Applications Flashcards
Name 4 insurance products with embedded options
- Equity Index Annuities
- Traditional Variable Annuities
- Immediate Variable Annuities
- Structured Product - Based Variable Annuities
Describe the main features of an Equity Index Annuity
QFIQ 134 22
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
Name 5 crediting methods for EIAs
- Point-to-Point PTP
- PTP with a cap rate
- Cliquet
- Cliquet with a cap rate
- High Water Mark
Give the maturity benefit of the 5 main crediting mechanisms
- Point-to-Point PTP: max{1+α (ST - S0)/S0 ; egT}
- PTP with a cap rate: max{min[1+α (ST - S0)/S0 ; eCAP]; egT}
- Cliquet: Π(i, T) max[1+α (Sk - Sk-1)/Sk-1 ; eg]
- Cliquet with a cap rate: Π(i, T) max[min{1+α (Sk - Sk-1)/Sk-1; ecap} ; eg]
- High Water Mark: max{1+α (SMax - S0)/S0 ; egT}
Describe some differences between EIA and trad VA
- 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
What are sources of revenue for the insurer for trad VA?
- Rider charge (margin offset)
- Insurance charge
- Admin expense charge
- Mortality and Expense fee (M&E)
- Investment management charge
- Surrender charge
What is the AV during the accumulation phase for a trad VA
Discrete time: Fk/n = F0 Sk/n/S0 (1-m/n)k
n = # periods per year, m = annual rate of charges
Cts time: Ft = F0 St/S0 e-mt
Name the 5 guarantees for traditional VAs
- GMMB
- GMDB
- GMAB
- GMWB
- GMWB for life (GLWB)
Name 4 mechanisms for adjusting the guaranteed amount during the VA life
- Reset option
- Rollup option
- Lifetime high step up option
- Annual high step up option
Define the Reset option for VA
Let T0 = 0 < T1 < … be renewal dates
Under a reset option, the guarantee base at time Tk is
GTk = Max(GTk-1, FTk)
Define the Rollup option for VA
Let ρ = rollup rate. Assume this is the nominal rate payable n times a year. The guarantee base at time k/n is
Gk/n = G0 (1 + ρ/n)k
<=> G(k+1)/n = Gk/n (1 + ρ/n)
Define the Lifetime high step up option for VA
G(k+1)/n = Max(Gk/n; F(k+1)/n)
Define the annual high step up option for VA
G(k+1)/n = Gk/n Max(Fk/n; F(k+1)/n)/Fk/n
This equation leads to:
Gk/n = G0 Π(0, k-1) Max(1; F(j+1)/n/Fj/n)
Describe the GMMB for VA
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
What is the PV of the gross/net liabilities under a GMMB
The PV of GROSS liability is e-rT (G-FT)+ I(Tx > T)
The NET liability has PV Le(Tx) = e-rT (G-FT)+ I(Tx > T) - MMin(T, T)
Mt = ∫(0,t) me e-rsFsds, me = cont rate of rider charge for GMMB
Describe the GMAB, what is the benefit payment at the end of the second period?
