Applications Flashcards

Question 1

Q

Name 4 insurance products with embedded options

Answer

A

Equity Index Annuities
Traditional Variable Annuities
Immediate Variable Annuities
Structured Product - Based Variable Annuities

Question 2

Q

Describe the main features of an Equity Index Annuity

QFIQ 134 22

Answer

A

The investor buys the contract with a single premium and in return:
* The product promises a payoff based on the greater of a minimum guaranteed floor and the performance of a reference index
* Usually lasts for 7-10 years
* If the index gains, the investor participates in the gains in excess of the guaranteed floor
* If the index loses, the payoff never sinks below a minimum level
* EIAs offer downside and protection and upside potential

Question 3

Q

Name 5 crediting methods for EIAs

Answer

A

Point-to-Point PTP
PTP with a cap rate
Cliquet
Cliquet with a cap rate
High Water Mark

Question 4

Q

Give the maturity benefit of the 5 main crediting mechanisms

Answer

A

Point-to-Point PTP: max{1+α (S_T - S₀)/S₀ ; e^gT}
PTP with a cap rate: max{min[1+α (S_T - S₀)/S₀ ; e^CAP]; e^gT}
Cliquet: Π(i, T) max[1+α (S_k - S_k-1)/S_k-1 ; e^g]
Cliquet with a cap rate: Π(i, T) max[min{1+α (S_k - S_k-1)/S_k-1; e^cap} ; e^g]
High Water Mark: max{1+α (S^Max - S₀)/S₀ ; e^gT}

Question 5

Q

Describe some differences between EIA and trad VA

Answer

A

There is no separate account for EIAs, all premiums go into the insurer’s general accounts and benefits come out of there, just like life insurance
With EIA, Equity risk lies largely with insurer, where VA places equity risk with p/h
EIA only allow some percentage of participation in equity returns while VA holders can choose and fully benefit from any equity returns
VA upside potential is limitless
VA holders can dynamically transfer funds btw investment strategies and EIA holders cannot
EIAs do not charge fees as they control cost through participation rates/caps/floors
In USA, EIAs are sold as fixed annuities not under SEC, VAs are regulated by SEC
EIAs usually involve a static hedge and VAs need a dynamic hedge
EIA is like a call option while a VA is a put
EIAs are short term, VAs are medium/long term
Equity index in EIA are not total return indices as used for separate account products

Question 6

Q

What are sources of revenue for the insurer for trad VA?

Answer

A

Rider charge (margin offset)
Insurance charge
Admin expense charge
Mortality and Expense fee (M&E)
Investment management charge
Surrender charge

Question 7

Q

What is the AV during the accumulation phase for a trad VA

Answer

A

Discrete time: F_k/n = F₀ S_k/n/S₀ (1-m/n)^k
n = # periods per year, m = annual rate of charges
Cts time: F_t = F₀ S_t/S₀ e^-mt

Question 8

Q

Name the 5 guarantees for traditional VAs

Answer

A

GMMB
GMDB
GMAB
GMWB
GMWB for life (GLWB)

Question 9

Q

Name 4 mechanisms for adjusting the guaranteed amount during the VA life

Answer

A

Reset option
Rollup option
Lifetime high step up option
Annual high step up option

Question 10

Q

Define the Reset option for VA

Answer

A

Let T₀ = 0 < T₁ < … be renewal dates
Under a reset option, the guarantee base at time T_k is
G_{T_k} = Max(G_{T_k-1}, F_{T_k})

Question 11

Q

Define the Rollup option for VA

Answer

A

Let ρ = rollup rate. Assume this is the nominal rate payable n times a year. The guarantee base at time k/n is
G_k/n = G₀ (1 + ρ/n)^k
<=> G_(k+1)/n = G_k/n (1 + ρ/n)

Question 12

Q

Define the Lifetime high step up option for VA

Answer

A

G_(k+1)/n = Max(G_k/n; F_(k+1)/n)

