31  Auto-differentiation and Asset Liability Management (AAD & ALM)

Alec Loudenback

31.1 Chapter Overview

Asset liability modeling requires computing derivatives of portfolio values with respect to yield curve changes. Traditional approaches use finite difference methods or analytical approximations, but automatic differentiation (“autodiff” or “AD”) provides exact derivatives with minimal additional computation. This chapter demonstrates how to implement ALM workflows using autodiff in Julia.

using FinanceCore              # provides Cashflow object
using DifferentiationInterface # autodiff 
import ForwardDiff             # specific autodiff technique
using CairoMakie               # plotting
using DataInterpolations       # yield curve interpolation
using Transducers              # data aggregation
using JuMP, HiGHS              # portfolio optimization
using LinearAlgebra            # math
using BenchmarkTools           # benchmarking
using OhMyThreads              # multi-threading

31.2 Interest Rate Curve Setup

We start by constructing a yield curve using cubic spline interpolation:

The curve function creates a discount factor curve from zero rates and time points. This curve will serve as input to our value function, which makes it straightforward to compute sensitivities by differentiating with respect to the rate parameters.

zeros = [0.01, 0.02, 0.02, 0.03, 0.05, 0.055] #continuous

times = [1., 2., 3., 5., 10., 20.]

function curve(zeros, times)
    DataInterpolations.CubicSpline([1.0; exp.(-zeros .* times)], [0.; times])
end

c = curve(zeros, times)

CubicSpline with 7 points
┌──────┬──────────┐
│ time │        u │
├──────┼──────────┤
│  0.0 │      1.0 │
│  1.0 │  0.99005 │
│  2.0 │ 0.960789 │
│  3.0 │ 0.941765 │
│  5.0 │ 0.860708 │
│ 10.0 │ 0.606531 │
│ 20.0 │ 0.332871 │
└──────┴──────────┘

31.3 Asset Valuation Framework

The core valuation function operates on any instrument that produces cashflows:

function value(curve, asset)
    cfs = asset(curve)
    mapreduce(cf -> cf.amount * curve(cf.time), +, cfs)
end

value (generic function with 1 method)

This design separates the valuation logic from the instrument definition. Each asset type implements a callable interface that generates cashflows given a yield curve. Note how the asset itself gets passed the curve (the asset(curve) statement) to determine the cashflows.

For fixed bonds, we create a structure that generates periodic coupon payments:

struct FixedBond{A,B,C}
    coupon::A
    tenor::B
    periodicity::C
end
function (b::FixedBond)(curve)
    map(1//b.periodicity:1//b.periodicity:b.tenor) do t
        Cashflow(b.coupon / b.periodicity + (t == b.tenor ? 1. : 0.), t)
    end
end

function par_yield(curve, tenor, periodicity)
    dfs = curve.(1//periodicity:1//periodicity:tenor)

    (1 - last(dfs)) / sum(dfs) * periodicity
end

The (b::FixedBond)(curve) function (sometimes called a ‘functor’, since we are using the b object itself as the function invocation) takes the curve and returns an array of Cashflows.

Note

Cashflow objects are part of the JuliaActuary suite. This allows the cashflows to be tied with the exact timepoint that they occur, rather than needing a bunch of logic to pre-determine a timestep (annual, quarterly, etc.) for which cashflows would get bucketed. This is more efficient in many cases and much simpler code.

The par_yield function computes the coupon rate that prices the bond at par, which we’ll use to construct our asset universe.

Here’s an example of bond cashflows and valuing that bond using the curve c that we constructed earlier.

FixedBond(0.08, 10, 2)(c)

20-element Vector{Cashflow{Float64, Rational{Int64}}}:
 Cashflow{Float64, Rational{Int64}}(0.04, 1//2)
 Cashflow{Float64, Rational{Int64}}(0.04, 1//1)
 Cashflow{Float64, Rational{Int64}}(0.04, 3//2)
 Cashflow{Float64, Rational{Int64}}(0.04, 2//1)
 Cashflow{Float64, Rational{Int64}}(0.04, 5//2)
 Cashflow{Float64, Rational{Int64}}(0.04, 3//1)
 Cashflow{Float64, Rational{Int64}}(0.04, 7//2)
 Cashflow{Float64, Rational{Int64}}(0.04, 4//1)
 Cashflow{Float64, Rational{Int64}}(0.04, 9//2)
 Cashflow{Float64, Rational{Int64}}(0.04, 5//1)
 Cashflow{Float64, Rational{Int64}}(0.04, 11//2)
 Cashflow{Float64, Rational{Int64}}(0.04, 6//1)
 Cashflow{Float64, Rational{Int64}}(0.04, 13//2)
 Cashflow{Float64, Rational{Int64}}(0.04, 7//1)
 Cashflow{Float64, Rational{Int64}}(0.04, 15//2)
 Cashflow{Float64, Rational{Int64}}(0.04, 8//1)
 Cashflow{Float64, Rational{Int64}}(0.04, 17//2)
 Cashflow{Float64, Rational{Int64}}(0.04, 9//1)
 Cashflow{Float64, Rational{Int64}}(0.04, 19//2)
 Cashflow{Float64, Rational{Int64}}(1.04, 10//1)

value(c, FixedBond(0.09, 10, 2))

1.3526976075662451

31.4 Liability Modeling

Deferred annuities require more complex modeling than fixed bonds due to policyholder behavior (optionality). The surrender rate depends on the difference between market rates and the guaranteed rate. The surrender function chosen below is arbitrary, but follows a typical pattern with much higher surrenders if the market rate on competing instruments is higher than what’s currently available. The account value accumulates at the guaranteed rate, and surrenders create negative cashflows representing benefit payments. Lastly, the annuities function is a wrapper function we will use to compute the portfolio value and ALM metrics later.

begin
    struct DeferredAnnuity{A,B}
        tenor::A
        rate::B
    end

    function (d::DeferredAnnuity)(curve)
        av = 1.
        map(1//12:1//12:d.tenor) do t
            mkt_rate = -log(curve(d.tenor) / curve(t)) / (d.tenor - t)
            av *= exp(d.rate / 12)
            rate_diff = mkt_rate - d.rate
            sr = t == d.tenor ? 1.0 : surrender_rate(rate_diff) / 12
            av_surr = av * sr
            av -= av_surr
            Cashflow(-av_surr, t)

        end
    end

    function surrender_rate(rate_diff)
        1 / (1 + exp(3 - rate_diff * 60))
    end
    function annuities(rates, portfolio)
        times = [1., 2., 3., 5., 10., 20.]

        c = curve(rates, times)

        # threaded map-reduce for more speed
        OhMyThreads.tmapreduce(+, 1:length(portfolio); ntasks=Threads.nthreads()) do i
            value(c, portfolio[i])
        end
        # mapreduce(l -> value(c,l),+,portfolio)
    end

end

annuities (generic function with 1 method)

Here’s what the surrender rate behavior looks like for different levels of market rates compared to the a 3% crediting rate.

let
    cred_rate = 0.03
    mkt_rates = 0.005:0.001:0.08
    rate_diff = mkt_rates .- cred_rate

    lines(rate_diff, surrender_rate.(rate_diff),
        axis=(
            title="Surrender rate by difference to market rate",
            xlabel="Rate Difference",
            ylabel="Annual Surrender Rate"
        ))
end

We model a large portfolio of these annuities with random tenors:

liabilities = map(1:100_000) do i
    tenor = rand(1:20)
    DeferredAnnuity(tenor, par_yield(c, tenor, 12))
end

100000-element Vector{DeferredAnnuity{Int64, Float64}}:
 DeferredAnnuity{Int64, Float64}(2, 0.0198981133775843)
 DeferredAnnuity{Int64, Float64}(12, 0.049761345150540474)
 DeferredAnnuity{Int64, Float64}(13, 0.05055368479631081)
 DeferredAnnuity{Int64, Float64}(14, 0.05108046813021651)
 DeferredAnnuity{Int64, Float64}(2, 0.0198981133775843)
 DeferredAnnuity{Int64, Float64}(2, 0.0198981133775843)
 DeferredAnnuity{Int64, Float64}(17, 0.051714618573584406)
 DeferredAnnuity{Int64, Float64}(18, 0.05177859152668137)
 DeferredAnnuity{Int64, Float64}(17, 0.051714618573584406)
 DeferredAnnuity{Int64, Float64}(20, 0.051933558828553925)
 ⋮
 DeferredAnnuity{Int64, Float64}(7, 0.03847750979086058)
 DeferredAnnuity{Int64, Float64}(9, 0.04476297466971608)
 DeferredAnnuity{Int64, Float64}(8, 0.0419258485355399)
 DeferredAnnuity{Int64, Float64}(14, 0.05108046813021651)
 DeferredAnnuity{Int64, Float64}(18, 0.05177859152668137)
 DeferredAnnuity{Int64, Float64}(19, 0.05183909988436309)
 DeferredAnnuity{Int64, Float64}(16, 0.051607152912174645)
 DeferredAnnuity{Int64, Float64}(15, 0.05141229637424002)
 DeferredAnnuity{Int64, Float64}(18, 0.05177859152668137)

NoteConsolidating Cashflows

Later on we will generate vectors of vectors of cashflows without any guarantee that the timepoints will line up, making aggregating cashflows by timepoints a non-obvious task. There are many ways to accomplish this, but I like Transducers.