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; FT1)
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; FT1) (1 - FT2/G1)= I(Tx > T2)
Describe the GMDB for VA
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
What is the PV of the gross/net liabilities under a GMDB
Assume a rollup option (Gt = G0eρt
The PV of Gross liability is e-rTx (GTx - FTx)+ I(Tx < T)
The PV of net liability is
Ld(Tx) = e-rTx (GTx - FTx)+ I(Tx < T) - MMin(T, Tx)
Mt = ∫(0,t) md e-rsFsds
What is the incremental AV under a GMWB VA
F(k+1)/n - Fk/n = [(S(k+1)/n - Sk/n)/Sk/n Fk/n] - m/n Fk/n - w/n
This can be written as F(k+1)/n = Max(0; [S(k+1)/n/Sk/n Fk/n] - m/n Fk/n - w/n
where w/n represents an actual withdrawal amount
Describe the liability of a GMWB from the insurer’s perspective
Let τ = min{k/n} s.t. Fk/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:
Lw(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 mw/n
Describe the GLWB for VA
- 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 (xWL = 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
What is the net liability of the insurer for a GLWB
Lw(n) = Σ(nτ, max[nτ-1, nTx]) e-r(k+1)/n Gk/n h/n - Σ(0, min[nτ-1, nTx]) e-rk/nGk/n mw/n
When n goes to infinity, we get
Lw(inf) = h ∫(τ, max[τ,Tx]) e-rtGtdt - mw ∫(0, min[τ, Tx]) e-rtGtdt
For an immediate VA, what is the P/h subaccount value at time k
Fk = (Fk-1 - Pk-1) Ik = Sk/Sk-1 (1-m)(Fk-1 - Pk-1)
Where Ik = [Sk-1 + (Sk - Sk-1)]/Sk-1 (1-m)
And Pk = P0 Sk/S0 [(1-m)/(1+i)]k
For a whole life immediate VA, what is the P/h subaccount value at time k
Let the intial payout rate be P0 = F0/äx
The time k value of the subaccount is
Fk = (Fk-1 - Pk-1) Ik = Sk/Sk-1 (1-m) (Fk-1 - Pk-1)
Or Fk = Sk/S0 (1-m)k (F0 - P0äk|)
How do we price the GMMB
The expected payoff of the GMMB is E[(G - FT)+] Tpx
The price at time t to charge for the GMMB is calculated via BSM as
Be(t, F) = Tpx BSPt(F, T-t, G, m)
How do we value the rider charges used to fund the GMMB
The time t value of all rider charges is ∫(t,T) e-r(s-t)meFs I(s < Tx)ds
Taking the expectation, we get Pe(t, F) = metpxF∫(0, T-t) e-msspx+tds
How do we price a GMAB
The first period benefit is similar to a GMMB of maturity T1:
Be(t, F) = T1px BSPt(f, T1-t, G, m)
The second period benefit is more complicated
Ba(2)(t, F) = T2pxe-r(T2-t) E[Max(G0; FT1)|Ft = F] E[(1-FT2/G1)+]
How do we price a GMDB
Since the GMDB is a put with variable expiry (time of death), we can integrate over the PDF of Tx
Bd(0, F) = ∫(0, T) BSP0(t, G, F) fTx(t)dt
How do we value the rider charges used to fund a GMDB
The time t value of all rider charges is given by
∫(t, Min(T,Tx)) e-r(s-t)mdFsds
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
Pd(t, F) = mdFt tpxāx+t:T-t;m
Describe the general flow of a hedging program for a GMMB
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 Δt 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
What are the steps for hedging a derivative with exogeneous cash flows
- Begin with 0 value. Estimate Δ0 and hold that many shares of the underlying asset
- Deposit net CF C0</sub in the MM account
- Collect and deposit net CFs over time Ct into MM account
- Update Δt 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
What are three objectives of new VA products
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
What 3 solutions were part of the first generation of volatility management
- 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
Describe Asset transfer programs
- 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
Describe capped volatility funds
- 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
Describe target volatility funds
- 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
Describe capital preservation funds
- 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
Describe VIX indexed fees
- 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
What are 3 challenges of all first generation volatility management solutions
- Performance benchmarking
- Loss of upside potential
- Lack of clarity of investment thesis
Describe the next generation solutions of volatility management
- 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
How does the next generation of volatililty management solutions perform
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
What are the benefits of the next generation solutions to volatility management for both insurer and client perspectives
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
Under a target volatility fund, what is the allocation to equity
At time t, we have wtE = equity weight at t = Min(σtarget/σtE, 100%)
What are the 5 steps to construct a VolTarget portfolio?
- 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 Vt1
- Assign equity weight αt11 = Min(V̅/Vt1, 200%)
- Liquidate the existing investment portfolio and use the proceeds Mt1 to create a VolTarget portfolio
How do ATM Euro calls on a VolTarget portfolio compare to those on the pure risky asset?
