During my studies, in certain modules there are calculations required related to financial investment instruments such as, for example, pricing a forward contract. Some of these calculations involve the implementation of relatively complicated formulas and a lot of careful manipulation on paper. I found that a lot of these processes are very repetitive and could be automated with the use of algorithms.
I’m going to share here some of my code I developed for solving these questions and for verifying my answers. The best use for this code is in an educational context as the problems I am working with, and the questions they come from, are derived from a university course and may or may not have great value in an actual investing context, because of course, much more complicated algorithms are already in use. However, simple problems involving finding the present value of a bond are in almost every business/accounting/finance/investing course out there, so while practicing these questions I found it very useful to be able to verify my answers, and in turn gain a more intuitive understanding of how such values can change drastically when the parameters of a problem are altered slightly.
The problems I will show here are as follows: finding the present value of a bond, of a swaption, of the expected loss over the lifetime of the corporate bond, pricing a forward contract, constructing a duration-immunising portfolio from two bonds, pricing a European put option contract, extending the LIBOR curve and using the extended LIBOR rates to find future forward rates. I have also included a Google Drive link where you can find my written solutions to these questions which may be useful if you are undertaking similar studies. My code is written in the statistical programming language R, the software to run this code is free to download at https://www.r-project.org or a cloud based option (my preferred method) is available at https://rstudio.cloud/ also for free. I will include my code below, but .txt files are also available in my Google Drive link.
1 – Finding the Present Value of a Bond
A typical question you can find asking for the present value of a bond could be:
A 3-year bond is issued with face value €1,000 that has coupon rate of 5% p.a., paid semi-annually. Assuming a flat yield curve of 4%p.a. for all maturities, find the cash value of the bond on that date, using continuous compounding.
The code sets up the discounting options (continuous vs p times per year) each as a function which can be called later. The “time” variable is created as a sequence of values which represent points in time where your coupons will be paid to you, given the parameters of the question. The value of the coupons are calculated below this. The code then initialises the value of the bond at time 0 (B_0) to be worth €0, and then in a for loop, the value will incrementally increased based on the discounted coupon values until the each time point has been accounted for. And finally adding the bond’s face value to the discounted coupons you arrive at the present value of the bond.
Using the below code, all you need to do is enter is the duration of the bond (3 years), the face value (1000), the coupon rate (0.05), the payments per year (2), the yield (0.04) and below that the type of compounding, in this case, continuous (cont).
A written solution can be found here: Simple Investment Modelling Algorithms
# PV of a Bond
# Find the cash value of the bond
T = 3 # T year bond
FV= 1000 # Face Value
c = 0.05 # coupon
p = 2 # payments per year (annually = 1, semiannually = 2, etc)
y = 0.04 # yield
# Setting up the discount options
cont = function(i){exp(-y*i)} # Continuous Compounding
ptimes = function(i){1/(1+y/p)^i} # P times per year Compounding
# "cont" vs "ptimes" - select which type of compounding is being used
disc = cont
# Formula to calculate the value
time = seq(1/p, T, by=1/p)
coupon = (FV*c)/p
coupon
B0 = 0
i = 0
for(i in time){
B0 = B0 + coupon*disc(i)
}
B0 = B0 + FV*disc(i)
B0
# B0 is the present value of the bond
# (in this case, €1026.861)
2 – Finding the Present Value of a Swaption
A typical question you can find asking for finding the present value of a swaption could be:
Find the present value of a swaption that gives the holder the right to enter into a 5-year annual-pay swap in 1 year where a fixed rate of 1.4% is paid and LIBOR is received. The swap principal is €200,000. Assume the yield curve is flat at 1.2% per annum, the forward swap rate is 1.6% and its volatility is 30%.
Note: the present value of a swaption, in the usual notation, is:
LA[s_0\Phi(d_1) − s_K\Phi(d_2)], \newline
where \quad d_1 = \frac{ln(s_0/s_K) + \sigma \frac{2T}{2}}{\sigma \sqrt{T}}, \newline
d_2 = d_1 − \sigma\sqrt{T} \quad and \quad A = \sum^{5}_{i=1}P(0, T_i), \newline
where \: T_i \: are \: the \: swap \: dates.We can see this type of question could cause a lot more problems when working out a solution on paper, however in terms of the code required to solve it, it doesn’t get much trickier than the previous example. The formula for finding the present value of a swaption includes a parameter associated with the value of an equivalent annuity, this is calculated in a similar way to the previous example and is denotes by the variable A. Calculating d_1 and d_2 is relatively simple when you know which values in the question are associated with s_0, s_k, etc. Then using these d values, we can use the pnorm() function which calculates the cumulative density function of the normal distribution. Plugging these values into the final formula gives the present value of a swaption.
