Applications Flashcards

Question

How do we price the GMMB

Answer 1

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)

Answer 2

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

Answer 3

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)₊]

Answer 4

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

Answer 5

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

Answer 6

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

Answer 7

1. Begin with 0 value. Estimate Δ₀ and hold that many shares of the underlying asset 2. Deposit net CF C₀t into MM account 4. Update Δ_t periodically and adjust stock holdings accordingly 5. 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

Answer 8

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

Answer 9

* 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

Answer 10

* 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

Answer 11

* 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

Answer 12

* 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

Answer 13

* 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

Answer 14

* 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

Answer 15

1. Performance benchmarking 2. Loss of upside potential 3. Lack of clarity of investment thesis

Answer 16

* 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

Answer 17

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

Answer 18

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

Answer 19

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

Answer 20

1. Choose the length of time h between rebalancings and the period H for estimating historical volatility 2. Set the volatility target level V̅ 3. Estimate the time t₁ historical volatility of the risky asset V_t₁ 4. Assign equity weight α_t₁¹ = Min(V̅/V_t₁, 200%) 5. Liquidate the existing investment portfolio and use the proceeds M_t₁ to create a VolTarget portfolio

Answer 21

* 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

Answer 22

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)

Answer 23

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

Answer 24

* 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

Answer 25

1. The strategy works well in specific market environments like low interest high volatility 2. It is a rule based dynamic allocation process, which may cause significant losses in nonstandard market environments 3. It may not be sufficient to solely define portfolio management decisions and should be considered in tandem with other asset allocation strategies

Answer 26

* 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

Answer 27

* 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

Answer 28

* 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

Answer 29

* 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

Answer 30

* 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

Answer 31

* 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

Answer 32

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))

Answer 33

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

Answer 34

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

Answer 35

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

Answer 36

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

Answer 37

1. Projected AV 2. Projected CFs 3. Projected distributable earnings The CFs from the contract can be viewed in an I/S perspective w/ 3 main components 1. Revenues: the amount the insurer receives as gross income 2. Expenses: the business costs incurred excluding the insurance liabilities 3. Benefits: the amount paid or set aside for the costs of insurance liabilities

Answer 38

* 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

Answer 39

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

Answer 40

1. NPV 2. IRR 3. Profit Margin

Answer 41

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 42

* 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 43

1. Buffer and capped payout 2. Principal protected note (PPN) payout

Answer 44

* 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 45

1. The stepped PPN 2. Capped PPN

Answer 46

* 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 47

* 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 48

1. The pro-rating approach 2. The valuation of the option

Answer 49

* 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 50

* 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 51

1. A ZCB 2. A short Euro put with strike = buffer level 3. A long Euro ATM call 4. A short Euro call with strike = cap level

Answer 52

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 53

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 54

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 55

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 56

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 57

* 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

Applications Flashcards

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