Transducers are unfamiliar to many people, and don’t let the presence deter you from the main points of this post. The details aren’t central to the point of this blog post so just skip over if confusing.

function consolidate(cashflows)

    cashflows |> # take the collection
    MapCat(identity) |> # flatten it out without changing elements
    # group by the time, and just keep and sum the amounts 
    GroupBy(x -> x.time, Map(last) ⨟ Map(x -> x.amount), +) |>
    foldxl(Transducers.right) # perform the aggregation and keep the final grouped result
end

consolidate (generic function with 1 method)

Example:

cashflow_vectors = [l(c) for l in liabilities]

100000-element Vector{Vector{Cashflow{Float64, Rational{Int64}}}}:
 [Cashflow{Float64, Rational{Int64}}(-0.004144410382128681, 1//12), Cashflow{Float64, Rational{Int64}}(-0.004315318017074089, 1//6), Cashflow{Float64, Rational{Int64}}(-0.004504093438676603, 1//4), Cashflow{Float64, Rational{Int64}}(-0.00471071623903657, 1//3), Cashflow{Float64, Rational{Int64}}(-0.004934713168882003, 5//12), Cashflow{Float64, Rational{Int64}}(-0.0051749370878303915, 1//2), Cashflow{Float64, Rational{Int64}}(-0.005429266292843056, 7//12), Cashflow{Float64, Rational{Int64}}(-0.005694197226230727, 2//3), Cashflow{Float64, Rational{Int64}}(-0.005964295051005587, 3//4), Cashflow{Float64, Rational{Int64}}(-0.006231455978636931, 5//6)  …  Cashflow{Float64, Rational{Int64}}(-0.007040339849056201, 5//4), Cashflow{Float64, Rational{Int64}}(-0.0070309918151435, 4//3), Cashflow{Float64, Rational{Int64}}(-0.006960391349463677, 17//12), Cashflow{Float64, Rational{Int64}}(-0.006830278507743372, 3//2), Cashflow{Float64, Rational{Int64}}(-0.006643759757520943, 19//12), Cashflow{Float64, Rational{Int64}}(-0.006405206284366148, 5//3), Cashflow{Float64, Rational{Int64}}(-0.006120116156813356, 7//4), Cashflow{Float64, Rational{Int64}}(-0.0057949431751937105, 11//6), Cashflow{Float64, Rational{Int64}}(-0.005436895608924536, 23//12), Cashflow{Float64, Rational{Int64}}(-0.9015948971297525, 2//1)]
 [Cashflow{Float64, Rational{Int64}}(-0.004961099093861061, 1//12), Cashflow{Float64, Rational{Int64}}(-0.005055658321176967, 1//6), Cashflow{Float64, Rational{Int64}}(-0.0051517659271625785, 1//4), Cashflow{Float64, Rational{Int64}}(-0.005248894972452931, 1//3), Cashflow{Float64, Rational{Int64}}(-0.0053464662084109875, 5//12), Cashflow{Float64, Rational{Int64}}(-0.005443845753613525, 1//2), Cashflow{Float64, Rational{Int64}}(-0.005540343217294943, 7//12), Cashflow{Float64, Rational{Int64}}(-0.0056352103608925804, 2//3), Cashflow{Float64, Rational{Int64}}(-0.005727640395032597, 3//4), Cashflow{Float64, Rational{Int64}}(-0.0058167680149368624, 5//6)  …  Cashflow{Float64, Rational{Int64}}(-0.00448186407908259, 45//4), Cashflow{Float64, Rational{Int64}}(-0.004420023164615087, 34//3), Cashflow{Float64, Rational{Int64}}(-0.004359312015469468, 137//12), Cashflow{Float64, Rational{Int64}}(-0.004299710096059417, 23//2), Cashflow{Float64, Rational{Int64}}(-0.004241197285320075, 139//12), Cashflow{Float64, Rational{Int64}}(-0.0041837538685546224, 35//3), Cashflow{Float64, Rational{Int64}}(-0.0041273605294409816, 47//4), Cashflow{Float64, Rational{Int64}}(-0.0040719983421960235, 71//6), Cashflow{Float64, Rational{Int64}}(-0.004017648763896468, 143//12), Cashflow{Float64, Rational{Int64}}(-0.3913747165242466, 12//1)]
 [Cashflow{Float64, Rational{Int64}}(-0.004988088840310257, 1//12), Cashflow{Float64, Rational{Int64}}(-0.0050771918288985704, 1//6), Cashflow{Float64, Rational{Int64}}(-0.005167564134824113, 1//4), Cashflow{Float64, Rational{Int64}}(-0.005258711609765578, 1//3), Cashflow{Float64, Rational{Int64}}(-0.005350095098399249, 5//12), Cashflow{Float64, Rational{Int64}}(-0.005441128704157761, 1//2), Cashflow{Float64, Rational{Int64}}(-0.0055311784406072055, 7//12), Cashflow{Float64, Rational{Int64}}(-0.005619561337812729, 2//3), Cashflow{Float64, Rational{Int64}}(-0.005705545077075528, 3//4), Cashflow{Float64, Rational{Int64}}(-0.005788348231031512, 5//6)  …  Cashflow{Float64, Rational{Int64}}(-0.0038503833423463286, 49//4), Cashflow{Float64, Rational{Int64}}(-0.003804934322852083, 37//3), Cashflow{Float64, Rational{Int64}}(-0.0037602800042857328, 149//12), Cashflow{Float64, Rational{Int64}}(-0.0037164078333552157, 25//2), Cashflow{Float64, Rational{Int64}}(-0.0036733054971266055, 151//12), Cashflow{Float64, Rational{Int64}}(-0.003630960918852027, 38//3), Cashflow{Float64, Rational{Int64}}(-0.0035893622538848845, 51//4), Cashflow{Float64, Rational{Int64}}(-0.0035484978856863665, 77//6), Cashflow{Float64, Rational{Int64}}(-0.003508356421914768, 155//12), Cashflow{Float64, Rational{Int64}}(-0.42567536516657395, 13//1)]
 [Cashflow{Float64, Rational{Int64}}(-0.004989026914463392, 1//12), Cashflow{Float64, Rational{Int64}}(-0.005072503613158184, 1//6), Cashflow{Float64, Rational{Int64}}(-0.005157007583915993, 1//4), Cashflow{Float64, Rational{Int64}}(-0.005242076337285006, 1//3), Cashflow{Float64, Rational{Int64}}(-0.0053272085834838116, 5//12), Cashflow{Float64, Rational{Int64}}(-0.005411862934794599, 1//2), Cashflow{Float64, Rational{Int64}}(-0.0054954569390823806, 7//12), Cashflow{Float64, Rational{Int64}}(-0.00557736649771718, 2//3), Cashflow{Float64, Rational{Int64}}(-0.005656925723839015, 3//4), Cashflow{Float64, Rational{Int64}}(-0.005733427299288154, 5//6)  …  Cashflow{Float64, Rational{Int64}}(-0.003404660033356735, 53//4), Cashflow{Float64, Rational{Int64}}(-0.0033708218012461827, 40//3), Cashflow{Float64, Rational{Int64}}(-0.003337581040356867, 161//12), Cashflow{Float64, Rational{Int64}}(-0.0033049298363434097, 27//2), Cashflow{Float64, Rational{Int64}}(-0.003272860428437546, 163//12), Cashflow{Float64, Rational{Int64}}(-0.0032413652074448113, 41//3), Cashflow{Float64, Rational{Int64}}(-0.003210436713800933, 55//4), Cashflow{Float64, Rational{Int64}}(-0.003180067635691048, 83//6), Cashflow{Float64, Rational{Int64}}(-0.0031502508072285204, 167//12), Cashflow{Float64, Rational{Int64}}(-0.4672090174521514, 14//1)]
 [Cashflow{Float64, Rational{Int64}}(-0.004144410382128681, 1//12), Cashflow{Float64, Rational{Int64}}(-0.004315318017074089, 1//6), Cashflow{Float64, Rational{Int64}}(-0.004504093438676603, 1//4), Cashflow{Float64, Rational{Int64}}(-0.00471071623903657, 1//3), Cashflow{Float64, Rational{Int64}}(-0.004934713168882003, 5//12), Cashflow{Float64, Rational{Int64}}(-0.0051749370878303915, 1//2), Cashflow{Float64, Rational{Int64}}(-0.005429266292843056, 7//12), Cashflow{Float64, Rational{Int64}}(-0.005694197226230727, 2//3), Cashflow{Float64, Rational{Int64}}(-0.005964295051005587, 3//4), Cashflow{Float64, Rational{Int64}}(-0.006231455978636931, 5//6)  …  Cashflow{Float64, Rational{Int64}}(-0.007040339849056201, 5//4), Cashflow{Float64, Rational{Int64}}(-0.0070309918151435, 4//3), Cashflow{Float64, Rational{Int64}}(-0.006960391349463677, 17//12), Cashflow{Float64, Rational{Int64}}(-0.006830278507743372, 3//2), Cashflow{Float64, Rational{Int64}}(-0.006643759757520943, 19//12), Cashflow{Float64, Rational{Int64}}(-0.006405206284366148, 5//3), Cashflow{Float64, Rational{Int64}}(-0.006120116156813356, 7//4), Cashflow{Float64, Rational{Int64}}(-0.0057949431751937105, 11//6), Cashflow{Float64, Rational{Int64}}(-0.005436895608924536, 23//12), Cashflow{Float64, Rational{Int64}}(-0.9015948971297525, 2//1)]
 [Cashflow{Float64, Rational{Int64}}(-0.004144410382128681, 1//12), Cashflow{Float64, Rational{Int64}}(-0.004315318017074089, 1//6), Cashflow{Float64, Rational{Int64}}(-0.004504093438676603, 1//4), Cashflow{Float64, Rational{Int64}}(-0.00471071623903657, 1//3), Cashflow{Float64, Rational{Int64}}(-0.004934713168882003, 5//12), Cashflow{Float64, Rational{Int64}}(-0.0051749370878303915, 1//2), Cashflow{Float64, Rational{Int64}}(-0.005429266292843056, 7//12), Cashflow{Float64, Rational{Int64}}(-0.005694197226230727, 2//3), Cashflow{Float64, Rational{Int64}}(-0.005964295051005587, 3//4), Cashflow{Float64, Rational{Int64}}(-0.006231455978636931, 5//6)  …  Cashflow{Float64, Rational{Int64}}(-0.007040339849056201, 5//4), Cashflow{Float64, Rational{Int64}}(-0.0070309918151435, 4//3), Cashflow{Float64, Rational{Int64}}(-0.006960391349463677, 17//12), Cashflow{Float64, Rational{Int64}}(-0.006830278507743372, 3//2), Cashflow{Float64, Rational{Int64}}(-0.006643759757520943, 19//12), Cashflow{Float64, Rational{Int64}}(-0.006405206284366148, 5//3), Cashflow{Float64, Rational{Int64}}(-0.006120116156813356, 7//4), Cashflow{Float64, Rational{Int64}}(-0.0057949431751937105, 11//6), Cashflow{Float64, Rational{Int64}}(-0.005436895608924536, 23//12), Cashflow{Float64, Rational{Int64}}(-0.9015948971297525, 2//1)]