Question 13

Q

Define the annual high step up option for VA

Answer

A

G_(k+1)/n = G_k/n Max(F_k/n; F_(k+1)/n)/F_k/n
This equation leads to:
G_k/n = G₀ Π(0, k-1) Max(1; F_(j+1)/n/F_j/n)

Question 14

Q

Describe the GMMB for VA

Answer

A

The guaranteed minimum maturity benefit
* It guarantees that the p/d receives the greater of the underlying investment fund or the pre-agree minimum maturity benefit G
* The guaranteed amount G plays the role of an option strike since it is fixed at inception
* GMMB is a put sold by the insurer on the underlying fund. It is European with maturity equal to the end of the VA contract

Question 15

Q

What is the PV of the gross/net liabilities under a GMMB

Answer

A

The PV of GROSS liability is e^-rT (G-F_T)₊ I(T_x > T)
The NET liability has PV L_e(T_x) = e^-rT (G-F_T)₊ I(T_x > T) - M_{Min(T, T)}
M_t = ∫(0,t) m_e e^-rsF_sds, me = cont rate of rider charge for GMMB

Question 16

Q

Describe the GMAB, what is the benefit payment at the end of the second period?

Answer

A

The GMAB works like the GMMB except that the guarantee is renewed at the end of the first term T₁ to a new guarantee value G₁ = Max(G₀; F_T₁)
If the separate account performs, the guarantee is set to account value, otherwise it remains at the inital level
The benefit payment at the end of the second period is Max(G₀; F_T₁) (1 - F_T₂/G₁)₌ I(T_x > T₂)

Question 17

Q

Describe the GMDB for VA

Answer

A

The guaranteed minimum death benefit
* This option is similar to the GMMB that it is a put option, but with a couple exceptions
* The excess of guarantee over investment fund account is payable on death, which is a random variable
* Valuation needs to account for the probability of death at any time t during the contract

Question 18

Q

What is the PV of the gross/net liabilities under a GMDB

Answer

A

Assume a rollup option (G_t = G₀e^ρt
The PV of Gross liability is e^-rT_x (G_{T_x} - F_{T_x})₊ I(T_x < T)
The PV of net liability is
L_d(T_x) = e^-rT_x (G_{T_x} - F_{T_x})₊ I(T_x < T) - M_{Min(T, T_x)}
M_t = ∫(0,t) m_d e^-rsF_sds

Question 19

Q

What is the incremental AV under a GMWB VA

Answer

A

F_(k+1)/n - F_k/n = [(S_(k+1)/n - S_k/n)/S_k/n F_k/n] - m/n F_k/n - w/n
This can be written as F_(k+1)/n = Max(0; [S_(k+1)/n/S_k/n F_k/n] - m/n F_k/n - w/n
where w/n represents an actual withdrawal amount

Question 20

Q

Describe the liability of a GMWB from the insurer’s perspective

Answer

A

Let τ = min{k/n} s.t. F_k/n <=0
The PV of insurer liabilities is PV of all withdrawals after ruin time and before maturity. They also collect M&E fees while the account has a balance to make profits. Therefore:
L_w⁽ⁿ⁾ = Σ(nτ, max[nτ-1, nT]) e^-rk/n w/n - Σ(1, min[nτ-1, nT]) e^-r(k-1)/nF_(k-1)/n m_w/n

Question 21

Q

Describe the GLWB for VA

Answer

A

As the name suggests, these withdrawals are guaranteed for life
The amount of each withdrawal is usually a percentage of AV at beginning of year (x_WL = h = guaranteed withdrawal rate)
The total amount of these withdrawals is unlimited
Often GLWB contain features to increase the guaranteed withdrawal amount if the fund performs well
On policy anniversary, the AV is compared to a fixed WBB = withdrawal benefit base
If AV > Benefit Base at anniversary, then the guaranteed withdrawal amount is increased

