Investment Science: The Basics of investing in Bonds

Tuesday, August 31, 2010

If there is one major thing I learnt from the Investment Science course, it's the investments in bonds and how to immunize your portfolio against interest rates, in this article, we will see the definition of a bond and  the types of bonds with a simple example.

A bond is an investment which pays you a fixed amount of money at it's maturity date and a coupon bond will pay you periodic payments (coupon payments) in addition to the fixed amount of money you will receive at the end of it's maturity.

The price of a bond (P) is the money you pay to buy the bond.
The face value of a bond (F) is the money you will receive at it's maturity date.
The coupon payment (C) is the money you will receive periodically till the maturity date of the bond (based on the coupon rate), this means in a coupon bond, you will receive (C + F) at it's maturity time.

Typically coupons are paid two periods in an year or every 6 months.

As obvious from the above discussion, bonds are classified into two types

1. Coupon bond - Which pay periodic coupons till maturity.

2. Zero coupon bond - Here there are no coupon payments involved, you buy a bond at a price P and get money equal to the face value of the bond (F) at maturity, therefore the Price (P) of a bond is the present value of the bond which pays F in future.

Let me answer some FAQ about bonds (More questions and answers will be added when I get additional information on bonds).

1. Who issues bonds and why? Bonds are issued by Governments, corporations, banks and other institutions whereby they can raise quick capital which can be used for several projects.

2. What if the bond issuer defaults? Typically Government bonds (like the one issued by US treasury) are essentially considered risk-free investments (they never default), however you may need to learn about the Bond credit ratings before buying a bond.

3. What factors influence bond market? The simple answer is interest rate, when you say the bond market is down, it means the interest rates are high and vice-versa, we will learn more about this in a future article.

Let's conclude this section with a small example.

Find the periodic coupon payment (occurs every 6 months or two periods in an year) of a $1000 bond at 5% coupon rate.

Here Coupon payments per year C = 1000 * 5% = $50

Number of periods in an year m = 2

Therefore periodic coupon payment (which occurs every 6 months) = C/m = 50/2 = $25.

Investment Science: Variable rate Mortgage

When you pay a mortgage/loan, the interest rates may not always be same, interest rates may vary in the due course of the loan depending on the market rates and a given interest rate may be applicable for only some period of time as per the contract, in this case what's the best thing to do when the interest rate changes (assume rates increase)? For answering this question we will look into a classic example from  Investment Science by Luenberger (Problem 3.8).

Problem 3.8

Variable rate mortgage: The smith family just took out a variable-rate mortgage on their new home. The mortgage value is $100,000, the term is 30 years and initially the interest rate is 8%. The interest rate is guaranteed for 5 years, after which time the rate will be adjusted according to prevailing rates. The new rate can be applied to their loan by either changing the payment amount or by changing the length of the mortgage.

(a) What is the original yearly mortgage payment? (Assume payments are yearly)
(b) What will be the mortgage balance after 5 years?
(c) If the interest rate rate on the mortgage changes to 9% after 5 years, what will be the new yearly payment that will keep the termination time the same?
(d) Under the interest change in (c), what will be the new term if payments remain the same?

We know that the present value (P) of an annuity that pays an amount A every individual period for 'm' periods for 'n' years at an yearly interest rate 'r' is given by


Given

(a) P= $100,000 n = 30, m = 1, n * m = 30, r = 8%, r/m = 8% = 0.08

Therefore


A = $8882.74 (The original yearly mortgage payment)

(b) The mortgage balance after 5 years is given by (refer this article)


Solving, we get, Mortgage balance after 5 years = $94,821.12

Now we have a new set of values after 5 years, which is

P = $94,821.12 (This becomes the new outstanding principal), m = 1, r = 9%, r/m = 9% = 0.09

(c) Here we need to keep the termination time the same (which is 30 - 5 = 25), therefore we need to find the new payment which will complete the loan at this termination time, or in other words,


A = $9653.4
The new yearly payment that will keep the termination time the same is $9653.4.

(d) Here we need to keep the same old monthly payments as we got in (a), but need to find the new term
(termination time) at which the loan will complete after the interest rate change (which increased to 9% from 8%, therefore if your payments are same at 9% interest rate, the new term is going to increase).


n = 37.56 years

In other words, it will take (37.56 + 5 = 42.56 years) to pay out the loan when you keep the payments same after 5 years at 9% interest rate.

The learning exercise: Which is beneficial, option (c) or (d)?