Using the below code, again, all you need to do is enter associate the values from the question to their variables labelled in the first section of the code.
A written solution can be found here: Simple Investment Modelling Algorithms
# Present Value of a Swaption
t = 5 # The length of the contract
T = 1 # When you will enter into the swap
sk = 0.014 # Rate to be paid
L = 200000 # Principle
y = 0.012 # Risk free rate ("yield curve")
s0 = 0.016 # Forward Swap Rate
vol = 0.30 # Volatility
# Annuity
A = 0
time = seq(T+1, T+t, 1)
for(i in time){
A = A + 1*exp(-y*i)}
A
d1 = (log(s0/sk) + 0.5*vol^2*T) / (vol*sqrt(T))
d2 = (log(s0/sk) - 0.5*vol^2*T) / (vol*sqrt(T))
d1
d2
# calculating the normal cdfs
phi1 = pnorm(d1)
phi2 = pnorm(d2)
phi1
phi2
PV = L*A*(s0*phi1 - sk*phi2)
PV
# PV = Present Value of the Swaption
# (in this case, €2822.835)
3 – Pricing a Forward Contract
A typical question asking you to find the price of a forward contract could be:
On May 7th, 2013, a 4-year bond is issued with face value €5,000 that has coupon rate of 2% p.a., paid annually. Assuming a flat yield curve of 1.2%p.a. for all maturities, find the cash value of the bond on that date.
Find the forward price in a long forward contract, with maturity three years later, i.e. a contract to buy the bond on May 7th, 2016.
Find the value of this forward contract two years later, on May 7th, 2015. Assume on that date the yield curve is flat at 0.8%p.a.
On that same day, the investor wants to unwind the contract, so she enters into an offsetting position by taking a short position in a new forward contract on offer to sell the bond on May 7th, 2016. Find the forward price of the new contract. Find the combined value of her position in both contracts on the maturity date of the contracts, May 7th, 2016.
Useful formula: the value of a forward contract during its lifetime is:
B(t) - I(t) - F_0 P(t, \tau )
To begin a question like this we must first find the present value of the bond at time 0 (2013) in the same way as in question 1. Then you find the forward price of the bond in 3 years (maturity three years), here we use an intuitive formula to find the forward price of a bond (see written solution) where we take into account the coupons we will miss out on in the first 3 years of the bonds life. To find the value of the contract 2 years later (2015) we first find the value of the bond at time 2 (2 years later) and account for the 1 coupon you will miss out on between year 2 and year 3. Using these values we can then find the value of the contract during the life of the contract. Finding the forward price of the new contract used to offset the first contract is relatively simple and uses the intuitive formula discussed before. Finding the combined value of the contracts simply means taking the forward price of the new contract and taking away the forward price of the original contract.
You are prompted in the code below when a new parameter is needed to solve this question.
A written solution can be found here: Simple Investment Modelling Algorithms
# Pricing a Forward Contract
# Find the cash value of the bond
T = 4 # T year bond
FV= 5000 # Face Value
c = 0.02 # coupon
p = 1 # payments per year (annually = 1, semiannually = 2)
y = 0.012 # yield
time = seq(1/p, T, by=1/p)
coupon = (FV*c)/p
coupon
B0 = 0
i = 0
for(i in time){
B0 = B0 + coupon*exp(-y*i)
}
B0 = B0 + FV*exp(-y*T)
B0
# Find the forward price in a long forward contract, with maturity **X** years later,
x = 3 # number of "years later"
time = seq(1/p, x, by=1/p)
time
I = 0
i = 0
for(i in time){
I = I + coupon*exp(-y*i)
