Investments, tenth edition

participation rate  or  multiplier,  in this case 70%; if the market 

falls, the investor is insured against loss. Just as clearly, the bank offering these CDs is in 

effect writing call options and can hedge its position by buying index calls in the options 

market.  Figure 20.14  shows the nature of the bank’s obligation to its depositors.  

 How might the bank set the appropriate multiplier? To answer this, note various fea-

tures of the option:

    1.  The price the depositor is paying for the options is the forgone interest on the 

conventional CD that could be purchased. Because interest is received at the end 

of the period, the present value of the interest payment on each dollar invested is 

 r  

 f 

    /(1  1   r  

 f  

 ). Therefore, the depositor trades a sure payment with present value per 

dollar invested of  r  

 f 

 

 

 /(1  1   r  

 f  

  ) for a return that depends on the market’s performance. 

Conversely, the bank can fund its obligation using the interest that it would have 

paid on a conventional CD.  

   2.  The option we have described is an at-the-money option, meaning that the exercise 

price equals the current value of the stock index. The option goes into the money as 

soon as the market index increases from its level at the inception of the contract.  

   3.  We can analyze the option on a per-dollar-invested basis. For example, the option 

costs the depositor  r  

 f  

 /(1  1   r  

 f  

 ) dollars per dollar placed in the index-linked CD. The 

market price of the option per dollar invested is  C / S  

0

 : The at-the-money option 

costs  C  dollars and is written on one unit of the market index, currently at  S  

0

 .    

 Now it is easy to determine the multiplier that the bank can offer on the CDs. It receives 

from its depositors a “payment” of  r  

 f 

    /(1  1   r  

 f  

 ) per dollar invested. It costs the bank  C / S  

0

   to 

purchase the call option on a $1 investment in the market index. Therefore, if  r  

 f 

    /(1  1   r  

 f  

 )  is, 

for example, 70% of  C / S  

0

 , the bank can purchase at most .7 call option on the $1 investment 

and the multiplier will be .7. More generally, the break-even multiplier on an index-linked 

CD is  r  

 f 

     /(1  1   r  

 f  

 ) divided by  C / S  

0

 .  

 Suppose that  

r  

 f 

    5   6% per year, and that 6-month maturity at-the-money calls on 

the market index currently cost $50. The index is at 1,000. Then the option costs 

50/1,000  5  $.05 per dollar of market value. The CD rate is 3% per 6 months, meaning 

that  r  

 f 

  /(1  1   r  

 f 

 )  5  .03/1.03  5  .0291. Therefore, the multiplier would be .0291/.05  5  .5825.