Financial Math Submodule

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 KeyRateZero) in the fixed income markets, even though the latter has more analytically attractive properties. See the discussion of KeyRateDuration in the FinanceModels.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 KeyRatePar) in the fixed income markets, even though the latter has more analytically attractive properties. See the discussion of KeyRateDuration in the FinanceModels.jl docs.

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

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_values(interest, cashflows, timepoints)

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(ForwardYield([0.1,0.2]), [10,20],[0,1]) # after `using FinanceModels`
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.

spread(curve1,curve2,cashflows)

Return the solved-for constant spread to add to curve1 in order to equate the discounted cashflows with curve2

Examples

spread(0.04, 0.05, cfs)
Rate{Float64, Periodic}(0.010000000000000009, Periodic(1))

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 FinanceModels.jl curve (e.g. does not have to be a curve constructed via FinanceModels.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 = FinanceModels.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(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.

Note that the calculated duration will depend on the periodicity convention of the interest_rate: a Periodic yield (or yield model with that convention) will be a slightly different computed duration than a Continous which follows from the present value differing according to the periodicity.

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

• Modified duration: the relative change per point of yield change.
• Macaulay: the cashflow-weighted average time.
• DV01: the absolute change per basis point (hundredth of a percentage point).

Examples

Using vectors of cashflows and times

julia> times = 1:5;

julia> cfs = [0,0,0,0,100];

julia> duration(0.03,cfs,times)
4.854368932038835

julia> duration(Periodic(0.03,1),cfs,times)
4.854368932038835

julia> duration(Continuous(0.03),cfs,times)
5.0

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