 [Cashflow{Float64, Rational{Int64}}(-0.004891215295878957, 1//12), Cashflow{Float64, Rational{Int64}}(-0.004958966692784059, 1//6), Cashflow{Float64, Rational{Int64}}(-0.005027230356527339, 1//4), Cashflow{Float64, Rational{Int64}}(-0.005095628499248275, 1//3), Cashflow{Float64, Rational{Int64}}(-0.005163757905081053, 5//12), Cashflow{Float64, Rational{Int64}}(-0.0052311893693252655, 1//2), Cashflow{Float64, Rational{Int64}}(-0.0052974673461451335, 7//12), Cashflow{Float64, Rational{Int64}}(-0.005362109830380762, 2//3), Cashflow{Float64, Rational{Int64}}(-0.0054246084999782515, 3//4), Cashflow{Float64, Rational{Int64}}(-0.005484429146402067, 5//6)  …  Cashflow{Float64, Rational{Int64}}(-0.00281549132409045, 65//4), Cashflow{Float64, Rational{Int64}}(-0.0028050122556286493, 49//3), Cashflow{Float64, Rational{Int64}}(-0.00279493303752313, 197//12), Cashflow{Float64, Rational{Int64}}(-0.0027852535075186717, 33//2), Cashflow{Float64, Rational{Int64}}(-0.0027759736354747693, 199//12), Cashflow{Float64, Rational{Int64}}(-0.002767093526022738, 50//3), Cashflow{Float64, Rational{Int64}}(-0.0027586134213474886, 67//4), Cashflow{Float64, Rational{Int64}}(-0.0027505337040978593, 101//6), Cashflow{Float64, Rational{Int64}}(-0.002742854900433503, 203//12), Cashflow{Float64, Rational{Int64}}(-0.6089189939944261, 17//1)]
 [Cashflow{Float64, Rational{Int64}}(-0.00484668200548985, 1//12), Cashflow{Float64, Rational{Int64}}(-0.0049100274641540355, 1//6), Cashflow{Float64, Rational{Int64}}(-0.004973771861878092, 1//4), Cashflow{Float64, Rational{Int64}}(-0.005037561154641696, 1//3), Cashflow{Float64, Rational{Int64}}(-0.005101018961600112, 5//12), Cashflow{Float64, Rational{Int64}}(-0.00516374613478892, 1//2), Cashflow{Float64, Rational{Int64}}(-0.005225320508363633, 7//12), Cashflow{Float64, Rational{Int64}}(-0.00528529684781004, 2//3), Cashflow{Float64, Rational{Int64}}(-0.005343207020245248, 3//4), Cashflow{Float64, Rational{Int64}}(-0.005398560407581078, 5//6)  …  Cashflow{Float64, Rational{Int64}}(-0.0028498899525714987, 69//4), Cashflow{Float64, Rational{Int64}}(-0.0028466477109700184, 52//3), Cashflow{Float64, Rational{Int64}}(-0.0028438381970336854, 209//12), Cashflow{Float64, Rational{Int64}}(-0.0028414643370289795, 35//2), Cashflow{Float64, Rational{Int64}}(-0.0028395292581497736, 211//12), Cashflow{Float64, Rational{Int64}}(-0.0028380362943343424, 53//3), Cashflow{Float64, Rational{Int64}}(-0.002836988992328804, 71//4), Cashflow{Float64, Rational{Int64}}(-0.002836391118006782, 107//6), Cashflow{Float64, Rational{Int64}}(-0.0028362466629578338, 215//12), Cashflow{Float64, Rational{Int64}}(-0.6510276535254121, 18//1)]
 [Cashflow{Float64, Rational{Int64}}(-0.004891215295878957, 1//12), Cashflow{Float64, Rational{Int64}}(-0.004958966692784059, 1//6), Cashflow{Float64, Rational{Int64}}(-0.005027230356527339, 1//4), Cashflow{Float64, Rational{Int64}}(-0.005095628499248275, 1//3), Cashflow{Float64, Rational{Int64}}(-0.005163757905081053, 5//12), Cashflow{Float64, Rational{Int64}}(-0.0052311893693252655, 1//2), Cashflow{Float64, Rational{Int64}}(-0.0052974673461451335, 7//12), Cashflow{Float64, Rational{Int64}}(-0.005362109830380762, 2//3), Cashflow{Float64, Rational{Int64}}(-0.0054246084999782515, 3//4), Cashflow{Float64, Rational{Int64}}(-0.005484429146402067, 5//6)  …  Cashflow{Float64, Rational{Int64}}(-0.00281549132409045, 65//4), Cashflow{Float64, Rational{Int64}}(-0.0028050122556286493, 49//3), Cashflow{Float64, Rational{Int64}}(-0.00279493303752313, 197//12), Cashflow{Float64, Rational{Int64}}(-0.0027852535075186717, 33//2), Cashflow{Float64, Rational{Int64}}(-0.0027759736354747693, 199//12), Cashflow{Float64, Rational{Int64}}(-0.002767093526022738, 50//3), Cashflow{Float64, Rational{Int64}}(-0.0027586134213474886, 67//4), Cashflow{Float64, Rational{Int64}}(-0.0027505337040978593, 101//6), Cashflow{Float64, Rational{Int64}}(-0.002742854900433503, 203//12), Cashflow{Float64, Rational{Int64}}(-0.6089189939944261, 17//1)]
 [Cashflow{Float64, Rational{Int64}}(-0.004783377181503488, 1//12), Cashflow{Float64, Rational{Int64}}(-0.004839628938067403, 1//6), Cashflow{Float64, Rational{Int64}}(-0.004896115454239143, 1//4), Cashflow{Float64, Rational{Int64}}(-0.004952522042686588, 1//3), Cashflow{Float64, Rational{Int64}}(-0.005008516357927531, 5//12), Cashflow{Float64, Rational{Int64}}(-0.005063748136459031, 1//2), Cashflow{Float64, Rational{Int64}}(-0.005117849072112032, 7//12), Cashflow{Float64, Rational{Int64}}(-0.005170432840042458, 2//3), Cashflow{Float64, Rational{Int64}}(-0.005221095283163374, 3//4), Cashflow{Float64, Rational{Int64}}(-0.0052694147752224245, 5//6)  …  Cashflow{Float64, Rational{Int64}}(-0.0034052147073842754, 77//4), Cashflow{Float64, Rational{Int64}}(-0.0034222845055589825, 58//3), Cashflow{Float64, Rational{Int64}}(-0.0034401086494695596, 233//12), Cashflow{Float64, Rational{Int64}}(-0.003458703030723001, 39//2), Cashflow{Float64, Rational{Int64}}(-0.003478084148257083, 235//12), Cashflow{Float64, Rational{Int64}}(-0.0034982691293209437, 59//3), Cashflow{Float64, Rational{Int64}}(-0.0035192757511736806, 79//4), Cashflow{Float64, Rational{Int64}}(-0.0035411224635144953, 119//6), Cashflow{Float64, Rational{Int64}}(-0.003563828411657379, 239//12), Cashflow{Float64, Rational{Int64}}(-0.697536591529004, 20//1)]
 ⋮
 [Cashflow{Float64, Rational{Int64}}(-0.004424258662627023, 1//12), Cashflow{Float64, Rational{Int64}}(-0.004531287849794747, 1//6), Cashflow{Float64, Rational{Int64}}(-0.004641425855601325, 1//4), Cashflow{Float64, Rational{Int64}}(-0.004753930805967418, 1//3), Cashflow{Float64, Rational{Int64}}(-0.004867946524016453, 5//12), Cashflow{Float64, Rational{Int64}}(-0.004982492484160319, 1//2), Cashflow{Float64, Rational{Int64}}(-0.005096454586812677, 7//12), Cashflow{Float64, Rational{Int64}}(-0.005208577175287752, 2//3), Cashflow{Float64, Rational{Int64}}(-0.005317456788965952, 3//4), Cashflow{Float64, Rational{Int64}}(-0.005421538222340484, 5//6)  …  Cashflow{Float64, Rational{Int64}}(-0.010779718099495577, 25//4), Cashflow{Float64, Rational{Int64}}(-0.010707757626314915, 19//3), Cashflow{Float64, Rational{Int64}}(-0.01063260474415043, 77//12), Cashflow{Float64, Rational{Int64}}(-0.010554332167721845, 13//2), Cashflow{Float64, Rational{Int64}}(-0.01047301604582292, 79//12), Cashflow{Float64, Rational{Int64}}(-0.010388735819327524, 20//3), Cashflow{Float64, Rational{Int64}}(-0.010301574071366447, 27//4), Cashflow{Float64, Rational{Int64}}(-0.010211616370121152, 41//6), Cashflow{Float64, Rational{Int64}}(-0.010118951104718079, 83//12), Cashflow{Float64, Rational{Int64}}(-0.48575081864113745, 7//1)]
 [Cashflow{Float64, Rational{Int64}}(-0.00469206916013884, 1//12), Cashflow{Float64, Rational{Int64}}(-0.004797944275763207, 1//6), Cashflow{Float64, Rational{Int64}}(-0.00490632362599739, 1//4), Cashflow{Float64, Rational{Int64}}(-0.0050165731210517475, 1//3), Cashflow{Float64, Rational{Int64}}(-0.005127976422810296, 5//12), Cashflow{Float64, Rational{Int64}}(-0.005239729406817578, 1//2), Cashflow{Float64, Rational{Int64}}(-0.00535093528564478, 7//12), Cashflow{Float64, Rational{Int64}}(-0.005460600611106549, 2//3), Cashflow{Float64, Rational{Int64}}(-0.005567632398211778, 3//4), Cashflow{Float64, Rational{Int64}}(-0.0056708366385607965, 5//6)  …  Cashflow{Float64, Rational{Int64}}(-0.008258197444484435, 33//4), Cashflow{Float64, Rational{Int64}}(-0.008152106726561411, 25//3), Cashflow{Float64, Rational{Int64}}(-0.008044931783812803, 101//12), Cashflow{Float64, Rational{Int64}}(-0.00793675620948152, 17//2), Cashflow{Float64, Rational{Int64}}(-0.007827663182837102, 103//12), Cashflow{Float64, Rational{Int64}}(-0.007717735341455881, 26//3), Cashflow{Float64, Rational{Int64}}(-0.007607054657307641, 35//4), Cashflow{Float64, Rational{Int64}}(-0.007495702316979209, 53//6), Cashflow{Float64, Rational{Int64}}(-0.0073837586063478095, 107//12), Cashflow{Float64, Rational{Int64}}(-0.37840410914288675, 9//1)]