Question 22

Q

What is the net liability of the insurer for a GLWB

Answer

A

L_w⁽ⁿ⁾ = Σ(nτ, max[nτ-1, nT_x]) e^-r(k+1)/n G_k/n h/n - Σ(0, min[nτ-1, nT_x]) e^-rk/nG_k/n m_w/n
When n goes to infinity, we get
L_w^(inf) = h ∫(τ, max[τ,T_x]) e^-rtG_tdt - mw ∫(0, min[τ, T_x]) e^-rtG_tdt

Question 23

Q

For an immediate VA, what is the P/h subaccount value at time k

Answer

A

F_k = (F_k-1 - P_k-1) I_k = S_k/S_k-1 (1-m)(F_k-1 - P_k-1)
Where I_k = [S_k-1 + (S_k - S_k-1)]/S_k-1 (1-m)
And P_k = P₀ S_k/S₀ [(1-m)/(1+i)]^k

Question 24

Q

For a whole life immediate VA, what is the P/h subaccount value at time k

Answer

A

Let the intial payout rate be P₀ = F₀/ä_x
The time k value of the subaccount is
F_k = (F_k-1 - P_k-1) I_k = S_k/S_k-1 (1-m) (F_k-1 - P_k-1)
Or F_k = S_k/S₀ (1-m)^k (F₀ - P₀ä_k|)

Question 25

Q

How do we price the GMMB

Answer

A

The expected payoff of the GMMB is E[(G - F_T)₊] _Tp_x
The price at time t to charge for the GMMB is calculated via BSM as
B_e(t, F) = _Tp_x BSP_t(F, T-t, G, m)

Question 26

Q

How do we value the rider charges used to fund the GMMB

Answer

A

The time t value of all rider charges is ∫(t,T) e^-r(s-t)m_eF_s I(s < T_x)ds
Taking the expectation, we get P_e(t, F) = m_e_tp_xF∫(0, T-t) e^-ms_sp_x+tds

Question 27

Q

How do we price a GMAB

Answer

A

The first period benefit is similar to a GMMB of maturity T₁:
B_e(t, F) = _T₁p_x BSP_t(f, T₁-t, G, m)
The second period benefit is more complicated
B_a⁽²⁾(t, F) = _T₂p_xe^-r(T₂-t) E[Max(G₀; F_T₁)|F_t = F] E[(1-F_T₂/G₁)₊]

Question 28

Q

How do we price a GMDB

Answer

A

Since the GMDB is a put with variable expiry (time of death), we can integrate over the PDF of T_x
B_d(0, F) = ∫(0, T) BSP₀(t, G, F) f_{T_x}(t)dt

Question 29

Q

How do we value the rider charges used to fund a GMDB

Answer

A

The time t value of all rider charges is given by
∫(t, Min(T,T_x)) e^-r(s-t)m_dF_sds
Taking an expecation, we see that the no arbitrage value of the income fee deductions of the GMDB at any point t during the contract is
P_d(t, F) = m_dF_t _tp_xā_x+t:T-t;m

Question 30

Q

Describe the general flow of a hedging program for a GMMB

Answer

A

A hedging strategy is typically implemented in a way similar to a discrete time model as follows
1. Begin with 0 value. Estimate Δ₀ and hold that many shares of the underlying asset
2. Update Δ_t periodically and adjust stock holdings accordingly
3. Increase (decrease) stock holdings by borrowing (lending) to the money market account
4. If Δ₀ is positive, the purchase shares by borrowing from money market account, else short that many shares and deposit into MM account

Question 31

Q

What are the steps for hedging a derivative with exogeneous cash flows

Answer

A

Begin with 0 value. Estimate Δ₀ and hold that many shares of the underlying asset
Deposit net CF C_{0</sub in the MM account}
Collect and deposit net CFs over time C_t into MM account
Update Δ_t periodically and adjust stock holdings accordingly
Increase (decrease) stock holdings by borrowing (lending) to the money market account. If Δ₀ is positive, the purchase shares by borrowing from money market account and/or using the initial CF C₀, else short that many shares and deposit into MM account

