Questions $5-9$ are about an equity$-$linked note. An equity$-$linked note is a debt instrument, usually a bond, that differs from a standard fixed$-$income security in that the final payout is based on the return of the underlying equity, which can be a single stock, a basket of stocks, or an equity index. Equity$-$linked notes are a type of structured products.

An equity$-$linked note combines two features that investors may find attractive. First, the note provides a guaranteed minimum payoff to its holder no matter how the underlying equity performs. Second, if the underlying equity appreciates in value during the life of the note, the payoff is higher than the minimum and positively related to the return of the underlying equity.

Consider a $5-$year equity$-$linked note whose payoff is tied to the return of the Meta Platforms Inc. $($Meta$)$ stock. An investor invests $$100$ in the note today. The note expires $5$ years from today. When the note expires $($i$.5$ years from today$),$ the investor receives a payoff that depends on the change in the Meta stock price between today and the expiration date. For simplicity, assume that the price of the Meta stock today is $$100.$

If the Meta stock price on the expiration date is below $$100,$ the investor$$s payoff is $$100.$ This provides the investor a form of capital protection: she will not lose money in this investment even if the Meta stock would fall in value $($or even if the company goes bankrupt and the stock price falls to $$0).$

If the Meta stock price on the expiration date is above $$100,$ the investor$$s payoff is

$$100$ x$(1+$rpa x k$),$

where rpa is the relative price appreciation of the Meta stock. Formally, and simply, it is defined as

rpa$=($P$5$P$0)/$P$0,$

where P$5$ is the price of the Meta stock on the note expiration date and P$0=$$$100$

is the stock price today. This is simply the return of the Meta stock ignoring any dividends. The multiplier k in the payoff function above is known as the participation rate. This rate tells how much the equity$-$linked note investor participates in the potential upside of the underlying equity. We don$$t know k

yet. Your job in these questions is to figure out what k is possible.

An equity$-$linked note is constructed, or engineered, out of two fairly simple financial securities: a zero$-$coupon bond and a call option on the underlying equity.

Let$$s start with the zero$-$coupon bond. In order to provide the capital guarantee for the $5-$year note linked to the Meta stock, the issuer of the note must buy a $5-$year zero coupon bond with par value equal to $$100.$ The yield to maturity of a $5-$year zero coupon bond today is $4\mathrm{.}5\%.$ After the purchase of the zero$-$coupon bond, the remainder of the $$100$ invested in the note is used to purchase as many $5-$year at$-$the$-$money call options on the Meta stock as possible. This is a call option with strike price equal to the current stock price of Meta, $$100,$ and expiration date coinciding with the expiration date of the note. The market price, premium, today of such call option $$31\mathrm{.}40.$ Assume that the option can be bought in fractional amounts $($i$.$e$.$ one can purchase, say, $0\mathrm{.}798$ options$).$

How many $5-$year at$-$the$-$money call options on the Meta stock can be purchased with the funds left over after the purchase of the zero$-$coupon bond?