Investment Science: Amortization schedule of a Mortgage/loan payment

Monday, August 30, 2010



In this post, we will look into the amortization schedule of a mortgage/loan payment, which will breakdown the loan into equal periodic payments (typically monthly payments for a car of home loan), also this schedule will contain the interest paid on a particular period or month, the principal paid, balance principal remaining after the current payment, the schedule repeats itself till the outstanding payment reaches zero.

We will illustrate the amortization schedule with a simple example.

Consider a loan of $1000 for 5 years at an yearly interest rate of 10% payable every year.

We know that the present value (P) of an annuity that pays an amount A every individual period for 'm' periods for 'n' years at an yearly interest rate 'r' is given by
For the above case,

P = $1000, n = 5, m = 1, r = 10%, r/m = 10%, therefore

A = $263.80

This is the yearly amount paid for the loan, the total interest paid = A * (n * m) - P = (263.80 * 5) - 1000 = $319

We will break this down into yearly schedule based on the below reusable spreadsheet for computing the amortization schedule for a given mortgage/loan payment.


The above spreadsheet can be used for any mortgage/loan payment for a principal 'P' for 'm' periods per year for 'n' years at an yearly interest rate r%.

From the above figure, we can infer the following properties.

1. The balance in principle at any period i is given by

For example, for the above loan schedule, the balance principal after two years is given by


Using the spreadsheet:

The above excel can be reused for any mortgage/loan payment for a principal 'P' for 'm' periods per year for 'n' years at an yearly interest rate r%, for using this, all you need to do is to copy the contents of row 15 (corresponding to period 1) of the spreadsheet to the next n * m - 1 rows (till the balance principal becomes zero).

Investment Science - Understanding the mathematics of Mortgage payments - 1

Sunday, August 29, 2010

In the last post, we have seen the annuity formula and an example on how you pay a car loan for a given interest rate for a fixed period of time, in this post we will look into an example to illustrate how the mortgage period affects your total interest payment, for this we will look into an example problem from Investment Science by Luenberger (Problem 3.6).

Problem 3.6

The biweekly mortgage: Here is a proposal that has been advanced as a way for homeowners to save thousands of dollars on mortgage payments: pay biweekly instead of monthly. Specifically if monthly payments are x, it is suggested that one instead pay x/2 every two weeks (for a total of 26 payments per year). This will pay down the mortgage faster, saving interest. The savings are surprisingly dramatic for this seemingly minor modification - often cutting the total interest payment by over one-third. Assume a loan amount of $100,000 for 30 years at 10% interest, compounded monthly.

(a) Under a monthly payment program, what are the monthly payments and the total interest paid over the course of the 30 years?
(b) Under the biweekly program, when will the loan be completely repaid and what are the savings in total interest paid over the monthly program (You may assume biweekly compounding for this part.)

We know that the present value (P) of an annuity that pays an amount A every individual period for 'm' periods for 'n' years at an yearly interest rate 'r' is given by


(a) In monthly payment program

P = 100000, r = 10%, n = 30, m = 12, r/m = 10%/12 = 0.1 / 12 = 0.0083333, therefore


Solving, we get A = $877.568

Total interest paid over the course of 30 years = A * (n * m) - P = 877.568 * 30 * 12 - 100000  =  $215924.48

(b) In biweekly payment program

A = $877.586 / 2 = $438.784
m = 26, r/m = 10% / 26 = 0.1 / 26 = 0.0038461, therefore


Solving, we get n = 20.95

In the biweekly program, the loan will be repaid in 20.95 years (compared to 30 years in the monthly payment program)

Total interest paid in biweekly payment program = A * (n * m) - P = 438.784 * 20.95 * 26 - 100000 = $139005.645

Therefore total interest savings in biweekly payment program compared to monthly payment program = 215924.48 - 139005.645 = $76,918.835 or 76918.835 / 215924.48 = 0.3562 = 35.62% (Wow! when we dig deep into financial mathematics, we can always find a treasure!)

Investment Science: Perpetual Annuity - How to compute your monthly loan payment

Friday, August 27, 2010

In this post on Investment Science, we will look into the perpetual annuity formula with a simple example.

What is an annuity? In simple words it's a stream of fixed payments over a fixed period of time, the best example is mortgage, consider a car loan or a home loan you may have taken, how to derive the monthly payments for your loan, how much interest you pay for over the period of your loan, how much interest your pay every month, how much balance principal you pay in your monthly loan  payment? Which is beneficial, paying additional mortgage for short term or paying less mortgage for long term? Questions galore. In the next few posts we will be exactly looking into these.

