ActuaryUtilities.jl

KeyRate(timepoints,shift=0.001)

A convenience constructor for KeyRateZero.

Extended Help

KeyRateZero is chosen as the default constructor because it has more attractive properties than KeyRatePar:

• rates after the key timepoint remain unaffected by the shift
  • e.g. this causes a 6-year zero coupon bond would have a negative duration if the 5-year par rate was used

KeyRatePar(timepoint,shift=0.001) <: KeyRateDuration

Shift the par curve by the given amount at the given timepoint. Use in conjunction with duration to calculate the key rate duration.

Unlike other duration statistics which are computed using analytic derivatives, KeyRateDurations are computed via a shift-and-compute the yield curve approach.

KeyRatePar is more commonly reported (than KayRateZero) in the fixed income markets, even though the latter has more analytically attractive properties. See the discussion of KeyRateDuration in the Yields.jl docs.

KeyRateZero(timepoint,shift=0.001) <: KeyRateDuration

Shift the par curve by the given amount at the given timepoint. Use in conjunction with duration to calculate the key rate duration.

Unlike other duration statistics which are computed using analytic derivatives, KeyRateDuration is computed via a shift-and-compute the yield curve approach.

KeyRateZero is less commonly reported (than KayRatePar) in the fixed income markets, even though the latter has more analytically attractive properties. See the discussion of KeyRateDuration in the Yields.jl docs.

CTE(v::AbstractArray,p::Real;rev::Bool=false)

The average of the values ≥ the pth percentile of the vector v is the Conditiona Tail Expectation. Assumes more positive values are higher risk measures, so a higher p will return a more positive number, but this can be reversed if rev is true.

May also be called with ConditionalTailExpectation(...).

Also known as Tail Value at Risk (TVaR), or Tail Conditional Expectation (TCE)

CTE

VaR(v::AbstractArray,p::Real;rev::Bool=false)

The pth quantile of the vector v is the Value at Risk. Assumes more positive values are higher risk measures, so a higher p will return a more positive number, but this can be reversed if rev is true.

Also can be called with ValueAtRisk(...).

VaR

accum_offset(x; op=*, init=1.0)

A shortcut for the common operation wherein a vector is scanned with an operation, but has an initial value and the resulting array is offset from the traditional accumulate.

This is a common pattern when calculating things like survivorship given a mortality vector and you want the first value of the resulting vector to be 1.0, and the second value to be 1.0 * x[1], etc.

Two keyword arguments:

• op is the binary (two argument) operator you want to use, such as * or +
• init is the initial value in the returned array

Examples

julia> accum_offset([0.9, 0.8, 0.7])
3-element Array{Float64,1}:
 1.0
 0.9
 0.7200000000000001

julia> accum_offset(1:5) # the product of elements 1:n, with the default `1` as the first value
5-element Array{Int64,1}:
  1
  1
  2
  6
 24

julia> accum_offset(1:5,op=+)
5-element Array{Int64,1}:
  1
  2
  4
  7
 11

breakeven(yield, cashflows::Vector)
breakeven(yield, cashflows::Vector,times::Vector)

Calculate the time when the accumulated cashflows breakeven given the yield.

Assumptions:

• cashflows occur at the end of the period
• cashflows evenly spaced with the first one occuring at time zero if times not given

Returns nothing if cashflow stream never breaks even.

julia> breakeven(0.10, [-10,1,2,3,4,8])
5

julia> breakeven(0.10, [-10,15,2,3,4,8])
1

julia> breakeven(0.10, [-10,-15,2,3,4,8]) # returns the `nothing` value

convexity(yield,cfs,times)
convexity(yield,valuation_function)

Calculates the convexity. - yield should be a fixed effective yield (e.g. 0.05). - times may be omitted and it will assume cfs are evenly spaced beginning at the end of the first period.

Examples

Using vectors of cashflows and times

julia> times = 1:5
julia> cfs = [0,0,0,0,100]
julia> duration(0.03,cfs,times)
4.854368932038834
julia> duration(Macaulay(),0.03,cfs,times)
5.0
julia> duration(Modified(),0.03,cfs,times)
4.854368932038835
julia> convexity(0.03,cfs,times)
28.277877274012614

Using any given value function:

julia> lump_sum_value(amount,years,i) = amount / (1 + i ) ^ years
julia> my_lump_sum_value(i) = lump_sum_value(100,5,i)
julia> duration(0.03,my_lump_sum_value)
4.854368932038835
julia> convexity(0.03,my_lump_sum_value)
28.277877274012617

duration(keyrate::KeyRateDuration,curve,cashflows)    
duration(keyrate::KeyRateDuration,curve,cashflows,timepoints)
duration(keyrate::KeyRateDuration,curve,cashflows,timepoints,krd_points)

Calculate the key rate duration by shifting the zero (not par) curve by the kwarg shift at the timepoint specified by a KeyRateDuration(time).

