LSMC valuation of life insurance contracts with surrender options

Antonio Edoardo Bargellini (speaker)
with Luca Gruppi and Giulia Spaletta

Department of Statistical Sciences, University of Bologna, Italy

(RowDefault[, (Spacer1), , (Spacer1), ])  #WolframTechConf, October 16-18, 2024

ABSTRACT

Principal aim: investigate role of surrender option in life insurance contracts

▪ Valuation framework (introducing the problem)
▪ Least Squares Monte Carlo (LSMC) approach 
▪ Numerical results

Valuation Framework

Aim: find  fair price for an endowment equity-linked insurance contract

Insured benefits linked to  performance of a reference asset  (Subscript[S, t])

((d(Subscript[S, t]) )/((Subscript[S, t]) )) =  r  dt + σ  d(Subscript[W, t])

▪r, σ, (Subscript[S, 0])  : deterministic interest rate,  volatility, asset initial value
▪(Subscript[W, t])    Wiener process
▪time  0≤ t ≤T maturity

Benefits of the contract

Contract provides 3 types of benefits :

(Subscript[B, t]^i)   = (Subscript[S, 0])  max {  ((Subscript[S, t])/(Subscript[S, 0])), (e^((r^i)t)) }                       i=s, d, w

▪(r^i)   min guaranteed interest rate
▪s   survival benefit (only defined for T)
▪d   death benefit
▪w  withdrawal benefit

Mortality Models

▪Weibull  force of mortality (Subscript[μ, t]) :              (Subscript[μ, x+t])   =  (a^(-b))  b (((x+t))^(b-1))
parameters  a,b∈ ℝ+  estimated from life tables from 1872 to 2020 
x  age of the insured

▪Lee-Carter :                                            Log[(Subscript[m, x,t])]   =  (Subscript[a, x]) + (Subscript[k, t])   (Subscript[b, x]) +  (Subscript[ϵ, x,t])   
(Subscript[m, x,t]) central mortality rate
(Subscript[a, x])  average  over  time  of  Log[(Subscript[m, x,t])]  for  each  age
(Subscript[k, t]) ,(Subscript[b, x])  from  a  Singular Value Decomposition of matrix (Log[(Subscript[m, x,t])] - (Subscript[a, x]))
(Subscript[ϵ, x,t])  stochastic component

▪Weibull diagonal : cohort-based mortality model 
A  Weibull  force  of  mortality for the mortality of each cohort  is assumed. 
The forecast is done w.r.t the parameters a,b∈ ℝ+ of the Weibull model. Moreover, we look at life tables in a diagonal perspective.

LSMC Approach

Presence of surrender option implies that there is no closed-form formula for Contract Value  at  time  t=0   [1,2]
⟹  Numerical  solution    (Subscript[V, 0])=(1/N) (∑_(k=1)^N)(Subscript[C, 0]^k)   must be sought  [2,3]

k=1, ..., N
▪ MC  simulation  of   N   paths   for   (Subscript[S, t])    
▪ MC  simulation  of   N  death  instants    (Subscript[τ, k])            
▪ Initialize ex-post realized benefits   (Subscript[C, (Subscript[τ, k])]^k)  =  (Subscript[B, (Subscript[τ, k])]^(d,k))  if  (Subscript[τ, k])≤T,  else   (Subscript[C, T]^k)  = (Subscript[B, T]^(s,k))

Contract   Value

LS approximation gives Continuation Values  (Subscript[(OverTilde[C]), t]^k)  [4]
(Subscript[(OverTilde[C]), t]^k) : value the contract would have if  surrender option were not exercised 
Based  on  withdrawal  benefit  (Subscript[B, t]^(w,k)) ,    (Subscript[(OverTilde[C]), t]^k)   is used to modify  (Subscript[C, t]^k)