Keep in mind that learning the mathematics of your mortgage payment will help you make informed decisions and to start with that, just remember this simple, yet powerful one, the annuity formula.

The present value (P) of an annuity which pays an amount A every individual period (an year) for 'n' years at an yearly interest rate 'r' is given by


Note: If the number of periods per year is > 1, say 'm' periods per year (m = 12 for mortgages since mortgages are paid monthly), the above formula for a total of n years becomes


This simple, yet powerful formula forms the basis for annuity payments, we will look a practical example.

Example:

Suppose you have taken a car loan for $15000 which is to be paid every month over 5 years at an interest rate of 5%, then how much you need to pay every month.

Here

P = $15000
m = 12 (Monthly payment)
Monthly interest rate = r/m = 5%/12 = 0.05/12 = 0.004166
Number of years the loan is to be paid, n = 5
Number of periods the loan is to be paid = n * m = 5 * 12 = 60

Therefore


A = $283.063

Therefore over the period of 5 years, the interest you would have paid = A * (n * m) - P = 283.063 * 60 - 15000 = $1983.78042


Troubleshooting: Method "/lib/svc/method/fs-local" failed with exit status 95 in Solaris - Possible Solution

Yesterday, when I was working with my Solaris system, after reboot, I couldn't ssh into the system, the ssh command failed with the "Connection refused" error, after some research (by logging into the serial console of the server and rebooting the system again through the serial console), I found the below system error message.

svc:/system/filesystem/local:default: Method "/lib/svc/method/fs-local" failed with exit status 95

Then everything came to standstill till I figured out the solution.

Solution:

1. The possible reason for ssh "Connection Refused" may be due to the fact that some filesystem, whose entry is in /etc/vfstab would have failed to mount properly.

A typical vfstab entry looks like the below one.

$ cat /etc/vfstab
#device         device          mount           FS      fsck    mount   mount
#to mount       to fsck         point           type    pass    at boot options
#
fd      -       /dev/fd fd      -       no      -
/proc   -       /proc   proc    -       no      -
/dev/dsk/c0t2d0s1       -       -       swap    -       no      -
/dev/dsk/c0t2d0s0       /dev/rdsk/c0t2d0s0      /       ufs     1       no      logging
...
...

2. To find which entry was causing the problem, I checked the system log, the entries were as follows.

$ cat /var/svc/log/system-filesystem-local:default.log
"/var/svc/log/system-filesystem-local:default.log" 149 lines, 9256 characters
[ Dec  3 01:25:35 Rereading configuration. ]
[ Dec  3 01:25:52 Executing start method ("/lib/svc/method/fs-local") ]
[ Dec  3 01:25:52 Method "start" exited with status 0 ]
[ Dec  3 01:31:54 Executing start method ("/lib/svc/method/fs-local") ]
....
....
WARNING: /sbin/mountall -l failed: exit status 33
[ Aug 27 07:33:14 Method "start" exited with status 95 ]
[ Aug 27 07:35:37 Leaving maintenance because clear requested. ]
[ Aug 27 07:35:37 Enabled. ]
[ Aug 27 07:35:37 Executing start method ("/lib/svc/method/fs-local") ]
mount: /var/js/1/boot: No such file or directory
WARNING: /sbin/mountall -l failed: exit status 1
[ Aug 27 07:35:37 Method "start" exited with status 95 ]
[ Aug 27 07:37:19 Leaving maintenance because clear requested. ]
[ Aug 27 07:37:19 Enabled. ]
[ Aug 27 07:37:19 Executing start method ("/lib/svc/method/fs-local") ]
mount: /var/js/1/boot: No such file or directory
WARNING: /sbin/mountall -l failed: exit status 33
[ Aug 27 07:37:19 Method "start" exited with status 95 ]
svc:/system/filesystem/local:default: Method "/lib/svc/method/fs-local" failed with exit status 95

Looks like there was an entry /var/js/1/boot in /etc/vfstab which didn't mount because the filesystem was not present, hence the error.

3. After finding the root cause for the problem, I removed the line which mounted the /var/js/1/boot filesystem in /etc/vfstab and rebooted the system, now I could login through ssh.

Therefore when you get ssh "Connection refused" error, chances are that some filesystem whose entry is in /etc/vfstab would have failed to mount, in which case check the system log, remove the line causing the problem from /etc/vfstab and ssh should work fine.