- 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
What is the payoff of guarantee structures with direct participation in an underlying asset
The payoff of 100% capital protection at T is
Pdp = Max{K; K(1+pdp(ST-S0)/S0)} = K + pdpKMax(ST/S0 - 1;0)
Describe how the participation rate relates to volatility and interest rates
pdp = RB/V0(gdp) = (K-PV(K))/V0(gdp)
* A lower interest rate increases PV(K) and lowers p
* Higher volatility increases option prices V0(gdp) 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
How do participation rates differ on S vs on VolTarget linked to S
- 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
List 3 drawbacks of the VolTarget mechanism
- 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
How does the BSM delta-rho hedge fare in the presence of model risk
- 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
What is the importance of hedging the rho risk
- 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
What impacts can rising interest rates have on guarantees (GMAB + GMWB)
- 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
What impacts can low interest rates have on guarantees (GMAB + GMWB)
- 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
What impacts can stock volatility have on guarantees (GMAB + GMWB)
- 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
What impact does rebalancing frequency have on guarantees (GMAB + GMWB)
- 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
What is the delta of the GMMB
Δt = ∂/∂s f(t,s)
For the GMMB, Δt = -TpxF/St e-m(T-t) N(-d1) - Pe(t, F)/St
d1 = [ln(u) + (r - m + 1/2 σ2)(T-t)]/(σsqrt(T-t))
What is the gamma of the GMMB
Γt = ∂/∂S(Δs)
Thus, Γt = TpxF/St2 e-m(T-t) N(d1)/(σsqrt(T-t))
What is the vega of the GMMB
Vegat = ∂/∂σ f(t,s)
For the GMMB
Vegat = TpxF e-m(T-t)N(d1)sqrt(T-t)
What is the rho of the GMMB
ρt = ∂/∂r f(t,s)
For the GMMB
ρt = Tpx[rGe-r(T-t)N(-d2) - mFe-m(T-t)N(-d1 - e-m(T-t)σFN(d1(T-t;Ft/G))/(2sqrt(T-t))] +mTpx
For profit testing and actuarial pricing, what are some factors we need to make assumptions on
- Equity returns
- Fees and charges
- Surrenders
- Annuitization
- Survival model
- Fund Values
- Expenses
- Interest on surplus
- STAT reserves
- Tax on surplus
- Target surplus
What are the 3 steps in profit testing
- 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
What is book profit vs distributable earnings in profit testing
- 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
How can we estimate distributable earnings in profit testing
Distributable Earnings = Profit after tax + After tax interest on target surplus - Increase in Target surplus
List three profit measures in profit testing
- NPV
- IRR
- Profit Margin
Give the equations for NPV, IRR, and Profit Margin
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
Describe Structured Product Variable Annuities
- 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
List the 2 crediting mechanisms for SPVA
- Buffer and capped payout
- Principal protected note (PPN) payout
Describe the buffer and capped payour for SPVA
- 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
List two types of PPN payouts for SPVA
- The stepped PPN
- Capped PPN
Describe the Stepped PPN Payout
- 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
Describe the Capped PPN Payout
- 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
Name two approaches for calculating the interim AV for SPVA
- The pro-rating approach
- The valuation of the option
Describe the pro-rating AV approach for SPVA
- 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
Describe the option valuation AV approach for SPVA
- 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
How can we decompose a buffered and capped payout SPVA
- A ZCB
- A short Euro put with strike = buffer level
- A long Euro ATM call
- A short Euro call with strike = cap level
How can we decompose a capped PPN and a stepped PPN SPVA
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
What is the EPV of the investment in a segment for an SPVA
EPV = e-r(tn-t0)I0Π(1,n)E[1+Ri]
I0 = initial investment, Ri = return for ith time period
What is the EPV of the payout for a buffered and capped segment
EPV = e-r(tn-t0)I0Π(1,n)E[1+Ri]
Under this mechanism, E[1+Ri] = 1 - FVPut(S0, Kb, τ, r, q, σ)/S0 + FVCall(S0, S0, τ, r, q, σ)/S0 - FVPut(S0, Kc, τ, r, q, σ)/S0
Kb = S0 * (1-buffer); Kc = S0 * (1+cap)
What is the EPV of the payout for a capped PPN segment
EPV = e-r(tn-t0)I0Π(1,n)E[1+Ri]
Under this mechanism, E[1+Ri] = 1 + FVCall(S0, S0, τ, r, q, σ)/S0 - FVPut(S0, Kc, τ, r, q, σ)/S0
Kb = S0 * (1-buffer); Kc = S0 * (1+cap)
What is the EPV of the payout for a stepped PPN segment
EPV = e-r(tn-t0)I0Π(1,n)E[1+Ri]
Under this mechanism, E[1+Ri] = 1 + FVBinary Call(S0, S0, τ, r, q, σ)/S0
What are some results of backtesting the SPVA (effects of cap and buffers on volatility)
- 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