Question 32

Q

What are three objectives of new VA products

Answer

A

P/hs, asset managers and insurers are looking for new ways to make product portfolios that
* Reduce balance sheet exposure to difficult to hedge risks like volatility spikes
* Enable insurers to support meaningful retirement income levels
* Minimize loss of investment upside potential perceived by clients

Question 33

Q

What 3 solutions were part of the first generation of volatility management

Answer

A

Asset transfer programs that reallocate client funds to bond funds based on the moneyness of contracts
Volatility manage funds (risk control funds) which rebalance allocation to equities depending on a target or trigger level of realized volatility
Market linked rider fees that can adjust as they are tied to a prevailing market index like the VIX or Treasury rates

Question 34

Q

Describe Asset transfer programs

Answer

A

Based on a pre-defined ratio of moneyness
The asset transfer programs reallocate funds between bond and equity strategies
When volatility increases, the AV of a traditional 60 equity 40 bond fund decreases and the option (put) becomes in the money, increasing the moneyness ratio. If this ratio exceeds some defined threshold, the allocation to equity is reduced

Question 35

Q

Describe capped volatility funds

Answer

A

These funds are set so that realized volatility of the fund is not to exceed some defined level (e.g. 30%)
This is a risk control fund that adjusts positions in response to the market signals of risk
The fund asjusts positions when the realized volatility exceeds the cap volatility
The goal is to leave the traditional allocations (60/40) intact except during crises
The equity allocation under this fund is maximized while being constrained to limiting the overal fund volatility
When market volatility increases, the fund volatility is capped at the threshold, but it can go below the threshold if markets are more stable

Question 36

Q

Describe target volatility funds

Answer

A

These funds target a specific volatility level
This fund dynamically adjusts positions so the level of volatility is always close to the target
When market volatility is low, the fund increases equity allocation and decreases it when volatility is high
Overall the realized volatility of the fund is close to the target

Question 37

Q

Describe capital preservation funds

Answer

A

Also known as the self-hedge strategy
It extends target volatility mechanics
The equity allocation from target volatility strategy is further reduced by using derivatives
The volatility of the cap. pres. fund fluctuates below the target volatility fund
It uses futures/derivatives to mitigate the risk of the fund following a market decline
The fund reduces its current risk positions if prior performance is poor

Question 38

Q

Describe VIX indexed fees

Answer

A

The target rider charge is determined and based upon prevailiing levels of the VIX
The fees are subject to a ceiling and floor but can scale between them for moderate levels of volatility
The goal is to adjust fees quarterly in line with changing market volatility

Question 39

Q

What are 3 challenges of all first generation volatility management solutions

Answer

A

Performance benchmarking
Loss of upside potential
Lack of clarity of investment thesis

Question 40

Q

Describe the next generation solutions of volatility management

Answer

A

These potential solutions combine the VIX fee structure with an underlying capped volatility fund
The design protects insurers’ balance sheets against volatility risk with a reduced challenge of investment performance attribution and benchmarking
The capped volatility fund kicks in to provide protection in volatility spike scenarios
VIX fees minimize impact of client investment performance

Question 41

Q

How does the next generation of volatililty management solutions perform

Answer

A

The joint risk solution operates relatively better than its individual components
The capped volatility fund provides the majority of the protection during volatility spikes
Incremental fees from VIX-indexing are not sufficient to completely offset P&L losses, but it does stabilize cash flows
During a crisis, both components are active

Question 42

Q

What are the benefits of the next generation solutions to volatility management for both insurer and client perspectives

Answer

A

Insurer
* Joint solution provides additive protection in reducing “volatility cost” and Vega
* The joint solution is more effective than its individual components
* Cash flows over time are more stable in response to volatility changes

Client
* The joint solution minimally affects client value metrics
* There is not a discernable impact on guaranteed income withdrawal rates

Question 43

Q

Under a target volatility fund, what is the allocation to equity

Answer

A

At time t, we have w_t^E = equity weight at t = Min(σ_target/σ_t^E, 100%)

Question 44

Q

What are the 5 steps to construct a VolTarget portfolio?