 [Cashflow{Float64, Rational{Int64}}(-0.004557385177676505, 1//12), Cashflow{Float64, Rational{Int64}}(-0.004664387545950275, 1//6), Cashflow{Float64, Rational{Int64}}(-0.0047742008193853965, 1//4), Cashflow{Float64, Rational{Int64}}(-0.00488614452367044, 1//3), Cashflow{Float64, Rational{Int64}}(-0.004999441950790049, 5//12), Cashflow{Float64, Rational{Int64}}(-0.0051132127602090665, 1//2), Cashflow{Float64, Rational{Int64}}(-0.005226466320959016, 7//12), Cashflow{Float64, Rational{Int64}}(-0.005338096093052123, 2//3), Cashflow{Float64, Rational{Int64}}(-0.005446875388809937, 3//4), Cashflow{Float64, Rational{Int64}}(-0.005551454897149994, 5//6)  …  Cashflow{Float64, Rational{Int64}}(-0.009612003204621645, 29//4), Cashflow{Float64, Rational{Int64}}(-0.009517691790623088, 22//3), Cashflow{Float64, Rational{Int64}}(-0.009421239317538287, 89//12), Cashflow{Float64, Rational{Int64}}(-0.009322735198487444, 15//2), Cashflow{Float64, Rational{Int64}}(-0.009222270238825802, 91//12), Cashflow{Float64, Rational{Int64}}(-0.009119936465989032, 23//3), Cashflow{Float64, Rational{Int64}}(-0.009015826957992887, 31//4), Cashflow{Float64, Rational{Int64}}(-0.008910035671122777, 47//6), Cashflow{Float64, Rational{Int64}}(-0.008802657267349105, 95//12), Cashflow{Float64, Rational{Int64}}(-0.4224849041397684, 8//1)]
 [Cashflow{Float64, Rational{Int64}}(-0.004989026914463392, 1//12), Cashflow{Float64, Rational{Int64}}(-0.005072503613158184, 1//6), Cashflow{Float64, Rational{Int64}}(-0.005157007583915993, 1//4), Cashflow{Float64, Rational{Int64}}(-0.005242076337285006, 1//3), Cashflow{Float64, Rational{Int64}}(-0.0053272085834838116, 5//12), Cashflow{Float64, Rational{Int64}}(-0.005411862934794599, 1//2), Cashflow{Float64, Rational{Int64}}(-0.0054954569390823806, 7//12), Cashflow{Float64, Rational{Int64}}(-0.00557736649771718, 2//3), Cashflow{Float64, Rational{Int64}}(-0.005656925723839015, 3//4), Cashflow{Float64, Rational{Int64}}(-0.005733427299288154, 5//6)  …  Cashflow{Float64, Rational{Int64}}(-0.003404660033356735, 53//4), Cashflow{Float64, Rational{Int64}}(-0.0033708218012461827, 40//3), Cashflow{Float64, Rational{Int64}}(-0.003337581040356867, 161//12), Cashflow{Float64, Rational{Int64}}(-0.0033049298363434097, 27//2), Cashflow{Float64, Rational{Int64}}(-0.003272860428437546, 163//12), Cashflow{Float64, Rational{Int64}}(-0.0032413652074448113, 41//3), Cashflow{Float64, Rational{Int64}}(-0.003210436713800933, 55//4), Cashflow{Float64, Rational{Int64}}(-0.003180067635691048, 83//6), Cashflow{Float64, Rational{Int64}}(-0.0031502508072285204, 167//12), Cashflow{Float64, Rational{Int64}}(-0.4672090174521514, 14//1)]
 [Cashflow{Float64, Rational{Int64}}(-0.00484668200548985, 1//12), Cashflow{Float64, Rational{Int64}}(-0.0049100274641540355, 1//6), Cashflow{Float64, Rational{Int64}}(-0.004973771861878092, 1//4), Cashflow{Float64, Rational{Int64}}(-0.005037561154641696, 1//3), Cashflow{Float64, Rational{Int64}}(-0.005101018961600112, 5//12), Cashflow{Float64, Rational{Int64}}(-0.00516374613478892, 1//2), Cashflow{Float64, Rational{Int64}}(-0.005225320508363633, 7//12), Cashflow{Float64, Rational{Int64}}(-0.00528529684781004, 2//3), Cashflow{Float64, Rational{Int64}}(-0.005343207020245248, 3//4), Cashflow{Float64, Rational{Int64}}(-0.005398560407581078, 5//6)  …  Cashflow{Float64, Rational{Int64}}(-0.0028498899525714987, 69//4), Cashflow{Float64, Rational{Int64}}(-0.0028466477109700184, 52//3), Cashflow{Float64, Rational{Int64}}(-0.0028438381970336854, 209//12), Cashflow{Float64, Rational{Int64}}(-0.0028414643370289795, 35//2), Cashflow{Float64, Rational{Int64}}(-0.0028395292581497736, 211//12), Cashflow{Float64, Rational{Int64}}(-0.0028380362943343424, 53//3), Cashflow{Float64, Rational{Int64}}(-0.002836988992328804, 71//4), Cashflow{Float64, Rational{Int64}}(-0.002836391118006782, 107//6), Cashflow{Float64, Rational{Int64}}(-0.0028362466629578338, 215//12), Cashflow{Float64, Rational{Int64}}(-0.6510276535254121, 18//1)]
 [Cashflow{Float64, Rational{Int64}}(-0.004808083969809224, 1//12), Cashflow{Float64, Rational{Int64}}(-0.004867578774020018, 1//6), Cashflow{Float64, Rational{Int64}}(-0.004927381341597908, 1//4), Cashflow{Float64, Rational{Int64}}(-0.004987158725210008, 1//3), Cashflow{Float64, Rational{Int64}}(-0.005046558205980105, 5//12), Cashflow{Float64, Rational{Int64}}(-0.005105206960483149, 1//2), Cashflow{Float64, Rational{Int64}}(-0.005162711883061914, 7//12), Cashflow{Float64, Rational{Int64}}(-0.005218659579946738, 2//3), Cashflow{Float64, Rational{Int64}}(-0.0052726165521757825, 3//4), Cashflow{Float64, Rational{Int64}}(-0.0053241295848203765, 5//6)  …  Cashflow{Float64, Rational{Int64}}(-0.0030295443688923046, 73//4), Cashflow{Float64, Rational{Int64}}(-0.00303493417322771, 55//3), Cashflow{Float64, Rational{Int64}}(-0.0030408607686393274, 221//12), Cashflow{Float64, Rational{Int64}}(-0.0030473319110749316, 37//2), Cashflow{Float64, Rational{Int64}}(-0.003054355701028045, 223//12), Cashflow{Float64, Rational{Int64}}(-0.003061940595233296, 56//3), Cashflow{Float64, Rational{Int64}}(-0.003070095418840538, 75//4), Cashflow{Float64, Rational{Int64}}(-0.0030788293780883934, 113//6), Cashflow{Float64, Rational{Int64}}(-0.003088152073497882, 227//12), Cashflow{Float64, Rational{Int64}}(-0.6829248274912895, 19//1)]
 [Cashflow{Float64, Rational{Int64}}(-0.004934341012856126, 1//12), Cashflow{Float64, Rational{Int64}}(-0.005006985379965156, 1//6), Cashflow{Float64, Rational{Int64}}(-0.005080281729186395, 1//4), Cashflow{Float64, Rational{Int64}}(-0.0051538260967086925, 1//3), Cashflow{Float64, Rational{Int64}}(-0.00522718538625404, 5//12), Cashflow{Float64, Rational{Int64}}(-0.00529989663220021, 1//2), Cashflow{Float64, Rational{Int64}}(-0.005371466505717817, 7//12), Cashflow{Float64, Rational{Int64}}(-0.005441371096268787, 2//3), Cashflow{Float64, Rational{Int64}}(-0.0055090560020900166, 3//4), Cashflow{Float64, Rational{Int64}}(-0.005573936764447176, 5//6)  …  Cashflow{Float64, Rational{Int64}}(-0.0029003255089834427, 61//4), Cashflow{Float64, Rational{Int64}}(-0.002882910770566691, 46//3), Cashflow{Float64, Rational{Int64}}(-0.0028659134998877214, 185//12), Cashflow{Float64, Rational{Int64}}(-0.0028493311657012427, 31//2), Cashflow{Float64, Rational{Int64}}(-0.0028331613441325196, 187//12), Cashflow{Float64, Rational{Int64}}(-0.002817401719508703, 47//3), Cashflow{Float64, Rational{Int64}}(-0.0028020500852618748, 63//4), Cashflow{Float64, Rational{Int64}}(-0.002787104344907078, 95//6), Cashflow{Float64, Rational{Int64}}(-0.0027725625130955237, 191//12), Cashflow{Float64, Rational{Int64}}(-0.5617474374389465, 16//1)]
 [Cashflow{Float64, Rational{Int64}}(-0.004969120405811204, 1//12), Cashflow{Float64, Rational{Int64}}(-0.005047050365765408, 1//6), Cashflow{Float64, Rational{Int64}}(-0.005125802688680543, 1//4), Cashflow{Float64, Rational{Int64}}(-0.005204945068597846, 1//3), Cashflow{Float64, Rational{Int64}}(-0.005284011648055461, 5//12), Cashflow{Float64, Rational{Int64}}(-0.005362502043055925, 1//2), Cashflow{Float64, Rational{Int64}}(-0.005439880651722219, 7//12), Cashflow{Float64, Rational{Int64}}(-0.005515576287951582, 2//3), Cashflow{Float64, Rational{Int64}}(-0.005588982183192956, 3//4), Cashflow{Float64, Rational{Int64}}(-0.005659456401106093, 5//6)  …  Cashflow{Float64, Rational{Int64}}(-0.003095135904573605, 57//4), Cashflow{Float64, Rational{Int64}}(-0.0030702310298667437, 43//3), Cashflow{Float64, Rational{Int64}}(-0.003045806631755013, 173//12), Cashflow{Float64, Rational{Int64}}(-0.0030218578210388625, 29//2), Cashflow{Float64, Rational{Int64}}(-0.0029983798227155747, 175//12), Cashflow{Float64, Rational{Int64}}(-0.0029753679754469797, 44//3), Cashflow{Float64, Rational{Int64}}(-0.0029528177310823808, 59//4), Cashflow{Float64, Rational{Int64}}(-0.0029307246542385214, 89//6), Cashflow{Float64, Rational{Int64}}(-0.0029090844219367973, 179//12), Cashflow{Float64, Rational{Int64}}(-0.513456896562433, 15//1)]
 [Cashflow{Float64, Rational{Int64}}(-0.00484668200548985, 1//12), Cashflow{Float64, Rational{Int64}}(-0.0049100274641540355, 1//6), Cashflow{Float64, Rational{Int64}}(-0.004973771861878092, 1//4), Cashflow{Float64, Rational{Int64}}(-0.005037561154641696, 1//3), Cashflow{Float64, Rational{Int64}}(-0.005101018961600112, 5//12), Cashflow{Float64, Rational{Int64}}(-0.00516374613478892, 1//2), Cashflow{Float64, Rational{Int64}}(-0.005225320508363633, 7//12), Cashflow{Float64, Rational{Int64}}(-0.00528529684781004, 2//3), Cashflow{Float64, Rational{Int64}}(-0.005343207020245248, 3//4), Cashflow{Float64, Rational{Int64}}(-0.005398560407581078, 5//6)  …  Cashflow{Float64, Rational{Int64}}(-0.0028498899525714987, 69//4), Cashflow{Float64, Rational{Int64}}(-0.0028466477109700184, 52//3), Cashflow{Float64, Rational{Int64}}(-0.0028438381970336854, 209//12), Cashflow{Float64, Rational{Int64}}(-0.0028414643370289795, 35//2), Cashflow{Float64, Rational{Int64}}(-0.0028395292581497736, 211//12), Cashflow{Float64, Rational{Int64}}(-0.0028380362943343424, 53//3), Cashflow{Float64, Rational{Int64}}(-0.002836988992328804, 71//4), Cashflow{Float64, Rational{Int64}}(-0.002836391118006782, 107//6), Cashflow{Float64, Rational{Int64}}(-0.0028362466629578338, 215//12), Cashflow{Float64, Rational{Int64}}(-0.6510276535254121, 18//1)]

