# How to prove with put-call parity: return multiplier ratio of ILCD (index linked certificate deposit ) < 1

Index-linked CDs pay interests based on a specific index, and a guaranteed payment of all principal.

The return multiplier ratio (multiplier) is defined as the rate of return of the derivative versus the market rate of return of the underlying asset.

Return multiplier ratio is defined as: $$\frac{r_f}{1+r_f}$$$$\frac{S_0}{C}$$
It is a ratio:
each dollar invested in the index-linked CD, gain riskless interest --- $$\frac{r_f}{1+r_f}$$
$$\frac{S-S_0}{C}$$,
divided by,
the profit generated by each dollar invested in the index, which is $$\frac{S-S_0}{S_0}$$.

$$r_f$$  is the riskless interest rate, the current index is $$S_0$$, the index at maturity is S, S > $$S_0$$, the cost of at-the-money index call is C.

Question is: show $$\frac{r_f}{1+r_f}$$*$$\frac{S_0}{C}$$ <1 with put‐call parity

Put‐Call Parity says:   payoff of the call‐plus bond portfolio is the same as the payoff of protective put position.

And by no arbitrage principle, the call‐plus‐bond portfolio (on left) must cost the same as the  stock‐plus‐put portfolio (on right): C+ PV(K) = $$S_0$$ + P.

PV() is the present value, P is the put price, K is the exercise price.

Since $$\frac{r_f}{1+r_f}$$ is more than price of put, then $$\frac{r_f}{1+r_f}$$*$$\frac{S-S_0}{C}$$ is less than $$\frac{S-S_0}{S_0}$$