Configuring a virtual interface in Solaris

Thursday, August 26, 2010

In this short article, we will see the steps to configure a virtual interface on Solaris, a virtual interface can be created for assigning multiple addresses to a physical interface and has several applications, the steps to create a virtual interface for a given physical interface is shown below.

1. Consider a physical interface

$ uname -a
SunOS qe9-proxy 5.10 Generic_139556-08 i86pc i386 i86pc
$ ifconfig -a
lo0: flags=2001000849 mtu 8232 index 1
        inet 127.0.0.1 netmask ff000000
e1000g0: flags=201000843 mtu 1500 index 2
        inet 10.1.150.151 netmask ffffff00 broadcast 10.1.150.255
        ether 0:14:4f:28:29:34

2. If we want to create a virtual interface e1000g0:1 for the physical interface e1000g0 and assign an address 10.1.150.152 to the virtual interface, the command would be as follows.

2.1 Plumb the virtual interface e1000g0:1

$ ifconfig e1000g0:1 plumb
$ ifconfig -a
lo0: flags=2001000849 mtu 8232 index 1
        inet 127.0.0.1 netmask ff000000
e1000g0: flags=201000843 mtu 1500 index 2
        inet 10.1.150.151 netmask ffffff00 broadcast 10.1.150.255
        ether 0:14:4f:28:29:34
e1000g0:1: flags=201000842 mtu 1500 index 2
        inet 0.0.0.0 netmask 0

2.2 Assign the IP address (10.1.150.152) to the virtual interface e1000g0:1

$ ifconfig e1000g0:1 10.1.150.152 netmask 255.255.255.0 broadcast + up
$ ifconfig -a
lo0: flags=2001000849 mtu 8232 index 1
        inet 127.0.0.1 netmask ff000000
e1000g0: flags=201000843 mtu 1500 index 2
        inet 10.1.150.151 netmask ffffff00 broadcast 10.1.150.255
        ether 0:14:4f:28:29:34
e1000g0:1: flags=201000843 mtu 1500 index 2
        inet 10.1.150.152 netmask ffffff00 broadcast 10.1.150.255

Note: There is no Ethernet address displayed for the virtual interface as shown above (obviously)

2.3 Test the reachability of the IP address served by the virtual interface

$ ping -s 10.1.150.152
PING 10.1.150.152: 56 data bytes
64 bytes from 10.1.150.152: icmp_seq=0. time=0.113 ms
64 bytes from 10.1.150.152: icmp_seq=1. time=0.101 ms
^C
----10.1.150.152 PING Statistics----
2 packets transmitted, 2 packets received, 0% packet loss
round-trip (ms)  min/avg/max/stddev = 0.101/0.107/0.113/0.0085
$

Using the above procedure, one can create several virtual interfaces (like e1000g0:2, e1000g0:3 and so on for the physical interface e1000g0), to disable a virtual interface, use the unplumb command (like ifconfig  e1000g0:1 unplumb), similar to the we used to unplumb a physical interface.

Investment Science: Incremental internal rate of return for analyzing alternate investment options

In the previous posts, we have seen the way to compute Net Present Value (NPV) and Internal rate of return (IRR) for cash flow streams and how they affect investment decisions, continuing that discussion, in this post we will see an example of how incremental internal rate of return can be used to analyze alternate investment options, for this we will look into an example from Investment Science by Luenberger (Problem 2.8) which goes as follows.

Problem 2.8

Two copy machines are available, both have useful lives of 5 years. One machine can be either leased or purchased outright; the other must be purchased. Hence there are a total of three options: A, B and C. The details are shown in the below table (The first year's maintenance is included in the initial cost. Then there are four additional maintenance payments, occurring at the beginning of each year, followed by revenues from resale). The present values of the expenses of these three options using a 10% interest rate are also indicated in the table. According to a present value analysis, the machine of least cost, as measured by the present value, should be selected, that is option B from the below table.

It is not possible to compute the IRR for any of these alternatives, because all cash flows are negative (except for the resale values). However it is possible to calculate the IRR on an incremental basis, Find the IRR corresponding to a change from A to B. Is a change from A to B justified on the basis of  the IRR?


Analysis:

Since IRR cannot be calculated for any of the above cash flow streams (remember IRR for a cash flow stream has unique positive solution only if the sign changes from negative to positive or vice versa, in the cash flow stream, see here). But we can analyze the IRR corresponding to a change from A to B and a change from A to B is justified if the IRR(Change from A to B) > 10% (interest rate).