And running consolidate groups the cashflows into timepoint => amount pairs.

consolidate(cashflow_vectors)

Transducers.GroupByViewDict{Rational{Int64},Float64,…}(...):
  20//3   => -577.461
  125//12 => -241.577
  29//4   => -499.886
  229//12 => -16.4751
  9//4    => -570.855
  71//4   => -44.7842
  10//3   => -712.354
  109//6  => -31.0602
  95//12  => -466.042
  19//6   => -688.738
  43//6   => -503.763
  175//12 => -93.8602
  143//12 => -178.893
  12      => -2142.5
  5//3    => -565.424
  19//4   => -747.042
  199//12 => -58.8193
  187//12 => -74.7678
  13//4   => -700.811
  ⋮       => ⋮

Here’s a visualization of the liability cashflows, showing that when the interest rates are bumped up slightly, that there is more surrenders that occur earlier on (so there’s fewer policies around at the time of each maturity). Negative cashflows are outflows:

let
    d = consolidate([p(c) for p in liabilities])
    ks = collect(keys(d)) |> sort!
    vs = [d[k] for k in ks]

    c2 = curve(zeros .+ 0.005, times)
    d2 = consolidate([p(c2) for p in liabilities])
    ks2 = collect(keys(d2)) |> sort!
    vs2 = [d2[k] for k in ks2]

    f = Figure(size = (900, 600))
    ax = Axis(f[1, 1], 
        xlabel = "Time (Years)",
        ylabel = "Cashflow Amount (cumulative)",
        title = "Cumulative Liability Cashflows: Base vs +50bp Rate Shock",
    )

    lines!(ax, ks, cumsum(vs), label = "Base Scenario")

    lines!(ax, ks2, cumsum(vs2), label = "+50bp Rate Shock")

    axislegend(ax, position = :rb)

    f
end

In the upwards shaped yield curve, without a surrender charge or market value adjustment, many mid-to-late-duration policyholders elect to surrender instead of hold to maturity.

31.5 Computing Derivatives with Autodiff

Rather than approximating derivatives through finite differences, autodiff computes exact values, gradients, and Hessians. The concepts and background are covered in Chapter 16.