▪ Discretize   [0,T] :    (Subscript[t, j])   = j (T/Δ)

j=n-1,...,0    &&   If  (Subscript[t, j])<(Subscript[τ, k])
▪Update    (Subscript[C, (Subscript[t, j])]^k)  =  (Subscript[C, (Subscript[t, j+1])]^k) (e^(-r Δ))    
▪ (Subscript[(OverTilde[C]), (Subscript[t, j])]^k)  = LS   regression  of   (Subscript[C, (Subscript[t, j])]^k)   w.r.t.  basis  { 1, (Subscript[S, t]) ,  (Subscript[S, t]^2) , (Subscript[S, t])  (Subscript[B, t]^w) , (Subscript[B, t]^w), ((((Subscript[B, t]^w)))^2) } 
▪ Update   (Subscript[C, (Subscript[t, j])]^k)  =  (Subscript[B, (Subscript[t, j])]^(w,k))  if  (Subscript[(OverTilde[C]), (Subscript[t, j])]^k)<(Subscript[B, (Subscript[t, j])]^(w,k))

▪ Sought  (Subscript[V, 0])  is  the  average  of  values  (Subscript[C, 0]^k)      (k=1,...,N)

Numerical  Applications

▪ User-friendly interface  to set up all parameters  in the presented LSMC approach

▪ All  information necessary to replicate numerical tests are saved in a file

▪ Results obtained are comparable to those in literature [1]

Interface

mydir = SetDirectory[NotebookDirectory[]];
<< LeastSquaresMonteCarloPackage.m

myfontsize=FontSize->10; 
mysize=10;
output="In attesa...";
seed=1;
ti=0;  
deltat=1/12;

Manipulate[
Column[{
Style["Selected parameters",Bold,mysize],
Style[Row[ {"r: ", r*100}],  Bold,mysize] ,Style[Row[ {"ri: ", ri*100}],  Bold,mysize] ,Style[Row[ {"σ: ", σ}],  Bold,mysize] ,Style[Row[ {"S0: ",S0}],  Bold,mysize],Style[Row[ {"Maturity: ",tf}],  Bold,mysize] ,
Style[Row[ {"N. samples: ",ncampioni}],  Bold,mysize] ,
Style[Row[ {"Year: ",anno}],  Bold,mysize] ,Style[Row[ {"Gender: ",gender}],  Bold,mysize] ,Style[Row[ {"Age: ",eta}],  Bold,mysize] ,Style[Row[ {"Mortality model: ",modello}],  Bold,mysize] ,
Style[Row[ {"N. executions ",nEsecuzioni}],  Bold,mysize] ,
Style[Row[ {"Seed: ",seed}],  Bold,mysize] ,

Button["Execute" ,
Clear[valoriSelezionati,output];
valoriSelezionati=Association[
"r"->r,
"ri"->ri,
"σ"->σ,
"S0"->S0,
 "Initial time"->ti,
"Maturity"->tf,
"N. samples"->ncampioni,
"Year"->anno,
"Gender"->gender,
"Age"->eta,
"Mortality model"->modello,
"N. executions"->nEsecuzioni,
"Save it on a file"->salvaSuFileTemp,
"File path"->mydir,
"Seed"->seed,
 "DeltaT"->deltat];

If[salvaSuFileTemp,
valoriSelezionati["Percorso file"]=mydir; ];

output="Calcolo in corso..."; 
	   Pause[0.5]; 
output=RunGenerationAndCalculation[valoriSelezionati];,
Appearance->Automatic,
BaseStyle->{myfontsize},
Evaluator->Automatic,
Method->"Queued",
Background->RGBColor[{1,0,0}]
]
},
     Dynamic[output,
TrackedSymbols:>{r,ri,σ,S0, tf,ncampioni,anno,gender,eta,nEsecuzioni,salvaSuFileTemp}],
Spacings->0.2
],

{{r,0.04,Style["r",myfontsize, Bold]},0.02,0.06,0.005,ContinuousAction->False},

{{ri,0.02,Style["ri",myfontsize, Bold]},0,Dynamic[r-0.005],0.005,ContinuousAction->False},