Answer

A

Choose the length of time h between rebalancings and the period H for estimating historical volatility
Set the volatility target level V̅
Estimate the time t₁ historical volatility of the risky asset V_t₁
Assign equity weight α_t₁¹ = Min(V̅/V_t₁, 200%)
Liquidate the existing investment portfolio and use the proceeds M_t₁ to create a VolTarget portfolio

Question 45

Q

How do ATM Euro calls on a VolTarget portfolio compare to those on the pure risky asset?

Answer

A

For options on S, prices increase with volatiltiy
For options on VT, option prices increase but much less. The rate of increase is much lower
When volatility of S is highest, the price of an option on S far exceeds the option on VT
When the volatility target V is <= volatility parameter in the GBM model, switching from options on S to options on VT does not create a significant loss

Question 46

Q

What is the payoff of guarantee structures with direct participation in an underlying asset

Answer

A

The payoff of 100% capital protection at T is
P_dp = Max{K; K(1+p_dp(S_T-S₀)/S₀)} = K + p_dpKMax(S_T/S₀ - 1;0)

Question 47

Q

Describe how the participation rate relates to volatility and interest rates

Answer

A

p_dp = RB/V₀(g_dp) = (K-PV(K))/V₀(g_dp)
* A lower interest rate increases PV(K) and lowers p
* Higher volatility increases option prices V₀(g_dp) which lowers p
* The combination of low rates and high uncertainties in the current environment makes it difficult to attract investors into the guarantee structures since the participation rate in the index gain is lower
* The participation rate may be increased by decreasing the option price which can be achieved from switching from S to a VolTarget linked to S

Question 48

Q

How do participation rates differ on S vs on VolTarget linked to S

Answer

A

On S: When volatility is highest and interest rates are low, the participation rates are very low ~20%
On VT: By lowering option price, we can get participation rates much higher, ~40%
At low volatility, option prices for VT are higher than S, so participation is worse in this case
VolTarget asses are helpful in cases where interest is low and volatility is high

Question 49

Q

List 3 drawbacks of the VolTarget mechanism

Answer

A

The strategy works well in specific market environments like low interest high volatility
It is a rule based dynamic allocation process, which may cause significant losses in nonstandard market environments
It may not be sufficient to solely define portfolio management decisions and should be considered in tandem with other asset allocation strategies

Question 50

Q

How does the BSM delta-rho hedge fare in the presence of model risk

Answer

A

The impact of model risk is significnt: we observe a huge increase in capital requirements from BS/1F (closest to BSM) to HE/3F (furthest)
The delta-rho hedge, nevertheless, results in an important risk reduction compared to no hedge
Although it is affected by model risk, it is still a large decrease in capital requirements
CTE increases by a similar factor between models, so model risk is not more pronounced in the tail of hedged loss distribution

Question 51

Q

What is the importance of hedging the rho risk

Answer

A

It provides significant reduction in the level of required capital
When there is little model risk, the rho hedge substantially improves the hedging quality up to ~85%
Model risk deteriorates the value of the rho hedge
The rho hedge contributes to reducing hedging risk in both GMAB and GMWB products by a similar margin

Question 52

Q

What impacts can rising interest rates have on guarantees (GMAB + GMWB)

Answer

A

The value of guarantees offered by the insurer will decrease which is a gain to the insurer if rho is not hedged
This is consistent with the argument that there is little incentive to hegde rho in the low rate environment
BSV delta only hedge results in significant gains under rising rates, but the opposite is true when rates decrease