The value_gradient_and_hessian function returns the present value, key rate durations (gradient), and convexities (Hessian diagonal) for the entire liability portfolio. We compute similar derivatives for each potential asset.

vgh_liab = let
    value_gradient_and_hessian(z -> annuities(z, liabilities), AutoForwardDiff(), zeros)
end

(-102337.8165948909, [11504.297366155115, 29870.011138629456, 29672.160372787308, 256573.36315787124, 122773.8683882791, 6317.157272228895], [-166687.66920132237 195891.40237708762 … 11057.700186961323 -42919.17049380064; 195891.40237708736 -1.6769719809453504e6 … 2.4755330051117623e6 1.2651095074186337e6; … ; 11057.700186961241 2.475533005111765e6 … -3.717194044503879e7 -2.7874235298093623e6; -42919.17049380067 1.2651095074186337e6 … -2.7874235298093683e6 -2.1251952156862594e7])

31.5.1 Gradients and Hessians in ALM

Let’s dive into the results here a little bit.

The first element of vgh_liab is the value of the liability portfolio using the yield curve constructed earlier:

vgh_liab[1]

-102337.8165948909

The second element of vgh_liab is the partial derivative with respect to each of the inputs (here, just the zeros rates that dictate the curve). The sum of the partials is the effective duration of the liabilities.

@show sum(vgh_liab[2])
vgh_liab[2]

sum(vgh_liab[2]) = 456710.8576959511

6-element Vector{Float64}:
  11504.297366155115
  29870.011138629456
  29672.160372787308
 256573.36315787124
 122773.8683882791
   6317.157272228895

This is the sensitivity relative to a full unit change in rates (e.g. 1.0). So if we wanted to estimate the dollar impact of a 50bps change, we would take 0.005 times the gradient/hessian. Also note these are ‘dollar durations’ but we could divide by the price to get effective or percentage durations:

-sum(vgh_liab[2]) / vgh_liab[1]

4.462777034845904

Additionally, note that this is the dynamic duration of the liabilities, not the static duration which ignores the effect of the interest-sensitive behavior of the liabilities.

let 
    dynamic(zeros) = value(curve(zeros,times),liabilities[1])
    cfs = liabilities[1](c)
    static(zeros) = let
        c = curve(zeros,times)
        # note that `cfs` are defined outside of the function, so 
        # will not change as the curve is sensitized
        mapreduce(cf -> c(cf.time) * cf.amount,+,cfs)
    end

    @show gradient(dynamic,AutoForwardDiff(),zeros) |> sum
    @show gradient(static,AutoForwardDiff(),zeros) |> sum
end

gradient(dynamic, AutoForwardDiff(), zeros) |> sum = 1.8285208753189608
gradient(static, AutoForwardDiff(), zeros) |> sum = 1.8740344387028733

1.8740344387028733

Due to the steepness of the surrender function, the policy exiting sooner, on average, results in a higher change in value than if the policy was not sensitive to the change in rates. The increase in value from earlier cashflows outweighs the greater discount rate.

The third element of vgh_liab is the Hessian matrix, containing all second partial derivatives with respect to the yield curve inputs:

vgh_liab[3]

6×6 Matrix{Float64}:
     -1.66688e5   1.95891e5  -2.19114e5  …  11057.7        -42919.2
      1.95891e5  -1.67697e6   2.87702e6         2.47553e6       1.26511e6
     -2.19114e5   2.87702e6  -9.91323e6        -4.19155e6      -2.75844e6
      6.54377e5  -3.32406e6   1.26992e7         1.88812e7       8.07891e6
  11057.7         2.47553e6  -4.19155e6        -3.71719e7      -2.78742e6
 -42919.2         1.26511e6  -2.75844e6  …     -2.78742e6      -2.1252e7

This matrix captures the convexity characteristics of the liability portfolio. The diagonal elements represent “key rate convexities”—how much the duration at each key rate changes as that specific rate moves:

@show diag(vgh_liab[3])
@show sum(diag(vgh_liab[3]))  # Total dollar convexity

diag(vgh_liab[3]) = [-166687.66920132237, -1.6769719809453504e6, -9.91322820830571e6, -2.8656568187110033e7, -3.717194044503879e7, -2.1251952156862594e7]
sum(diag(vgh_liab[3])) = -9.88373486474638e7

-9.88373486474638e7

Like duration, we can convert dollar convexity to percentage convexity by dividing by the portfolio value:

sum(diag(vgh_liab[3])) / vgh_liab[1]

965.7949713615264

The off-diagonal elements show cross-convexities—how the sensitivity to one key rate changes when a different key rate moves. For most portfolios, these cross-terms are smaller than the diagonal terms but can be significant for complex instruments.

This convexity measurement is also dynamic, capturing how the surrender behavior changes the second-order interest rate sensitivity:

let 
    dynamic(zeros) = value(curve(zeros,times),liabilities[1])
    cfs = liabilities[1](c)
    static(zeros) = let
        c = curve(zeros,times)
        mapreduce(cf -> c(cf.time) * cf.amount,+,cfs)
    end

    @show hessian(dynamic,AutoForwardDiff(),zeros) |> diag |> sum
    @show hessian(static,AutoForwardDiff(),zeros) |> diag |> sum
end

(hessian(dynamic, AutoForwardDiff(), zeros) |> diag) |> sum = -57.707315937727834
(hessian(static, AutoForwardDiff(), zeros) |> diag) |> sum = -3.653003043786183

-3.653003043786183

The dynamic convexity differs from static convexity because the surrender function creates path-dependent behavior. As rates change, not only do the discount factors change, but the timing and magnitude of cashflows shift as well. This interaction between discount rate changes and cashflow timing changes produces the additional convexity captured in the dynamic measurement. Note how the convexity is larger in the dynamic case.

For ALM purposes, this convexity information helps quantify how well a duration-matched hedge will perform under large rate movements.

31.6 Optimizing an Asset Portfolio

31.6.1 Define Asset Universe

We will create a set of par bonds and select a portfolio of assets that matches the liabilities, subject to duration and KRD constraints:

asset_universe = [
    FixedBond(par_yield(c,t,4),t,4)
    for t in 1:20
]

20-element Vector{FixedBond{Float64, Int64, Int64}}:
 FixedBond{Float64, Int64, Int64}(0.009998004795647176, 1, 4)
 FixedBond{Float64, Int64, Int64}(0.019932158064569137, 2, 4)
 FixedBond{Float64, Int64, Int64}(0.01997170973543043, 3, 4)
 FixedBond{Float64, Int64, Int64}(0.02388292451655035, 4, 4)
 FixedBond{Float64, Int64, Int64}(0.02952635925170046, 5, 4)
 FixedBond{Float64, Int64, Int64}(0.03446708593197626, 6, 4)
 FixedBond{Float64, Int64, Int64}(0.038602669873830896, 7, 4)
 FixedBond{Float64, Int64, Int64}(0.04207416430705521, 8, 4)
 FixedBond{Float64, Int64, Int64}(0.04493178862970366, 9, 4)
 FixedBond{Float64, Int64, Int64}(0.04717539935714873, 10, 4)
 FixedBond{Float64, Int64, Int64}(0.04881213298628277, 11, 4)
 FixedBond{Float64, Int64, Int64}(0.04996932107971216, 12, 4)
 FixedBond{Float64, Int64, Int64}(0.050768208246499, 13, 4)
 FixedBond{Float64, Int64, Int64}(0.051299382922034086, 14, 4)
 FixedBond{Float64, Int64, Int64}(0.051633982591327676, 15, 4)
 FixedBond{Float64, Int64, Int64}(0.051830458100536235, 16, 4)
 FixedBond{Float64, Int64, Int64}(0.051938805267308895, 17, 4)
 FixedBond{Float64, Int64, Int64}(0.05200329735010632, 18, 4)
 FixedBond{Float64, Int64, Int64}(0.0520643078824987, 19, 4)
 FixedBond{Float64, Int64, Int64}(0.052159576852118396, 20, 4)

And we capture the measures for each of the available assets for the portfolio selection:

vgh_assets= [value_gradient_and_hessian(x->value(curve(x,times),a),AutoForwardDiff(), zeros) for a in asset_universe]

