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using Dates
include("jlFiles/printmat.jl")
include("jlFiles/printTable.jl")
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In [2]:
using Plots
#pyplot(size=(600,400))
gr(size=(480,320))
default(fmt = :svg)
Consider an AR(1) with mean $\mu$
$r_{t+1}-\mu=\rho\left( r_{t}-\mu\right) +\varepsilon_{t+1}$
The forecast (based on information in $t$) for $t+s$ is
$\text{E}_{t}r_{t+s}=\left( 1-\rho^{s}\right) \mu+\rho ^{s}r_{t}.$
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"""
AR1Prediction(x0,ρ,μ,s)
Calculate forecast from AR(1)
x0,ρ,μ,s: scalars
"""
function AR1Prediction(x0,ρ,μ,s)
E0xs = (1-ρ^s)*μ + ρ^s*x0
return E0xs
end
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In [4]:
ρ = 0.975
μ = 0.05
printlnPs("Prediction for t+50, assuming current r=0.07: ",
AR1Prediction(0.07,ρ,μ,50))
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sMax = 120
sM = 1:sMax
x0 = [0.03;0.05;0.07]
xPred = fill(NaN,(sMax,length(x0))) #one column for each value in x0
for s in sM, j = 1:length(x0)
xPred[s,j] = AR1Prediction(x0[j],ρ,μ,s)
end
plot( sM,xPred,
linecolor = [:black :red :blue],
linewidth = 2,
linestyle = [:solid :dash :dot],
label = ["3%" "5%" "7%"],
title = "Forecast of short interest rate",
xlabel = "forecast horizon",
ylabel = "short interest rate",
annotation = [(80,0.038,text("(mu,rho) = ($μ,$ρ)",8)),
(15,0.068,text("(based on different values\nof current short rate)",8,:left))] )
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The Vasicek model (without risk premia) implies that
$ y_{t}(n) =a(n)+b(n)r_{t} \: \text{, where } $
$ b(n) =(1-\rho^{n})/[(1-\rho)n], $
$ a(n) = \mu\left[ 1-b(n)\right]. $
If the periods are one month, then $y_{t}(36)$ is the (annualized) continuously compounded interest rate for a bond maturing in 36 months (...3 years).
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"""
Vasicek(r,ρ,μ,n)
Vasicek model: calculate interest rate and (a,b) coeffs
r,ρ,μ,n: scalars
"""
function Vasicek(r,ρ,μ,n)
if ρ == 1.0
b = 1.0
else
b = (1-ρ^n)/((1-ρ)*n)
end
a = μ*(1-b)
y = a + b*r
return y,a,b
end
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yM = fill(NaN,(sMax,length(x0))) #interest rates, different starting values in columns
ab = fill(NaN,(sMax,2)) #a and b coefs
for n in sM, j = 1:length(x0)
#local a, y, b #only needed in REPL/script
(y,a,b) = Vasicek(x0[j],ρ,μ,n)
yM[n,j] = y
if j == 1
ab[n,:] = [a b] #the same across x0 values
end
end
printblue("a and b for the first few horizons (months):")
printTable([ab[1:4,1]*100 ab[1:4,2]],["a*100","b"],string.(sM[1:4]),cell00="horizon")
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plot( sM,ab,
linecolor = [:black :red],
linestyle = [:dot :solid],
linewidth = 2,
label = ["a(m)" "b(m)"],
title = "Vasicek model: coefficients",
xlabel = "maturity (months)",
annotation = [(80,0.2,text("(mu,rho) = ($μ,$ρ)",8,:left)),
(25,0.95,text("in y(n) = a(n) + b(n)r",8,:left))] )
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plot( sM,yM,
linecolor = [:black :red :blue],
linewidth = 2,
linestyle = [:solid :dash :dot],
label = ["3%" "5%" "7%"],
title = "Vasicek model: yield curves",
xlabel = "maturity (months)",
ylabel = "interest rate",
annotation = [(80,0.038,text("(mu,rho) = ($μ,$ρ)",8)),
(25,0.069,text("for different current short rates",8,:left))] )
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Recall that the change of the (value of the) hedge portfolio is
$\begin{equation} \Delta V=v\Delta P_{H}-\Delta P_{L} \end{equation}$
A. For an initial value of the short log interest rate $r$, use the Vasicek model to calculate all spot rates needed to value the bond portfolios. Repeat for a somewhat different short rate.
B. Use the two sets of spot interest rates ($y(m)$) to calculate (two different) prices of both the liability and the hedge bond.
C. Calculate the $v$ value that makes $\Delta V = 0$, that is, $v=\Delta P_{L}/\Delta P_{H}$.
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"""
y: scalar or K vector of continuously compounded interest rates
cf: scalar or K vector of cash flows
m: K vector of times of cash flows
"""
function BondPrice3b(y,cf,m) #cf is a vector of all cash flows
cdisc = cf./exp.(m.*y)
P = sum(cdisc) #price
return P
end
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ρ = 0.975 #experiment with this
μ = 0.05
r₀ = 0.05 #initial (day 0) short log interest rate
r₁ = 0.049 #another possible short rate
nL = 1:120 #times of the cash flows of liability, months
nH = 1:36 #same, but for hedge bond
nMax = maximum([nL;nH]) #calculate yield curves for at least this maturity
(y₀,y₁) = [fill(NaN,nMax) for i=1:2] #yield curve, before and after
for n in 1:nMax
y₀[n] = Vasicek(r₀,ρ,μ,n)[1]
y₁[n] = Vasicek(r₁,ρ,μ,n)[1]
end
In the example below, the liability pays 0.2 every 12 months for 10 years. The hedge bond is a 3-year zero coupon bond.
The Vasicek model is here implemented such that periods are months, that is, y[36] is the (annualized, continuously compounded) interest rate for a 36 month bond.
With the two yield curves (different short rate values) from the Vasicek model, calculate prices of the hedge bond and the liability.
Recall, the bond price $P$ is the present value of the future cash flows $cf_k$
$P = \sum_{k=1}^{K} \frac{cf_{k}}{\exp\left[m_{k} y(m_{k})\right] }$,
where $y(m_k)$ is the continuously compounded (annualized) interest rate on a bond maturing in $m_k$ years.
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cfL = repeat([zeros(11);0.2],10) #cash flow of liability
cfH = [zeros(35);1] #cash flow of hedge bond, 3-year zero-coupon bond
PL₀ = BondPrice3b(y₀[nL],cfL,nL/12) #liability, before, /12 to get years
PL₁ = BondPrice3b(y₁[nL],cfL,nL/12) #after
ΔPL = PL₁ - PL₀
PH₀ = BondPrice3b(y₀[nH],cfH,nH/12) #hedge bond
PH₁ = BondPrice3b(y₁[nH],cfH,nH/12)
ΔPH = PH₁ - PH₀
printblue("Bond prices (according to the Vasicek model) at different r values")
xy = [PL₀ PH₀;PL₁ PH₁]
printTable(xy,["PL","PH"],["$r₀","$r₁"])
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v = ΔPL/ΔPH #change PL/change PH
printblue("Hedge ratio from the Vasicek model:")
xy = [v;v*PH₀/PL₀]
printTable(xy,[""],["v","v*PH₀/PL₀"])
printred("Notice: ρ is important for the hedge ratio. Try also ρ=1 to see how h changes")
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