Question 53

Q

What impacts can low interest rates have on guarantees (GMAB + GMWB)

Answer

A

The insurer can substantially benefit from a rho hedge as its exposure to large losses is reduced
When interest rates are low, there is an incentive to hedge interest rate risk
Although delta-rho hedging is affected by model risk, it does reduce capital requirements

Question 54

Q

What impacts can stock volatility have on guarantees (GMAB + GMWB)

Answer

A

The hedged loss of these products increase with increasing volatility
Stochastic volatility has a large impact on the effectiveness of the BSV delta-rho hedge
Although volatility parameters would be calibrated to market data, the hedge is not vega hedged and is exposed to stochastic volatility
However, high levels of volatility do not have a disproportionate adverse impact on the BSV delta rho hedge

Question 55

Q

What impact does rebalancing frequency have on guarantees (GMAB + GMWB)

Answer

A

In the presence of model risk, computing the Greeks too often with the wrong model could lead to an accumulation of hedging errors
Frequent rebalancing comes with higher monitoring and transaction costs
There are still benefits, but they can be balanced with these 2 downsides

Question 56

Q

What is the delta of the GMMB

Answer

A

Δ_t = ∂/∂s f(t,s)
For the GMMB, Δ_t = -_Tp_xF/S_t e^-m(T-t) N(-d₁) - P_e(t, F)/S_t
d₁ = [ln(u) + (r - m + 1/2 σ²)(T-t)]/(σsqrt(T-t))

Question 57

Q

What is the gamma of the GMMB

Answer

A

Γ_t = ∂/∂S(Δ_s)
Thus, Γ_t = _Tp_xF/S_t² e^-m(T-t) N(d₁)/(σsqrt(T-t))

Question 58

Q

What is the vega of the GMMB

Answer

A

Vega_t = ∂/∂σ f(t,s)
For the GMMB
Vega_t = _Tp_xF e^-m(T-t)N(d₁)sqrt(T-t)

Question 59

Q

What is the rho of the GMMB

Answer

A

ρ_t = ∂/∂r f(t,s)
For the GMMB
ρ_t = _Tp_x[rGe^-r(T-t)N(-d₂) - mFe^-m(T-t)N(-d₁ - e^-m(T-t)σFN(d₁(T-t;F_t/G))/(2sqrt(T-t))] +m_Tp_x

Question 60

Q

For profit testing and actuarial pricing, what are some factors we need to make assumptions on

Answer

A

Equity returns
Fees and charges
Surrenders
Annuitization
Survival model
Fund Values
Expenses
Interest on surplus
STAT reserves
Tax on surplus
Target surplus

Question 61

Q

What are the 3 steps in profit testing

Answer

A

Projected AV
Projected CFs
Projected distributable earnings
The CFs from the contract can be viewed in an I/S perspective w/ 3 main components
Revenues: the amount the insurer receives as gross income
Expenses: the business costs incurred excluding the insurance liabilities
Benefits: the amount paid or set aside for the costs of insurance liabilities

Question 62

Q

What is book profit vs distributable earnings in profit testing

Answer

A

Both measure profitability of insurances business
Book profit is determined by total revenues - total expenses - total benefits
Distributable earnings is the amount an insurer can distribute while retaining the level of capital needed to continue its current operations
The difference between the two is due to taxes and holding any extra capital beyond statutory reserves

Question 63

Q

How can we estimate distributable earnings in profit testing

Answer

A

Distributable Earnings = Profit after tax + After tax interest on target surplus - Increase in Target surplus

Question 64

Q

List three profit measures in profit testing

Answer

A

NPV
IRR
Profit Margin

Answer 65

A

NPV = Σ(1, T) Distributable earnings(i)/(1+j)ⁱ
j = yield rate required by shareholders
IRR is the j such that NPV = 0
Profit margin = NPV/Single Premium, the NPV of distributable earnings as a percent of premium