20-element Vector{Tuple{Float64, Vector{Float64}, Matrix{Float64}}}:
 (0.9999999999999999, [-0.9976397847116376, 0.001889581158300676, -0.0005911493114439049, 7.570359994056739e-5, -6.627009344962502e-6, 4.849306454692366e-7], [0.9976397847116376 0.0 … 0.0 0.0; 0.0 -0.003779162316601352 … 0.0 0.0; … ; 0.0 0.0 … 6.627009344962502e-5 0.0; 0.0 0.0 … 0.0 -9.698612909384728e-6])
 (1.0, [-0.023388491193219983, -1.9440744792877025, 0.002357046582052256, -0.00030184744874882244, 2.6423391558419248e-5, -1.9335286336438875e-6], [0.023388491193219983 0.0 … 0.0 0.0; 0.0 3.888148958575405 … 0.0 0.0; … ; 0.0 0.0 … -0.00026423391558419245 0.0; 0.0 0.0 … 0.0 3.867057267287775e-5])
 (0.9999999999999999, [-0.021831553704588873, -0.04038472750022178, -2.8579627416823383, 0.0013610088406224487, -0.00011914120745198929, 8.718144131725877e-6], [0.021831553704588873 0.0 … 0.0 0.0; 0.0 0.08076945500044357 … 0.0 0.0; … ; 0.0 0.0 … 0.0011914120745198928 0.0; 0.0 0.0 … 0.0 -0.0001743628826345176])
 (1.0, [-0.12086175598272814, 0.6873410858480586, -2.580077722810009, -1.9182858491821426, 0.12039440671851541, -0.00880984684370108], [0.12086175598272814 0.0 … 0.0 0.0; 0.0 -1.3746821716961173 … 0.0 0.0; … ; 0.0 0.0 … -1.2039440671851547 0.0; 0.0 0.0 … 0.0 0.1761969368740217])
 (0.9999999999999999, [-0.03587182758225417, -0.03178847835869419, -0.1623244800675958, -4.434521498439157, 0.004426971666740553, -0.00032394314178212753], [0.03587182758225417 0.0 … 0.0 0.0; 0.0 0.06357695671738837 … 0.0 0.0; … ; 0.0 0.0 … -0.04426971666740108 0.0; 0.0 0.0 … 0.0 0.006478862835641435])
 (1.0, [0.030718124690857655, -0.6006841432615017, 1.4677638724025694, -5.637585375627018, -0.7347382844153851, 0.05016595986898516], [-0.030718124690857655 0.0 … 0.0 0.0; 0.0 1.2013682865230033 … 0.0 0.0; … ; 0.0 0.0 … 7.347382844153855 0.0; 0.0 0.0 … 0.0 -1.003319197379703])
 (1.0000000000000002, [0.04918723421089006, -0.7875296020508475, 1.9813722618339165, -5.475677497528941, -2.004491225500428, 0.11757953091355329], [-0.04918723421089006 0.0 … 0.0 0.0; 0.0 1.575059204101695 … 0.0 0.0; … ; 0.0 0.0 … 20.044912255004277 0.0; 0.0 0.0 … 0.0 -2.3515906182710657])
 (1.0, [0.033435294105167324, -0.7017190688236229, 1.6989628644341668, -4.366584268206587, -3.585974828657458, 0.16298608630113406], [-0.033435294105167324 0.0 … 0.0 0.0; 0.0 1.4034381376472458 … 0.0 0.0; … ; 0.0 0.0 … 35.859748286574586 0.0; 0.0 0.0 … 0.0 -3.259721726022681])
 (1.0, [-0.002431911053734727, -0.453291445963555, 0.943680562936937, -2.736926924650531, -5.249469475354395, 0.14536286030892792], [0.002431911053734727 0.0 … 0.0 0.0; 0.0 0.90658289192711 … 0.0 0.0; … ; 0.0 0.0 … 52.49469475354395 0.0; 0.0 0.0 … 0.0 -2.907257206178559])
 (1.0, [-0.043736737346666915, -0.1561960575179688, 0.05060552918717243, -1.0331954789936535, -6.749501521945598, 0.021211022722198228], [0.043736737346666915 0.0 … 0.0 0.0; 0.0 0.3123921150359376 … 0.0 0.0; … ; 0.0 0.0 … 67.49501521945596 0.0; 0.0 0.0 … 0.0 -0.42422045444396633])
 (1.0, [-0.07783333114293098, 0.09131520428519538, -0.691404940606247, 0.358435555117127, -7.871988981058315, -0.24640615516201977], [0.07783333114293098 0.0 … 0.0 0.0; 0.0 -0.18263040857039076 … 0.0 0.0; … ; 0.0 0.0 … 78.71988981058314 0.0; 0.0 0.0 … 0.0 4.928123103240393])
 (0.9999999999999998, [-0.10282370981723697, 0.27316914850365254, -1.236189221863321, 1.3810544521498946, -8.590793469976155, -0.6595197816774591], [0.10282370981723697 0.0 … 0.0 0.0; 0.0 -0.5463382970073051 … 0.0 0.0; … ; 0.0 0.0 … 85.90793469976155 0.0; 0.0 0.0 … 0.0 13.190395633549183])
 (1.0, [-0.11949556676658321, 0.39420696816817774, -1.5990337099647314, 2.061593933011007, -8.929700855238488, -1.2105657614617258], [0.11949556676658321 0.0 … 0.0 0.0; 0.0 -0.7884139363363555 … 0.0 0.0; … ; 0.0 0.0 … 89.29700855238488 0.0; 0.0 0.0 … 0.0 24.21131522923451])
 (1.0, [-0.12861217374499395, 0.45940241183877495, -1.795349002854901, 2.4278333854267613, -8.91526236179936, -1.8910239053244644], [0.12861217374499395 0.0 … 0.0 0.0; 0.0 -0.9188048236775499 … 0.0 0.0; … ; 0.0 0.0 … 89.15262361799358 0.0; 0.0 0.0 … 0.0 37.82047810648928])
 (1.0000000000000002, [-0.13095883074250508, 0.47410464862949026, -1.8414811453964168, 2.509733511219679, -8.577552928649126, -2.6913836723507663], [0.13095883074250508 0.0 … 0.0 0.0; 0.0 -0.9482092972589805 … 0.0 0.0; … ; 0.0 0.0 … 85.77552928649126 0.0; 0.0 0.0 … 0.0 53.82767344701532])
 (1.0000000000000002, [-0.12736808399407992, 0.4441629661639336, -1.755136918280806, 2.3401183445834253, -7.95046134051263, -3.601221533618457], [0.12736808399407992 0.0 … 0.0 0.0; 0.0 -0.8883259323278672 … 0.0 0.0; … ; 0.0 0.0 … 79.50461340512629 0.0; 0.0 0.0 … 0.0 72.02443067236912])
 (1.0, [-0.11873204577234525, 0.3759779476315208, -1.5555709896240153, 1.9549504787382388, -7.071672780590235, -4.609356718883685], [0.11873204577234525 0.0 … 0.0 0.0; 0.0 -0.7519558952630416 … 0.0 0.0; … ; 0.0 0.0 … 70.71672780590235 0.0; 0.0 0.0 … 0.0 92.18713437767369])
 (1.0000000000000002, [-0.10600658486565401, 0.2765052903118995, -1.263620816446502, 1.3933438321898368, -5.982443557400675, -5.704063303455876], [0.10600658486565401 0.0 … 0.0 0.0; 0.0 -0.553010580623799 … 0.0 0.0; … ; 0.0 0.0 … 59.824435574006756 0.0; 0.0 0.0 … 0.0 114.0812660691175])
 (0.9999999999999998, [-0.09021040840946391, 0.15322952181447289, -0.901645146872998, 0.6974094494321246, -4.727239626091724, -6.873321888975546], [0.09021040840946391 0.0 … 0.0 0.0; 0.0 -0.30645904362894577 … 0.0 0.0; … ; 0.0 0.0 … 47.27239626091725 0.0; 0.0 0.0 … 0.0 137.46643777951095])
 (1.0, [-0.07242116850655933, 0.014120467762457808, -0.49340697871628564, -0.08799425539267751, -3.353297361284719, -8.105097006013095], [0.07242116850655933 0.0 … 0.0 0.0; 0.0 -0.028240935524915616 … 0.0 0.0; … ; 0.0 0.0 … 33.53297361284719 0.0; 0.0 0.0 … 0.0 162.1019401202619])

31.6.2 Optimization Routine

This optimization function uses functionality from JuMP, a robust optimization library in Julia.

With derivatives available, we can optimize the asset portfolio to match liability characteristics.The optimization maximizes asset yield while constraining the difference between asset and liability key rate durations. This ensures that small yield curve movements don’t create large changes in surplus.

function optimize_portfolio(assets, vgh_assets, liabs, vgh_liabs, constraints)
    n = length(assets)

    # Create model
    model = Model(HiGHS.Optimizer)
    set_silent(model)  # Suppress solver output

    # Decision variables: weight vector w
    @variable(model, w[1:n])

    @constraint(model, w .>= 0)  # Long-only constraint
    # Budget/asset value constraint
    budget_sum = sum(w .* [a[1] for a in vgh_assets]) + vgh_liabs[1]
    @constraint(model, budget_sum <= 1e2)
    @constraint(model, budget_sum >= -1e2)

    # Objective: Maximize total yield
    @objective(model, Max, sum(w[i] * assets[i].coupon for i in 1:n))

    # Gradient component (krd) constraints
    for j in 1:length(vgh_liabs[2])
        gradient_sum = sum(w[i] * vgh_assets[i][2][j] for i in 1:n) - sum(vgh_liabs[2][j])

        @constraint(model, gradient_sum >= constraints[:krd][:lower])
        @constraint(model, gradient_sum <= constraints[:krd][:upper])
    end

    # total duration constraint 
    duration_gap = sum(w[i] * sum(vgh_assets[i][2]) for i in 1:n) + sum(vgh_liabs[2])
    @constraint(model, duration_gap <= constraints[:krd][:upper])
    @constraint(model, duration_gap >= constraints[:krd][:lower])

    # Solve
    optimize!(model)

    # Return results
    if termination_status(model) == MOI.OPTIMAL
        return (
            status=:optimal,
            weights=JuMP.value.(w),
            objective_value=objective_value(model),
        )
    else
        return (status=termination_status(model), weights=nothing)
    end
end