}
I # this is the value of the coupons you will miss in the "x" years
disc = exp(-y*x) # discount factor for the "x" years
F0 = (B0 - I)/disc
F0
# Find the value of this forward contract **t** year later, on __ __th ____.
# Assume on that date the yield curve is flat at **y2**%p.a.
t = 2
y2 = 0.008
# Bt = discounted coupons and FV starting at time t
# (similar to B0 calc, but youre missing the coupons up to time t)
time = seq(1/p, T-t, by=1/p)
time
Bt = 0
for(i in time){
Bt = Bt + coupon*exp(-y2*i)
}
Bt = Bt + FV*exp(-y2*(T-t))
Bt
# I will miss out on the coupons issued between now (time t) and when I buy it (time x)
# I will miss out on the coupons issued for this amount of time: **(x - t)**
It = 0
time = seq(1/p, x-t, by=1/p)
time
for(i in time){
It = It + coupon*exp(-y2*i)
}
It
disc = exp(-y2*(x-t))
Value = Bt - It - F0*disc
Value
# Forward price of the new contract
Ft = (Bt - It)/disc
Ft
# Combined Value of her position
CombValue = Ft - F0
CombValue
# Combined Value = €20.19703 in this example)
# Optional check
# Value*exp(y2*(x-t)) - CombValue
# should = almost zero
4 – Finding the Present Value of the Expected Loss over the Lifetime of a Corporate Bond
A typical question on this topic could be:
A 4-year corporate bond, with face value €100, provides a coupon of 1.6% per year, payable annually, and has a yield of 1%. The risk-free yield curve is flat at 0.7% p.a.
Find the present value of the corporate bond and the corresponding risk-free bond with the same payments; hence find the present value of the expected loss over the lifetime of the corporate bond.
Assume that defaults of the corporate bond can only occur at the end of each year, just prior to each coupon payment, and that the recovery amount is €35. Estimate the risk-neutral default probability q per year, assuming it is the same each year.
To find the present value of the corporate bond, you calculate this in the usual way using 1% as your discount rate. Then calculate the present value of the risk-free bond, only this time use 0.7% as your discount rate. The present value of the expected loss is the PV of the riskless bond minus the PV of the corporate bond.
Then taking this further, we will calculate the expected loss at different time points over the lifetime of the bond. To do this, as it says in the question, we assume that defaults only occur at the end of each year, just prior to each coupon payment. Using a recovery amount of €35, we will estimate the risk-neutral probability of default for each time point. To do this, we find the value of the bond just prior to each coupon payment. My code brings you up to this point, you will then enter these values into a table and using discount rates you can find the present value of the expected loss times q (the probability of default) for each time point. Taking the total value of the expected loss times q along with the original value for the expected loss found before, we can solve for q, the probability of default each year.
A written solution can be found here: Simple Investment Modelling Algorithms
# Present Value of Expected Loss
# Find the cash value of the Corporate bond
T = 4 # T year bond
FV= 100 # Face Value
c = 0.016 # coupon
p = 1 # payments per year (annually = 1, semiannually = 2)
y = 0.01 # yield
time = seq(1/p, T, by=1/p)
coupon = (FV*c)/p
coupon
Bc0 = 0
i = 0
for(i in time){
Bc0 = Bc0 + coupon*exp(-y*i)
}
Bc0 = Bc0 + FV*exp(-y*T)
Bc0
# PV of Corporate Bond
# Find the cash value of the Risk Free bond
y = 0.007 # The risk-free yield
time = seq(1/p, T, by=1/p)
coupon = (FV*c)/p
coupon
Bf0 = 0
i = 0
for(i in time){
Bf0 = Bf0 + coupon*exp(-y*i)
}
Bf0 = Bf0 + FV*exp(-y*T)
Bf0
# PV of Risk Free Bond
# Present Value of Expected Loss
round(Bf0,2) - round(Bc0,2)
# We need to find the value of the RISKLESS BOND
# just before these default dates
A = 35 # this is the "recovery amount"
y = 0.007 # yield
# this code calculates the value of the Riskless Bond just
# BEFORE the default date that you will mark as **t**
t = 2 # change this number every time
# this wont work on the last run (when T = t) in this case,
# Bft should = coupon + FV
time = seq(1/p, T-t, by=1/p)
time
coupon = (FV*c)/p
coupon
Bft = coupon
i = 0
for(i in time){
Bft = Bft + coupon*exp(-y*i)
}
Bft = Bft + FV*exp(-y*(T-t))
Bft
5 – Constructing a Duration-Immunising Portfolio from Two Bonds
A typical question asking you to construct a duration-immunising portfolio from two bonds could be:
Assume a yield curve for all maturities of 1.1% p.a. You have a liability of €100,000 to be paid in seven years time. You wish to immunise this debt using two available bonds: a 5-year €100 zero-coupon bond and a 10-year €100 zero-coupon bond. Construct a duration-
immunising portfolio from these bonds (fractions of bonds are permitted). If after exactly two years, all rates fall by 0.3%, find the value of your bond holding (i) using a linear approximation in the rate change and (ii) exactly. Compare these with the new exact value of the liability.
