the bonds controlled by one futures contract is $63,330. If we were to equate the market
values of the portfolio and the futures contract, we would sell
However, we must adjust this “naive” hedge for the price volatility of the bond portfolio
relative to the futures contract. Price volatilities differ according to both the duration and the
the naive hedge for relative duration and relative yield volatility, we obtain the adjusted
hedge position
The duration ratio is 7.2/8.0, and the relative yield volatility is 1.25. Therefore, the hedge
requires the treasurer to take a long position in
33025.1
0.8
2.7
330,63
000,600,18
contracts
8. a. % change in T-bond price = Modified duration Change in YTM
= 7.0 0.50% = 3.5%
b. When the YTM of the T-bond changes by 50 basis points, the predicted change in the
% change in KC price = Modified duration Change in YTM