# Define gradient constraints
constraints = Dict(
    :krd => Dict(:lower => -0.35e6, :upper => 0.35e6),
    :duration => Dict(:lower => -0.05e6, :upper => 0.05e6)
)

# Optimize
result = optimize_portfolio(asset_universe, vgh_assets, liabilities, vgh_liab, constraints)

(status = :optimal, weights = [-0.0, -0.0, 7138.4690690590905, 39595.27306834956, -0.0, -0.0, -0.0, -0.0, -0.0, 13308.977466166176, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, -0.0, 42395.09699131608], objective_value = 3927.3849967794354)

31.6.3 Results

The optimization produces asset weights that hedge the liability portfolio. We can visualize both the resulting cashflow patterns and the key rate duration matching:

let
    d = consolidate([p(c) for p in liabilities])
    ks = collect(keys(d)) |> sort!
    vs = -cumsum([d[k] for k in ks])

    f = Figure(size = (900, 600))
    ax = Axis(f[1, 1], 
        xlabel = "Time (Years)",
        ylabel = "Cashflow Amount (cumulative)",
        title = "Cumulative Asset vs Liability Cashflows",
    )

    lines!(ax, ks, vs, label = "Liabilities")


    asset_cfs = map(1:length(asset_universe)) do i
        cfs =
            result.weights[i] * asset_universe[i](c)
    end

    d = consolidate(asset_cfs)
    ks2 = collect(keys(d)) |> sort!
    vs2 = cumsum([d[k] for k in ks2])
    lines!(ax, ks2, vs2, label = "Assets")

    axislegend(ax, position = :rb)

    f
end    

let
    asset_krds = sum(getindex.(vgh_assets,2) .* result.weights)
    liab_krds = -vgh_liab[2]

    f = Figure(size = (800, 500))
    ax = Axis(f[1, 1], 
        xlabel = "Tenor (Years)",
        ylabel = "Key Rate Dollar Duration",
        title = "Asset vs Liability Key Rate Dollar Duration Profile",
    )
    
    scatter!(ax, times, asset_krds, label = "Optimized Assets")
    
    scatter!(ax, times, liab_krds, label = "Liabilities")
    
    axislegend(ax, position = :rt)
    
    f
end

The first plot shows the distribution of asset cashflows over time. The second compares the key rate duration profiles of the optimized asset portfolio and the liability portfolio, demonstrating how well the hedge performs across different points on the yield curve.

31.7 Computational Benefits

Autodiff provides several advantages over traditional finite difference approaches:

• Exact derivatives rather than approximations
• Single function evaluation computes value and all derivatives
• No tuning of step sizes or dealing with numerical artifacts
• Scales efficiently to high-dimensional parameter spaces

For ALM applications, this means more accurate risk measurement and the ability to optimize portfolios with complex constraints that would be computationally expensive using traditional methods.

Here, we value 100,000 interest-sensitive policies with a monthly timestep for up to 20 years and compute 1st and 2nd order partial sensitives extremely quickly:

@btime value_gradient_and_hessian(z -> annuities(z, liabilities), AutoForwardDiff(), zeros)

  5.230 s (127027389 allocations: 36.45 GiB)

(-102337.8165948909, [11504.297366155115, 29870.011138629456, 29672.160372787308, 256573.36315787124, 122773.8683882791, 6317.157272228895], [-166687.66920132237 195891.40237708762 … 11057.700186961323 -42919.17049380064; 195891.40237708736 -1.6769719809453504e6 … 2.4755330051117623e6 1.2651095074186337e6; … ; 11057.700186961241 2.475533005111765e6 … -3.717194044503879e7 -2.7874235298093623e6; -42919.17049380067 1.2651095074186337e6 … -2.7874235298093683e6 -2.1251952156862594e7])

However, there’s still some performance left on the table! the (d::DeferredAnnuity)(curve) function defined above is not type stable. In the appendix to this post, we’ll cover a way to improve the performance even more.

31.8 Conclusion

The Julia ecosystem supports this workflow through packages like DifferentiationInterface for autodiff, JuMP for optimization, and FinanceCore for financial mathematics. This combination enables sophisticated ALM implementations that are both mathematically precise and computationally efficient.

31.9 Appendix: Even more performance (Advanced)

Julia is fastest when all functions are type stable (i.e. the return type can be inferred at compile time). Looking back at the function defined above, the issue is that the av function is defined outside of the scope used within the map block. This means that the compiler can’t be sure that av won’t be modified while being used within the map. Therefore, av get’s ‘boxed’ and held as an Any type. This type uncertainty propagates to the value returned from the (d::DeferredAnnuity)(curve) function:

function (d::DeferredAnnuity)(curve)
    av = 1. 
    map(1//12:1//12:d.tenor) do t
        mkt_rate = -log(curve(d.tenor) / curve(t)) / (d.tenor - t)
        av *= exp(d.rate / 12)
        rate_diff = mkt_rate - d.rate
        sr = t == d.tenor ? 1.0 : surrender_rate(rate_diff) / 12
        av_surr = av * sr
        av -= av_surr
        Cashflow(-av_surr, t)
    end
end

An alterative would be to write a for loop and initialize an array to hold the cashflows. The challenge with that is to concretely define the output type of the resulting array. Particularly when combine with AD, the types within the program are no longer basic floats and integers, as we have dual numbers and more complex types running through our functions.

To maintain most of the simplicity, an alternative approach¹ is to use small, immutable containers from MicroCollections.jl and combine them with BangBang.jl. Then, instead of using map we will write a regular loop. The macro @unroll is defined to unroll the first N iterations of the loop. This means that the macro transforms the source code to explicitly write out the first two loops. An example of this might be as follows where two iterations of the loop are unrolled.

function basic_loop()
    out = []
    for i ∈ 1:10
        push!(out,i)
    end
    out
end

function partially_unrolled_loop()
    out = []
    push!(out,1)
    push!(out,2) # two steps unrolled
    for i ∈ 3:10
        push!(out,i)
    end
    out
end

Here’s the macro that does this (expand to see the full definition):

Note@unroll macro

"""
    @unroll N for_loop

Unroll the first `N` iterations of a for loop, with remaining iterations handled by a regular loop.

This macro takes a for loop and explicitly expands the first `N` iterations, which can improve 
performance and type stability, particularly when building collections where the first few 
iterations determine the container's type.

# Arguments
- `N::Int`: Number of loop iterations to unroll (must be a compile-time constant)
- `for_loop`: A standard for loop expression

"""
macro unroll(N::Int, loop)
    Base.isexpr(loop, :for) || error("only works on for loops")
    Base.isexpr(loop.args[1], :(=)) || error("This loop pattern isn't supported")
    val, itr = esc.(loop.args[1].args)
    body = esc(loop.args[2])
    @gensym loopend
    label = :(@label $loopend)
    goto = :(@goto $loopend)
    out = Expr(:block, :(itr = $itr), :(next = iterate(itr)))
    unrolled = map(1:N) do _
        quote
            isnothing(next) && @goto loopend
            $val, state = next
            $body
            next = iterate(itr, state)
        end
    end
    append!(out.args, unrolled)
    remainder = quote
        while !isnothing(next)
            $val, state = next
            $body
            next = iterate(itr, state)
        end
        @label loopend
    end
    push!(out.args, remainder)
    out
end

Main.Notebook.@unroll

Then, we re-write and redefine (d::DeferredAnnuity)(curve) to utilize this technique.

using BangBang, MicroCollections

function (d::DeferredAnnuity)(curve)
    times = 1//12:1//12:d.tenor
    out = UndefVector{Union{}}(length(times)) # 1
    av = 1.0
    @unroll 2 for (i, t) ∈ enumerate(times) # 2
        mkt_rate = -log(curve(d.tenor) / curve(t)) / (d.tenor - t)
        av *= exp(d.rate / 12)
        rate_diff = mkt_rate - d.rate
        sr = t == d.tenor ? 1.0 : surrender_rate(rate_diff) / 12
        av_surr = av * sr
        av -= av_surr
        cf = Cashflow(-av_surr, t)
        out = setindex!!(out, cf, i) # 3
    end
    out
end;

1. We tell the out vector how many elements to expect
2. We unroll two iterations of the loop so that the compiler can use the calculated result to determine the type of the output container.
3. We use setindex!! from BangBang to efficiently update the output vector and it’s type.

Using this technique, we can see that we achieve a significant speedup (less than half the runtime) from the earlier version due to improving the type stability of the code:

@btime value_gradient_and_hessian(z -> annuities(z, liabilities), AutoForwardDiff(), zeros)

  1.857 s (600199 allocations: 5.91 GiB)

(-102337.8165948909, [11504.297366155115, 29870.011138629456, 29672.160372787308, 256573.36315787124, 122773.8683882791, 6317.157272228895], [-166687.66920132237 195891.40237708762 … 11057.700186961323 -42919.17049380064; 195891.40237708736 -1.6769719809453504e6 … 2.4755330051117623e6 1.2651095074186337e6; … ; 11057.700186961241 2.475533005111765e6 … -3.717194044503879e7 -2.7874235298093623e6; -42919.17049380067 1.2651095074186337e6 … -2.7874235298093683e6 -2.1251952156862594e7])

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1. With thanks to the helpful persons on the Julia Zulip and in particular Mason Protter for this approach.