Here you have a liability and a time when it has to be paid, you can create a portfolio consisting of a combination of these two bonds available, such that liability of your portfolio matches the liability that must be paid, in the same way the the duration of this portfolio matches the time at which the liability has to be paid. To do this you allow your liability to equal the present values of your bonds, each bond is multiplied by a constant denoting how many of each bond you will purchase, this value is unknown. Similarly, allow the duration of your liability equal the duration of your bonds, again each multiplied by the amounts. You can then take these simultaneous equations and solve them to find how many of each bond must be purchased to immunise your liability. My code finds these amounts for you.
Market conditions are susceptible to change overt this time span so we then consider the case where after two years all rates drop by 0.3%. We will find the value of our portfolio in this case using two methods. The first is a linear approximation in the rate of change using a formula used to calculate the change in the value of a bond by taking the negative duration multiplied by the change in yield multiplied by the value of the bond, this is denoted: ΔB = -D(Δy)B
The second finds value of our portfolio by recalculating using the new yield.
Both of these methods involve a lot of manipulation of equations, my code below solves for the value of the portfolio in both cases.
A written solution can be found here: Simple Investment Modelling Algorithms
# Duration Imunising Portfolio
y = 0.011 # risk free rate
# Liability of **L** to be paid in **T** years
L = 100000
T = 7
# 2 bonds available
# Bond 1:
t1 = 5 # length of bond 1
FV1 = 100
# Bond 2:
t2 = 10
FV2 = 100
# Present Value of Liability (then PV of Bond 1 and Bond 2)
PVL = L*exp(-y*T)
PVL
PVB1 = FV1*exp(-y*t1)
PVB2 = FV2*exp(-y*t2)
PVB1
PVB2
# Duration of Liability (then D of Bond 1 and Bond 2)
DL = L*T*exp(-y*T)
DL
DB1 = FV1*t1*exp(-y*t1)
DB2 = FV2*t2*exp(-y*t2)
DB1
DB2
# Then you should manipulate these simultaneous equations until
# you find the number of Bond 1's and Bond 2's you need.
# This code will give you the answer
# Number of Bond 2's to buy
NoB2 = (DL - (DB1*PVL)/PVB1) / (DB2 - (DB1*PVB2)/PVB1)
NoB2
# Number of Bond 1's to buy
NoB1 = (PVL - PVB2*NoB2) / PVB1
NoB1
# after exactly **x** years, all rates FALL by **y1%**
x = 2
y1 = 0.003
# Linear approx. of rate of change
#Bond price will rise by:
rise = (DL/PVL)*y1
rise # now write this as a percentage
( NoB1*FV1*exp(-y*t1) + NoB2*FV2*exp(-y*t2) ) * (1+rise)
# this is the value of the bond holding using linear approx rate of change
# Exactly
newy = y - y1
NoB1*FV1*exp(-newy*t1) + NoB2*FV2*exp(-newy*t2)
# exact value of bond holding using the new rate
6 – Pricing a European Put Option Contract
A typical question asking you to price a European put option contract could be:
Consider a 2.5-year coupon bond with a face value of €100 and a coupon rate of 4% paid
semi-annually. The zero curve is flat at 3.6%. Find the present value of the bond.
Consider a European call option on this bond with strike price €98 and expiration date in one year. The forward yield volatility is 60%. Find the present value of the interest paid during the life of the option and hence calculate the forward price of the bond in one year.
Find the duration of the bond option at maturity (i.e. in one year) and hence find the bond price volatility at that time.
Using Black’s model, calculate the present value of the call option.
Some relevant formulae are:
c = P(0,T)[F_B\Phi(d_1)-K\Phi(d_2)], \newline
where \; d_1 = \frac{ln(\frac{F_B}{K})+\frac{\sigma^2_BT}{2}}{\sigma_B\sqrt{T}} \newline
and \; d_2 = d_1 - \sigma_B \sqrt{T}\newline
\sigma_B = Dy\sigma_yOf course we find the present value of the bond in the usual way. The present value of the interest paid is equivalent to the present value of the coupons missed, we will miss two coupons in the first year of the life of the bond. We then use these two values to find the forward price of the bond in one year. To find the duration of the bond option at maturity (of the option) we recognise that at this time there will be 1.5 years still left until the maturity of the bond. We use this knowledge in applying the duration calculation (see written example). Using this duration value, we plug this into the formula to find the bond price volatility (σB=Dyσy). Taking all of this together, we can input these values into Black’s model to find the present value of the call option c (see formula above).