To calculate the IRR from A to B, we subtract the cash flow streams A and B (or B and A) and then compute the IRR of the resulting cash flow stream as shown in the below figure.



Note: In the below figure, each cash flow stream has 6 cash flows, the first year's maintenance + initial cost (negative cash flow), followed by 4 additional maintenance cost payments (all negative cash flows) followed by a resale value (>=0), for example, for option A, the cash flow stream would be like (-6000, -8000, -8000, -8000, -8000, 0) based on the above data in the table.

From the spreadsheet computation, we find that the internal rate of return for a change from A to B is 11.84%, since this value is greater than the 10% interest rate, therefore a change from A to B is justified based on IRR, in general, incremental IRR calculation an be used to analyze different investment alternatives like the above example, this will be handy in cases where IRR cannot be calculated for individual cash flow streams.

Investment Science: Net Present Value vs Internal rate of return analysis in investment decisions

Wednesday, August 25, 2010


Now we have a solid understanding of Net Present Value (NPV) and Internal rate of return (IRR) concepts, both can be effectively put into use for analyzing investment decisions, now an interesting question arises, will NPV and IRR always agree? For answering this question, we will look into a sample problem (Problem 2.11) from Investment Science by Luenberger.

Problem 2.11 Luenberger:

Conflicting recommendations: Consider the two projects whose cash flows are shown in the table below. Find the IRRs of the two projects and the NPVs at 5%. Show that the IRR and NPV figures yield different recommendations. Can you explain this?

From the formulas for calculating Net Present Value (NPV) and Internal rate of return (IRR), and the spreadsheet way of solving them, we get

NPV(Project 1) = 29.88, NPV(Project 2) = 31.84

IRR(Project 1) = 15.24%, IRR(Project 2) = 12.38%

Therefore based on the Net Present Value Analysis, one would recommend Project 1, but based on Internal rate of return calculation, one would recommend Project 2, therefore the IRR and NPV figures yield different recommendations.

But why?

The Net Present Value (NPV) and IRR (Internal rate of return) may yield consistent results if cash flows are reinvested (at IRR), IRR calculation assumes that the cash flows are reinvested at IRR which is not the case in the above scenario, hence the conflict in recommendation, in general it's advisable to analyze an investment based on NPV (Net Present Value analysis).

Don't worry if the above statement is not clear to you, we should understand one thing from this article, Net Present Value analysis and IRR computations would yield consistent recommendations if the cash flows are reinvested at IRR, else they may differ.


Investment Science: Net Present Value and Internal rate of return calculations in spreadsheet (OpenOffice)

Tuesday, August 24, 2010

In the previous posts, we have seen ways to calculate the Net Present Value and Internal rate of returns using Texas Instruments BA II Plus Financial Calculator, in this short post, we will see the way to compute NPV and IRR using the openoffice spreadsheet program.

Given a cash flow stream (-2, 1, 1, 1) at an yearly interest rate of 10%, we know that

Net Present Value (NPV) = -2 + 1 / (1.1) + 1 / (1.1 ^ 2) + 1 / (1.1 ^ 3) = 0.487

Internal rate of return (IRR) is computed from

-2 + 1 / (1 + IRR) + 1 / (1 + IRR) ^ 2 + 1 / (1 + IRR) ^ 3 = 0

Internal rate of return IRR = 23.38%

To do these computations, enter the cash flow streams in columns, let's assume that we entered the above cash flow stream (-2, 1, 1, 1) from rows B7 to B10, then

Net Present value (NPV) = NPV(0.1, B8:B10) + B7
Internal rate of return (IRR) = IRR(B7:B10)



Figure 1: Calculating NPV and IRR in openoffice spreadsheet program

Note: While calculating the Net Present Value, never include the initial cash flow (-2 in the above example) in the NPV function, but enter the subsequent cash flow streams which are to be discounted in the NPV function along with the interest rate and then add the initial cash flow to the result of the NPV function.

Investment Science: Computing the Internal rate of return of a cash flow stream using a financial calculator

Saturday, August 21, 2010

In this short article, we will see how to use a financial calculator to compute the Internal rate of return (IRR) of a cash flow stream, the financial calculator used for illustrating this example is Texas Instruments BA II Plus.

For more details on computing internal rate of return of a cash flow stream, refer this article.

