4. NUMERICAL ILLUSTRATIONS
4.2 Parameter Analyses
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4. NUMERICAL ILLUSTRATIONS
4.1 Valuation Examples
We consider the case that the domestic currency is Australian dollar and the linked index is S&P 500, which is denominated in US dollar. Using the monthly data from January 2000 to June 2010, we estimate the volatility and correlation parameters as follows: σ = 16.47% (the volatility of S&P 500) , S σ = 13.84% (the volatility of C
the exchange rate USD/AUS), ρ = -0.52 (the correlation coefficient of log
( )
S( )
t and log(
C( )
t)
)A typical contract usually has maturity 3 to 7 years. We thus select T = 5 years. We set annual ceiling rate c = 30%, annual floor rate f = 0%, participate rate α= 100%. We use 5-year treasury rates of June 30, 2010 to proxy the risk free rates. Therefore, the 5-year risk free rate of Australian dollar r is set to 4.78% and the 5-year risk free rate of US dollar rf is set to 1.83%. We also set the number of averaging in a year m = 4 (when applicable). Above combination of model parameters and contract features is our benchmark example.
4.2 Parameter Analyses
In this section, we use the previous benchmark example to illustrate how various
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contract features and model parameters may affect the value of the contract. For each
set of parameters we examine six product specifications:
• SR: Simple version of Ratchet EIAs
• CR: Compound version of ratchet EIAs
• SR G1: Simple version of Ratchet EIAs with G1 averaging scheme
• CR G1: Compound version of ratchet EIAs with G1 averaging scheme
• SR G2: Simple version of Ratchet EIAs with G2 averaging scheme
• CR G2: Compound version of ratchet EIAs with G2 averaging scheme
4.2.1 Impact of return cap
The value of the contract with various return cap shows in Figure 1. The contract value increases with the return cap, as expected, because capping the return that can be credited to the contract truncates the upside potential. The value increases at a diminishing rate (i.e., all curves are concave). This is reasonable because the probability of hitting the upper bound decreases at an increasing rate when the upper bound rises as long as the probability density of positive returns is a decreasing function of returns. We further observe that the impact of return cap is the most significant when there is no return averaging and is the least significant when returns are averaged by the first type of scheme. The underlying reason is that the first type
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of averaging scheme has the most significant averaging effect. It averages over non-overlapping sub-periods while the second type averages on cumulative returns of sub-periods. The stronger return averaging effect decreases the probability of hitting the upper bound more and thus reduces the impact of return cap.
The impact of return cap is more significant when returns are accumulated compoundedly than the corresponding case when returns are accumulated additively as we see from Figure 1. This is also reasonable because the compound version generates higher returns in our current parameter settings and thus is bounded more by return caps.
Figure 1: Impact of Return Cap c
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4.2.2 Impact of Return Floor Rate
The value of the contract increases with the return floor as Figure 2 shows. The impact of return floor is more significant when returns are accumulated compoundedly than the corresponding case when returns are accumulated additively.
Figure 2: Impact of return floor rate f
We observe that return floor has the least impact on the contract without return averaging and has the greatest impact on the contract with the G1 averaging, given the same way of return accumulation. More specifically, the percentage change of the contract value given a change in the return floor is the smallest when there is no return
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averaging and is the largest when returns are averaged by the first type of scheme.
The underlying reason is that the value contributed by the return volatility decreases with the floor rate. The reduction in the contract value due to the volatility dampening of return averaging thus decreases with the floor rate as well. Therefore we observe that the value increase the fastest/slowest with the floor rate for the contract with the strongest/weakest return averaging scheme.
4.2.3 Impact of Participation Rate
The value of the contract increases with the participation rate as Figure 3 shows. It is interesting to seeing that the contract value is nearly linear function of participation rate for 0.5 ≤ α ≤ 1.2. Also, the impact of participation rate is more significant when returns are accumulated compoundedly than the corresponding case when returns are accumulated additively. Besides, the participation has the greatest impact on the contract with no return averaging but has the least impact on the contract with the G1 averaging scheme. The rationale is that the participating rate amplifies/condenses the effect of return averaging since it is the multiplier to the annual return in equation (4). The reduction in the contract value due to return averaging thus increases with the participating rate.
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Figure 3: The impact of Participation rate
4.2.4 Impact of Return Averaging
Figure 4 shows that the contract value decreases with the frequency of averaging.
The impact of return averaging can be rather significant. The frequency of return averaging would decrease the contract value because higher frequencies produce stronger averaging effects and reduce the volatilities of annual returns. The reduced volatilities decrease the value of the options embedded in the ratchet EIA products.
The impact of return averaging is more significant for the compound version than for the simple version.
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Figure 4: Impact of Return Averaging
The averaging frequency has more impact on the G1 averaging scheme than on G2. Remember that G1 averages returns over non-overlapping sub-periods while G2 averages on cumulative returns of sub-periods. The marginal effect of increasing the number of sub-periods is thus larger for G1.
4.2.5 Impact of the Volatility of Linked Index
The value of the contract increases with the volatility of the linked index as Figure 5 shows. The impact of with the volatility of the linked index is also more significant
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when returns are accumulated compoundedly than the corresponding case when returns are accumulated additively. When the volatility of the linked index is greater than 30%, the increase in contract value of SR and CR becomes very minor. This is because the annual return is capped at 30%.
Figure 5: Impact of the volatility of the linked index
4.2.6 Impact of the Volatility of Exchange Rate
The value of the contract increases with the volatility of exchange rate as Figure 6 shows. It is interesting to seeing that the contract value is nearly linear function of the
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volatility of exchange rate. The impact of the volatility of exchange has no big difference for CR and SR versions.
Figure 6: Impact of the volatility of exchange rate
4.2.7 Impact of the correlation coefficient of log(S(t)) and log(C(t))
The value of the contract decrease with the correlation coefficient of log(S(t)) and log(C(t)) as Figure 7 shows. It is interesting to noting that the contract value is nearly linear function of the correlation coefficient of log(S(t)) and log(C(t)). The impact of
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ρ has no big difference for CR and SR versions. Please note that ρ = 0 is
corresponding to the “non”-quanto case. From Figure 7, it is clear that the contacts are mispriced if the quanto feature has been ignored.
Figure 7: Impact of the correlation coefficient of log(S(t)) and log(C(t))
4.2.8 Impact of the Domestic Risk-Free Rate
The value of the contract decreases with the domestic risk-free rate as Figure 8 shows, because the present value of the cash flow at maturity is a decreasing function of the domestic risk-free rate. The curves show little convexity since the contract maturity is
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merely 5 years. The impacts of r on the contract values look to be similar across return accumulation methods and return averaging schemes.
Figure 8: Impact of the domestic risk free rate
4.2.9 Impact of the Foreign Risk-Free Rate
The value of the contract increases with the foreign risk-free rate at a moderately increasing speed as Figure 9 shows. This effect is the most appearing when there is no return averaging and is the least significant with the G1 return averaging. Figure 9 further shows that the differences in the contract values between the compound and simple versions increase with the foreign risk-free rate.
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Figure 9: Impact of the foreign risk free rate