{{σ,0.2,Style["σ",myfontsize, Bold]},0.1,0.3,0.05,ContinuousAction->False},

{{S0,100,Style["S0",myfontsize, Bold]},100,500,100,ContinuousAction->False},

{{tf,20,Style["Maturity",myfontsize, Bold]},10,30,0.5,ContinuousAction->False},

{{ncampioni,2000,Style["N. samples",myfontsize, Bold]},2000,100000,1000,ContinuousAction->False},

{{anno,2020,Style["Year",myfontsize, Bold]},1872,2020,1,ContinuousAction->False},

{{gender,"All",Style["Gender",myfontsize, Bold]},{Style["All",myfontsize],
Style["Female",myfontsize],
Style["Male",myfontsize]},
ContinuousAction->False},

{{eta,40,Style["Age",myfontsize, Bold]},0,110,1,
ContinuousAction->False},

{{modello,"W",
Style["Mortality model",myfontsize, Bold]},{Style["W",myfontsize],
Style["W_d",myfontsize],
Style["LC_d",myfontsize]},
ContinuousAction->False},

{{nEsecuzioni,1,Style["N. executions",myfontsize, Bold]},1,20,1,ContinuousAction->False},

{{salvaSuFileTemp,True,Style["Save it on a file",myfontsize, Bold]},
{True,False}},

Row[{
Style["Enter the Seed",myfontsize, Bold],InputField[Dynamic[seed],
Number,
BaseStyle->myfontsize]
}],
SaveDefinitions->True ]

Numerical  Results (I)

▪As r increases, both EU and American prices decrease
▪As σ increases, both European and American prices increase
▪As T increases, European has no clear trend, American prices increase

Grid[{{Style["T", Bold],Style["EU - W", Bold],Style["A - W", Bold], Style["EU - W_d", Bold],Style["A - W_d", Bold], Style["EU - LC_d", Bold],Style["A - LC_d", Bold]},(*Titoli delle colonne*){"15",119.05

,121.25

, 121.12

,122.29

, 120.70

,122.01

},(*Prezzi Europei*){"20",117.50
,122.34

, 122.10

, 124.23

, 121.56

, 123.92


} ,{"25",113.00

,122.64

, 122.07

, 125.65

, 121.48

, 125.23


}(*Prezzi Americani*)},Frame->All]

Insert[%,{Dividers->All,Spacings->1.5` {1,1}},2]

Numerical  Results (II)

▪ Both European and American prices are higher in the Female case

Essential Bibliography

[1] Bacinello, Biffis, Millossovich,  Regression-based algorithms for life insurance contracts with surrender guarantees, Quantitative Finance, Vol. 10, Issue 9, pp. 1077-1090, 2010. https://doi.org/10.1080/14697680902960242

[2] Bacinello, Millossovich, Montealegre,  A Comparison between Different Numerical Schemes for the Valuation of Unit-Linked  Contracts Embedding a Surrender Option, Mathematical and Statistical Methods for Actuarial Sciences and Finance (M.Corazza & C.Pizzi Editors), Springer, 2014. https://link.springer.com/chapter/10.1007/978-3-319-02499-8_3

[3] Bargellini,  Least Squares Monte Carlo valuation of life insurance contracts involving surrender options,  M.Sc. Thesis, University of Bologna, 2023.

[4] Longstaff, Schwartz, Valuing American Options by Simulation: A Simple Least-Squares Approach, The Review of Financial Studies, Vol. 14, Number 1, pp. 113-147, 2001. https://econpapers.repec.org/RePEc:oup:rfinst:v:14:y:2001:i:1:p:113-47

Thank you for your attention!

(RowDefault[, (Spacer1), , (Spacer1), ])  #WolframTechConf, October 16-18, 2024

Essential Bio

Dr.Bargellini holds an Economics & Business Bachelor (Florence University 2020) and a Quantitative Finance M.Sc. (Bologna Univ. 2023). He won an Erasmus scholarship (Athens Univ. of Economics & Business 2019-2020) and a PhD scholarship (Bologna Statistics Department 2023-current). He successfully completed an Intensive Programme on Advances Topics in Energy Markets & Finance, an Advanced Training Course in Actuarial Sciences. He worked in private firms as Junior Analyst and Accounting Assistant.