Consider a cash flow stream (-2, 1, 1, 1) , the internal rate of return of this cash flow stream is given by


Solving the above equation, we get IRR = 23.38%.

To calculate the internal rate of return of the above cash flow stream with a Texas Instruments BA II Plus Financial Calculator, follow the below steps.

1. Reset previous cash flow streams if any (read the -> below as a sequence)

Use the following keys: 2ND -> RESET -> ENTER

2. Set the number of decimal places in the financial calculator

By default, the number of decimal digits displayed is 2 and often you need a better precision for a computation like IRR, therefore, set the number of decimal places to atleast 4, for doing this

Use the following keys: 2ND -> FORMAT -> 4 -> ENTER

3. Compute Internal rate of return: Now we are ready to enter the stream of inputs for the cash flow stream (-2, 1, 1, 1)  for computing the internal rate of return, for this use the following keys

3.1.   CF -> 2 -> - (Minus key) -> ENTER ->

3.2.   (CF1 appears) 1 -> ENTER ->

3.3.   (F01 appears with a default value 1, accept it as it is) ENTER ->

3.4.   (CF2 appears) 1 -> ENTER ->

3.5.   (F02 appears with a default value 1, accept it as it is) ENTER ->

3.6.   (CF3 appears) 1 -> ENTER ->

3.7.   (F03 appears with a default value 1, accept it as it is) ENTER ->

3.8.   (CF4 appears) Now Press IRR -> CPT keys, IRR = 23.3752% will be displayed.

Unique positive value of internal rate of return: When a cash flow stream sign doesn't change from negative to positive or vice-versa, trying to compute the internal rate of return result in error, for more details, refer here.

Investment Science: Internal rate of return of a cash flow stream


In this article, we will see the definition of Internal rate of return or IRR, Internal rate of return is often used to analyze investment decisions in that given two cash flow streams, generally the one with the highest internal rate of return is preferred.

Definition: To keep it simple, just keep this in mind, the internal rate of return of a cash flow stream is that return which will make the present value of the cash flow stream equal to zero.

From this definition, it's very obvious that Internal rate of return calculation doesn't depend on interest rate, whereas Net Present Value calculation is.

We know that the Net Present Value of a cash flow stream (x1, x2, x3,....,xn) at an yearly interest rate 'r' for 'n' periods is given by

The internal rate of return is that value of 'r' (IRR) which will make the Net Present value of the cash flow stream equal to zero.


Example:

Find the internal rate of return of a cash flow stream (-2, 1, 1, 1) at an annual interest rate 10% (does this matter to our computation).

The internal rate of the above cash flow stream is given by

-2  + 1 / (1 + IRR) + 1 / (1+ IRR) ^ 2 + 1 / (1 + IRR) ^ 3 = 0

Solving the above equation, we get IRR = 0.2338 or 23.38%

Net Present Value and Internal rate of return are often compared to analyze different investments which raises a question, given a cash flow stream, which analysis would yield the right results, well we will look into that in a future article, it's fun.

The last question/thought of the day: Is there an unique positive solution for Internal rate of return? There is one if the cash flow stream changes from negative to positive or vice versa, for example internal rate of return for the below cash flow streams are unique, positive and equal, please refer Investment Science by Luenberger if you want to explore more on this.

1. Cash flow stream 1: (-1, 0, 3), Internal rate of return IRR = 73.21%

2. Cash flow stream 2: (1, 0, -3) (This equivalent to -(CF1)), Internal rate of return IRR = 73.21% (Obviously, but just remember the sign change in cash flows)

Investment Science: Computing Net Present Value of a cash flow stream using a financial calculator

Thursday, August 19, 2010

In this short article, we will see how to use a financial calculator to compute the Net Present Value of a cash flow stream, the financial calculator used for illustrating this example is Texas Instruments BA II Plus.

For more details on computing Net Present Value of a cash flow stream, refer my  previous article.

Consider a cash flow stream (-2, 1, 1, 1) , the Net Present Value of this cash flow stream at an annual interest rate 10% is given by

NPV = -2 + 1 / (1.1) + 1 / (1.1 ^ 2) + 1 / (1.1 ^ 3)  = 0.48685

To calculate the Net Present Value of the above cash flow stream with a Texas Instruments BA II Plus Financial Calculator, follow the below steps.