A written solution can be found here: Simple Investment Modelling Algorithms
# Pricing a European Put Option Contract
# Bond 1
t = 2.5 # duration of bond
c = 0.04 # coupon rate
FV = 100 # Face Value
y = 0.036 # risk free rate ("the zero curve is flat at __%")
# European Call Option / Bond 2
k = 98 # strike price
T = 1 # expiration date of contract / duration of second bond
sigmay = 0.60 # forward yield volatility
# other variables
p = 2 # payments per year (annually = 1, semiannually = 2)
# PV of the Bond
time = seq(1/p, t, by=1/p)
coupon = (FV*c)/p
coupon
B0 = 0
i = 0
for(i in time){
B0 = B0 + coupon*exp(-y*i)
}
B0 = B0 + FV*exp(-y*t)
B0
# Find the present value of coupons paid during the life
# of the option and hence find the forward price,
x = T # length of option
time = seq(1/p, x, by=1/p)
time
I = 0
i = 0
for(i in time){
I = I + coupon*exp(-y*i)
}
I # this is the value of the coupons you will miss in the "x" years
disc = exp(-y*x) # discount factor for the "x" years
F0 = (B0 - I)/disc
F0
# forward price
# Find the duration of the bond option at maturity
time = seq(1/p, t-T, by=1/p)
time
coupon = (FV*c)/p
coupon
top = 0
i = 0
for(i in time){
top = top + i*coupon*exp(-y*i)
}
top = top + (t-T)*FV*exp(-y*(t-T))
top
bottom = 0
i = 0
for(i in time){
bottom = bottom + coupon*exp(-y*i)
}
bottom = bottom + FV*exp(-y*(t-T))
bottom
Duration = top/bottom
Duration
# find the bond price volatility at this time
sigmab = Duration*y*sigmay
sigmab
# use Black's Model to find the present value of the call option
d1 = (log(F0/k) + 0.5 * sigmab^2 * T) / sigmab*sqrt(T)
d2 = (log(F0/k) - 0.5 * sigmab^2 * T) / sigmab*sqrt(T)
d1
d2
phi1 = pnorm(d1)
phi2 = pnorm(d2)
c = exp(-y*T)*(F0*phi1 - k*phi2)
c
# c = the present value of the call option
7 – Extending the LIBOR Curve and Using the Extended LIBOR Rates to Find Future Forward Rates
A typical question on this topic could be:
Use six-monthly compounding for all rates. The 6-month LIBOR spot rate is 0.4% and 12-month LIBOR spot rate is 0.6%. A bank trades swaps where a fixed rate of interest is exchanged for 6-month LIBOR, with payments being exchanged biannually. The 18-month and 24-month swap rates are 0.8% and 1.2% per annum. Estimate the 18-month and 24-month LIBOR rates. (Hint: construct par value bonds with coupon rates equal to the relevant swap rates). Using the extended LIBOR rates, find the 6-month forward rates starting in 6, 12 and 18 months.
Using these, show that the present value of 18-month swap is zero.
To find the LIBOR estimates, we construct par value bonds with coupon rates equal to the relevant swap rates, this involves some manipulation on paper (see written solutions), but my code below is configured to find these estimates. Finding the 6-month forward rates for the three dates is done using a formula which represented in the code below. See written solutions to see how you can show that the present value of 18-month swap is zero.
A written solution can be found here: Simple Investment Modelling Algorithms
# Extend the LIBOR Curve
L6 = 0.004 # 6 month LIBOR
L12 = 0.006 # 12 month LIBOR
S18 = 0.008 # 18 month Swap
S24 = 0.012 # 24 month Swap
c1 = S18*100
c1
c2 = S24*100
c2
# Construct par value bonds with coupons equal to the
# relevant Swap rates
# 1.5 year LIBOR estimate (18 month LIBOR est)
L18=(((100+c1/2)/(100-(c1/2)/(1+L6/2)-(c1/2)/(1+L12/2)^2))^(1/3)-1)*2
L18
# this is your 1.5 year LIBOR est
# 2 year LIBOR estimate (24 month LIBOR est)
xx = 100-(c2/2)/(1+L6/2)-(c2/2)/(1+L12/2)^2-(c2/2)/(1+L18/2)^3
L24=(((100+c2/2)/xx)^(1/4)-1)*2
L24
# Using the extended LIBOR rates, find the 6-month
# forward rates starting in 6, 12 and 18 months.
# Using these, show that the present value of 18-month swap is zero.
# 6-month forward rates starting in 6 Months
F6 = ( ( ((1+L12/2)^2) / ((1+L6/2)^1) ) -1 )*2
F6
# 6-month forward rates starting in 12 Months
F12 = ( ( ((1+L18/2)^3) / ((1+L12/2)^2) ) -1 )*2
F12
# 6-month forward rates starting in 18 Months
F18 = ( ( ((1+L24/2)^4) / ((1+L18/2)^3) ) -1 )*2
F18