The approach is to carve up the curve into krd_points (default is the unit steps between 1 and the last timepoint of the casfhlows). The zero rate corresponding to the timepoint within the KeyRateDuration is shifted by shift (specified by the KeyRateZero or KeyRatePar constructors. A new curve is created from the shifted rates. This means that the "width" of the shifted section is ± 1 time period, unless specific points are specified via krd_points.

The curve may be any Yields.jl curve (e.g. does not have to be a curve constructed via Yields.Zero(...)).

!!! Experimental: Due to the paucity of examples in the literature, this feature does not have unit tests like the rest of JuliaActuary functionality. Additionally, the API may change in a future major/minor version update.

Examples

julia> riskfree_maturities = [0.5, 1.0, 1.5, 2.0];

julia> riskfree    = [0.05, 0.058, 0.064,0.068];

julia> rf_curve = Yields.Zero(riskfree,riskfree_maturities);

julia> cfs = [10,10,10,10,10];

julia> duration(KeyRate(1),rf_curve,cfs)
8.932800152336995

Extended Help

Key Rate Duration is not a well specified topic in the literature and in practice. The reference below suggest that shocking the par curve is more common in practice, but that the zero curve produces more consistent results. Future versions may support shifting the par curve.

References:

• Quant Finance Stack Exchange: To compute key rate duration, shall I use par curve or zero curve?
• (Financial Exam Help 123](http://www.financialexamhelp123.com/key-rate-duration/)

duration(d1::Date, d2::Date)

Compute the duration given two dates, which is the number of years since the first date. The interval [0,1) is defined as having duration 1. Can return negative durations if second argument is before the first.

julia> issue_date  = Date(2018,9,30);

julia> duration(issue_date , Date(2019,9,30) ) 
2
julia> duration(issue_date , issue_date) 
1
julia> duration(issue_date , Date(2018,10,1) ) 
1
julia> duration(issue_date , Date(2019,10,1) ) 
2
julia> duration(issue_date , Date(2018,6,30) ) 
0
julia> duration(Date(2018,9,30),Date(2017,6,30)) 
-1

duration(Macaulay(),interest_rate,cfs,times)
duration(Modified(),interest_rate,cfs,times)
duration(DV01(),interest_rate,cfs,times)
duration(interest_rate,cfs,times)             # Modified Duration
duration(interest_rate,valuation_function)    # Modified Duration

Calculates the Macaulay, Modified, or DV01 duration. times may be ommitted and the valuation will assume evenly spaced cashflows starting at the end of the first period.

• interest_rate should be a fixed effective yield (e.g. 0.05).

When not given Modified() or Macaulay() as an argument, will default to Modified().

Examples

Using vectors of cashflows and times

julia> times = 1:5
julia> cfs = [0,0,0,0,100]
julia> duration(0.03,cfs,times)
4.854368932038834
julia> duration(Macaulay(),0.03,cfs,times)
5.0
julia> duration(Modified(),0.03,cfs,times)
4.854368932038835
julia> convexity(0.03,cfs,times)
28.277877274012614

Using any given value function:

julia> lump_sum_value(amount,years,i) = amount / (1 + i ) ^ years
julia> my_lump_sum_value(i) = lump_sum_value(100,5,i)
julia> duration(0.03,my_lump_sum_value)
4.854368932038835
julia> convexity(0.03,my_lump_sum_value)
28.277877274012617

eurocall(;S=1.,K=1.,τ=1,r,σ,q=0.)

Calculate the Black-Scholes implied option price for a european call, where:

• S is the current asset price
• K is the strike or exercise price
• τ is the time remaining to maturity (can be typed with \tau[tab])
• r is the continuously compounded risk free rate
• σ is the (implied) volatility (can be typed with \sigma[tab])
• q is the continuously paid dividend rate

Rates should be input as rates (not percentages), e.g.: 0.05 instead of 5 for a rate of five percent.

!!! Experimental: this function is well-tested, but the derivatives functionality (API) may change in a future version of ActuaryUtilities.

Extended Help

This is the same as the formulation presented in the dividend extension of the BS model in Wikipedia.

Other general comments:

• Swap/OIS curves are generally better sources for r than government debt (e.g. US Treasury) due to the collateralized nature of swap instruments.
• (Implied) volatility is characterized by a curve that is a function of the strike price (among other things), so take care when using
• Yields.jl can assist with converting rates to continuously compounded if you need to perform conversions.

europut(;S=1.,K=1.,τ=1,r,σ,q=0.)