1. Reset previous cash flow streams if any in the calculator (read the -> below as a sequence)

Use the following keys: 2ND -> RESET -> ENTER

2. Set the number of decimal places in the financial calculator

By default, the number of decimal digits displayed is 2 and often you need a better precision for a computation like NPV, therefore, set the number of decimal places to atleast 4, for doing this

Use the following keys: 2ND -> FORMAT -> 4 -> ENTER

3. Compute Net Present Value: Now we are ready to enter the stream of inputs for the cash flow stream (-2, 1, 1, 1) with interest rate 10% for computing Net Present Value, for this use the following keys

3.1   CF -> 2 -> - (Minus key) -> ENTER ->

3.2   (CF1 appears) 1 -> ENTER ->

3.3   (F01 appears with a default value 1, accept it as it is) ENTER ->

3.4   (CF2 appears) 1 -> ENTER ->

3.5   (F02 appears with a default value 1, accept it as it is) ENTER ->

3.6   (CF3 appears) 1 -> ENTER ->

3.7   (F03 appears with a default value 1, accept it as it is) ENTER ->

3.8   (CF4 appears) Now Press NPV key

3.9   (I= appears, now enter the yearly interest rate) 10 -> ENTER ->

3.10 (NPV appears) Press CPT key, now NPV = 0.4869 will be displayed.

Common mistake when entering interest rate: At times when you are in hurry, it's easy to enter the interest rate as 10/100 or 0.1, but note that the interest rate is already accepted as a percentage value, also when the interest is compounded more than once in a year, enter the interest rate value as yearly interest rate divided by the number of periods (for a 10% annual interest rate compounded monthly, this will be 10/12).


Investment Science: Present value analysis of cash flow streams

Tuesday, August 17, 2010

In the last post, we saw the formula for calculating the present value and future value of a stream, in this section, we will see an example on how to apply the present value analysis to arrive at an investment  decision, for illustrating this, we will solve an exercise problem from by Investment Science David G. Luenberger (Problem 2.7) to make things clear.

The investment scenario is as follows.

Suppose there exists an opportunity to plant trees which can be harvested in future and there are two choices to cut the tree

1. Cut early with an associated cash flow stream of (-1, 2) (Get quick returns)
2. Cut later with an associated cash flow stream of (-1, 0, 3) (Wait an extra year so that the trees will grow, hence there is a chance for a greater revenue).

Suppose the interest rate is 10% and cash flows occur at the end of the year.

Case 1: Cut early with an associated cash flow stream (-1, 2)


Case 2: Cut later with an associated cash flow stream (-1, 0, 3)


As obvious, based on the present value analysis, it's better to cut later, but now a different case arises.

Problem 2.7: Gavin Jones is inquisitive and and determined to learn both the theory and the application of investment theory. He pressed the tree farmer for additional information and learned that it was possible to delay cutting of trees for another year. The farmer said that from a present value perspective, it was not worthwhile to do so. Gavin instantly deduced that the revenue obtained must be less than x, What is x?

Let's first calculate the present value this cash flow stream.

In case Gavin decided to wait one more year, then the present value of this cash flow stream would be (assume that the cash flow of this stream is (-1, 0, 0, x) where x > 0)

Case 3: Wait one more year than in case 2 and cut the tree with a cash flow stream (1, 0, 0, x)


But since the farmer mentioned that from the present value analysis, it's not worthwhile to do so, therefore

Therefore the revenue obtained must be less than 3.3 (or the cash flow stream at the end of three years should be <3.3)

In short, one can analyze different cash flow streams based on computing their present values to arrive at an investment decision.

References: Investment Science by Gavin G. Luenberger.

Investment Science: Cash flow streams and their present and future values


In this article, we will look into what a cash flow stream looks like and how to calculate the present value and future value of a cash flow stream, let's first define a cash flow stream and the present value and future value of that cash flow stream.

A cash flow stream is of the form (x1, x2, x3, x4,....,xn)  where x1, x2, x3, x4,....,xn are the cash flows corresponding to compounding periods 1, 2, 3, 4,....,n.

Typically cash flows occur at the end of each year or at the end of each compounding period.

The present value of the above cash flow stream (x1, x2, x3, x4,....,xn) at an interest rate 'r' for 'n' periods is given by


The future value of a cash flow stream (x1, x2, x3, x4,....,xn) at an interest rate 'r' for 'n' periods is given by


Example:

Lets's keep it simple and try to calculate the present value and future value of a cash flow stream, consider that you invested $100 in a bank for two years at an interest rate 10% compounded annually.