Answer 66

A

With an SPVA, investors pick a fixed number of market indices (e.g. S&P 500, Russell 2k)
Investors decide to allocate capital to selected indices known as segments
A segment term is agree on as the maturity date
The crediting mechanism involves a cap rate to limit upsides and a buffer rate to limit losses
At the term, the insurer credits the VA with a return based on index, cap, and buffer
All payouts use price returns, not total returns. This excludes any income from dividends
These products protect buyers from small losses (up to buffer) in exchange for capping gains
The cap rate is the max percentage of return that may be credited at the end of the term

Answer 67

A

Buffer and capped payout
Principal protected note (PPN) payout

Answer 68

A

This is the most common structure
Investors are exposed to losses beyond the buffer up to a maximum of 100% - buffer
Issuers do not publish cap rates, which vary over time and with different buffer levels and underlying assets

Answer 69

A

The stepped PPN
Capped PPN

Answer 70

A

The contract pays a fixed amount (the step rate) if returns on the underlying are positive, else it returns the amount invested
This is like a PTP EIA with a minimum guaranteed rate of 0%
MetLife and Allianz offer Step PPN segments, only for the S&P 500

Answer 71

A

This method is equivalent to a 100% buffer
The segment returns the principal paid if the reference asset has a negative return and pays any positive returns up to a cap
This is only available for the S&P 500

Answer 72

A

The pro-rating approach
The valuation of the option

Answer 73

A

This method pro-rates the cap and/or buffer level based on the fraction of time into the segment term
E.g., if 6 months of a 1 year segment passed, the cap/buffer would be halved and then applied to the to-date returns of the index
MetLife’s SLS uses this approach

Answer 74

A

The issuer will decompose the segment down into its component options, value them separately, and sum them up along with the embedded ZCB
Allianz uses this method
AXA takes the lesser of this method and the pro-rated value. Although they only pro-rate the cap, not buffer

Answer 75

A

A ZCB
A short Euro put with strike = buffer level
A long Euro ATM call
A short Euro call with strike = cap level

Answer 76

A

Capped PPN: The payoff of a capped PPN can be replicated using a ZCB, a long Euro ATM call, and a short Euro call with strike = cap

Stepped PPN: This is different, we use a ZCB and a long ATM cash-or-nothing binary call option

Answer 77

A

EPV = e^-r(t_n-t₀)I₀Π(1,n)E[1+R_i]
I₀ = initial investment, R_i = return for i^th time period

Answer 78

A

EPV = e^-r(t_n-t₀)I₀Π(1,n)E[1+R_i]
Under this mechanism, E[1+R_i] = 1 - FV_Put(S₀, K_b, τ, r, q, σ)/S₀ + FV_Call(S₀, S₀, τ, r, q, σ)/S₀ - FV_Put(S₀, K_c, τ, r, q, σ)/S₀
K_b = S₀ * (1-buffer); K_c = S₀ * (1+cap)

Answer 79

A

EPV = e^-r(t_n-t₀)I₀Π(1,n)E[1+R_i]
Under this mechanism, E[1+R_i] = 1 + FV_Call(S₀, S₀, τ, r, q, σ)/S₀ - FV_Put(S₀, K_c, τ, r, q, σ)/S₀
K_b = S₀ * (1-buffer); K_c = S₀ * (1+cap)

Answer 80

A

EPV = e^-r(t_n-t₀)I₀Π(1,n)E[1+R_i]
Under this mechanism, E[1+R_i] = 1 + FV_{Binary Call}(S₀, S₀, τ, r, q, σ)/S₀

Answer 81

A

The crediting formulas of the SPVA segments reduce the volatility of realized returns compared to underlying index
As the buffer increases, the average return of the strategy decreases and the volatility increases. Thus the fair cap level decreases
As the buffer increases, the cap decreases and the percentage of returns that are capped increases

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