Calculate the Black-Scholes implied option price for a european call, where:

• S is the current asset price
• K is the strike or exercise price
• τ is the time remaining to maturity (can be typed with \tau[tab])
• r is the continuously compounded risk free rate
• σ is the (implied) volatility (can be typed with \sigma[tab])
• q is the continuously paid dividend rate

Rates should be input as rates (not percentages), e.g.: 0.05 instead of 5 for a rate of five percent.

!!! Experimental: this function is well-tested, but the derivatives functionality (API) may change in a future version of ActuaryUtilities.

Extended Help

This is the same as the formulation presented in the dividend extension of the BS model in Wikipedia.

Other general comments:

• Swap/OIS curves are generally better sources for r than government debt (e.g. US Treasury) due to the collateralized nature of swap instruments.
• (Implied) volatility is characterized by a curve that is a function of the strike price (among other things), so take care when using
• Yields.jl can assist with converting rates to continuously compounded if you need to perform conversions.

internal_rate_of_return(cashflows::vector)::Yields.Rate
internal_rate_of_return(cashflows::Vector, timepoints::Vector)::Yields.Rate

Calculate the internalrateof_return with given timepoints. If no timepoints given, will assume that a series of equally spaced cashflows, assuming the first cashflow occurring at time zero and subsequent elements at time 1, 2, 3, ..., n.

Returns a Yields.Rate type with periodic compounding once per period (e.g. annual effective if the timepoints given represent years). Get the scalar rate by calling Yields.rate() on the result.

Example

julia> internal_rate_of_return([-100,110],[0,1]) # e.g. cashflows at time 0 and 1
0.10000000001652906
julia> internal_rate_of_return([-100,110]) # implied the same as above
0.10000000001652906

Solver notes

Will try to return a root within the range [-2,2]. If the fast solver does not find one matching this condition, then a more robust search will be performed over the [.99,2] range.

The solution returned will be in the range [-2,2], but may not be the one nearest zero. For a slightly slower, but more robust version, call ActuaryUtilities.irr_robust(cashflows,timepoints) directly.

irr(cashflows::vector)
irr(cashflows::Vector, timepoints::Vector)

An alias for `internal_rate_of_return`.

moic(cashflows<:AbstractArray)

The multiple on invested capital ("moic") is the un-discounted sum of distributions divided by the sum of the contributions. The function assumes that negative numbers in the array represent contributions and positive numbers represent distributions.

Examples

julia> moic([-10,20,30])
5.0

present_value(interest, cashflows::Vector, timepoints)
present_value(interest, cashflows::Vector)

Discount the cashflows vector at the given interest_interestrate, with the cashflows occurring at the times specified in timepoints. If no timepoints given, assumes that cashflows happen at times 1,2,...,n.

The interest can be an InterestCurve, a single scalar, or a vector wrapped in an InterestCurve.

Examples

julia> present_value(0.1, [10,20],[0,1])
28.18181818181818
julia> present_value(Yields.Forward([0.1,0.2]), [10,20],[0,1])
28.18181818181818 # same as above, because first cashflow is at time zero

Example on how to use real dates using the DayCounts.jl package

using DayCounts 
dates = Date(2012,12,31):Year(1):Date(2013,12,31)
times = map(d -> yearfrac(dates[1], d, DayCounts.Actual365Fixed()),dates) # [0.0,1.0]
present_value(0.1, [10,20],times)

# output
28.18181818181818

present_value(interest, cashflows::Vector, timepoints)
present_value(interest, cashflows::Vector)

Efficiently calculate a vector representing the present value of the given cashflows at each period prior to the given timepoint.

Examples

julia> present_values(0.00, [1,1,1])
[3,2,1]

julia> present_values(Yields.Forward([0.1,0.2]), [10,20],[0,1])
2-element Vector{Float64}:
 28.18181818181818
 18.18181818181818

price(...)

The absolute value of the present_value(...).

Extended help

Using price can be helpful if the directionality of the value doesn't matter. For example, in the common usage, duration is more interested in the change in price than present value, so price is used there.

pv()

An alias for `present_value`.

Years_Between(d1::Date, d2::Date)

Compute the number of integer years between two dates, with the first date typically before the second. Will return negative number if first date is after the second. Use third argument to indicate if calendar anniversary should count as a full year.

Examples

julia> d1 = Date(2018,09,30);

julia> d2 = Date(2019,09,30);

julia> d3 = Date(2019,10,01);

julia> years_between(d1,d3) 
1
julia> years_between(d1,d2,false) # same month/day but `false` overlap
0 
julia> years_between(d1,d2) # same month/day but `true` overlap
1 
julia> years_between(d1,d2) # using default `true` overlap
1