After two years, the growth of the investment would be

FV = 100 * (1 + 10/100) ^ 2 = $121

Here we can define two equivalent cash flows

Cash flow 1: (100, 0, 0) = You invest $100 today for a period of two years ($100 is the initial investment) at an interest rate r = 10%

The present value and future value of this cash flow stream based on the above formula is given by

PV  = 100 + 0 / (1.1) + 0 / (1.1) ^ 2 = 100

FV = 100 * (1.1) ^ 2 + 0 * 1.1 + 0 = 121

Cash flow 2: (0, 0, 121) = You get $121 after two periods from the bank (at an interest rate r = 10%), the present value of this cash flow stream based on the above formula is given by

PV = 0 + 0 / (1.1)  + 121 / (1.1) ^ 2 = 100

FV = 0 * (1.1) ^ 2 + 0 * 1.1 + 121 = 121

Therefore the cash flows (100, 0, 0) and (0, 0, 121) are equivalent, you invest $100 today in a bank at 10% interest rate and the cash flow at the end of 2 years would be $121and this cash flow can be defined as (100, 0, 0) or (0, 0, 121).

The above cash flow is quite simple, in general different investments produce different cash flow streams at the end of every year (they can be negative, meaning negative returns at the end of a period or a positive value), for any cash flow stream, the above formula can be used to compute the present value and the future value of the stream.

In the next article, we will see an example of a cash flow stream and how the present value value analysis affects an investment decision.

Investment Science: Present Value and Future Value of an investment

Monday, August 16, 2010

This article answers the questions

1. What is the present value and future value of an investment?
2. What is the relationship between the present value and future value of an investment?

1. What is the present value and future value of an investment?

Lets take an example, assume that you invest $1000 in a bank for 2 years at an interest rate 10% (compounded yearly), after 2 years, the growth of the investment is

Growth = 1000 * (1 + 10/100) ^ 2 = 1210
 
The growth of an investment (under compound interest) is known as the "Future value" of an investment.
The amount invested initially to attain that growth is called the "Present value" of an investment.
 
Suppose that you have a current bank balance of $1210 that pays 10% interest, what's the investment's worth two years back, clearly it's worth 1210 / (1 + 10/100) ^ 2 = $1000

Now we have a relationship between present value and the future value of an investment.

2. What is the relationship between the present value and future value of an investment?

For an investment compounded at an yearly interest rate 'r' for 'n' years at a present value PV, the future value FV is given by, 


 where (1 + r) ^ n is called as the growth factor.



where 1 / (1 + r) ^ n is called the discount factor.

If the investment is compounded at 'm' periods per year at an interest rate 'r', then after 'n' years,



and


An investor would always make a decision based on the present value analysis (or maximizing the Net Present Value), for a simple investment like the one shown above, it's easy to calculate the present value, but for an investment which involves a cash flow stream which is not uniform, it's not as trivial as shown above, which brings us to our next article, which will be on understanding more on cash flow streams.

Investment Science: Effective and Nominal interest rates

Sunday, August 15, 2010

In this post, we will see the difference between the effective and nominal interest rates, if you are a beginner, I would strongly recommend you to refer to the concept of compound interest in this article, note that the understanding of interest rates and how it's compounded is very crucial in investments as investment science makes sense only if there is an interest rate involved, isn't it?

As mentioned earlier, interest rate can be compounded yearly, monthly or quarterly, therefore if you have an investment which is compounded 'm' periods per year, then it's easy to ask a question "What is the yearly interest rate for an investment compounded 'm' periods per year", accordingly, we define two interest rates

1. Nominal interest rate 'r' compounded 'm' periods per year - Here the investment grows by (1 + r/m) per period and (1 + r/m) ^ m per year.
2. Effective interest rate (r') which is the equivalent yearly rate for the nominal interest rate 'r' - Here the investment grows by (1 + r') per year.

Therefore the relationship between effective interest rate and nominal interest rate should be derived from

For example, given an interest rate of 10% compounded quarterly, what is the effective interest rate?

Here r = 10%, m = 4, therefore the effective interest rate

r' = (1 + 0.1 / 4) ^ 4 - 1 = 10.381%

Also, when it comes to continuous compounding (where the compounding period m -> Infinity), then the relationship between effective interest rate (r') and nominal interest rate (r) is given by


Note, when compounded continuously with an interest rate of 10%, the effective interest rate will be

r' = e ^ (10/100) - 1 = 10.517%

Therefore continuous compounding will always be beneficial.

In the next post, we will see look into the present value and the future value of an investment.


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