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Q: calculating current value of a bond ( Answered,   0 Comments )
Question  
Subject: calculating current value of a bond
Category: Business and Money > Accounting
Asked by: thanksmate-ga
List Price: $20.00
Posted: 28 Sep 2004 08:29 PDT
Expires: 28 Oct 2004 08:29 PDT
Question ID: 407428
BACKGROUND
There are bonds valued at $50,000 each and they are due in 20 years.
They pay interest every 4 months (tri-annually) and have a "coupon
rate" of 12%

QUESTIONS
What is a coupon rate and what is the current value of a bond if the
required rate of return (yield) on the bonds are 15%?

Please show all working, calculations, definitions etc.

Thank you very much.

Request for Question Clarification by omnivorous-ga on 28 Sep 2004 09:53 PDT
TM --

This question can be answered without a spreadsheet but it's MUCH
easier using Microsoft Excel.  Even if you don't have a copy of Excel,
you'd be able to view the calculations in your web browser.

Not, let me show you why: the PRICE function is what Excel uses to
value bonds.  It is based on the calculation about 2/3's of the way
down this page, labelled PRICE =
Microsoft Office Online
"PRICE"
http://office.microsoft.com/en-us/assistance/HP052092191033.aspx 

It's a little complicated to be doing in a text-only screen on Google
Answers.  Let me know if it's okay to present the answer in Excel --
or if it really needs be done without the assistance of a spreadsheet.

Best regards,

Omnivorous-GA
Answer  
Subject: Re: calculating current value of a bond
Answered By: omnivorous-ga on 28 Sep 2004 11:34 PDT
 
Thanksmate --


The first part of this problem is to determine what a 12% bond, paid
in quarterly coupons, is really paying?  Since you've received $1,500
each quarter, you've been able to invest the following first-year
payments and compound the interest:

April: $1,500 (re-invested for 9 months)
July: $1,500 (re-invested for 6 months)
October: $1,500 (re-invested for 3 months)

THE HARDEST DECISION:

Here's the hardest decision of the day: should the money be
re-invested at 12% (your bond yield) or 15% (your required rate of
return).   For consistency, I'm going to use 15% to calculate what the
ADDITION to your 12% return would be:

APRIL: Percent return paid (3%) * percent of year remaining (75%) *
reinvestment rate (15%) = 0.0034

JUNE:  Percent return paid (3%) * percent of year remaining (50%) *
reinvestment rate (15%) = 0.0023

OCTOBER: Percent return paid (3%) * percent of year remaining (25%) *
reinvestment rate (15%) = 0.0011


COUPON EQUIVALENT YIELD
=========================

What you have now is the "coupon equivalent yield" or annual yield of
your 12% bond:
12.00% + .34% + .23% + .11% = 12.68%

(Incidentally, if you'd done this same internal return calculation
using 12% on your re-investments, the number would have been 12.52%)


VALUE OF THE BOND
===================

Rather than going through the complex PRICE evaluations in Microsoft
Excel, we can now calculate the value of the bond through 20
iterations.  Your required rate of return is 15% and the bond is
already providing 12.68% in coupon payments -- so you can simply
discount every year by the following to get your Net Present Value or
bond price:

15.00% - 12.68% = 2.32%



Year 20: $50,000
Year 19: $50,000/1.0232 = $48,866.30
Year 18: $50,000/(1.0232)^2 = $47,758.31
Year 17: $50,000/(1.0232)^3 = $46,675.44
.
.
.
Year 0: $50,000/(1.0232)^20 = $31,605.26

So, that's what the bond should be priced at: $31,605.26

As always, if any aspect of this is confusing, please let us know via
a clarification request before rating this answer.  And good to see
you here again Mate!

Best regards,

Omnivorous-GA

Request for Answer Clarification by thanksmate-ga on 02 Oct 2004 05:37 PDT
As far as I understand, the answer requires the use of the following formula:

c(1 + r)-1 + c(1 + r)-2 + . . . + c(1 + r)-n + B(1 + r)-n = P  
    where 
 c = annual coupon payment (in dollars, not a percent) 
 n = number of years to maturity 
 B = par value 
 P = purchase price 

I found the formula at this URL:
http://www.moneychimp.com/articles/finworks/fmbondytm.htm

Where is the definition of the formula you used?

Thanks

Clarification of Answer by omnivorous-ga on 02 Oct 2004 06:13 PDT
TM --

That formula is missing a definition of r, though I think it's the
required rate of return.  If it is, it's not accounting for years 2-n
properly.

You should be using something like the following net present value
(NPV)  calculation:
http://www.prenhall.com/divisions/bp/app/cfldemo/CB/NetPresentValue.html

Here is how I did it:
1.  calculate annual return on the coupon -- which simply provides an
annual interest rate from 4 coupon payments.  I trust that's easily
understood.

Annual return = I (for interest rate)

2,  calculate the gap between the annual return and the required rate
of return.  Required rate of return = R

3.  used that to calculate the discount necessary in the bond price. 
So it becomes, with 20 being the number of years to bond maturity:

P = B/(1 + [R-I])^20

This pricing is simply an NPV of the bond using the "gap" between
annual interest rate and expected returns.  It highly simplifies the
NPV calculation of the Prentice Hall definition above (which is also
used by Microsoft Excel) by turning the calculation on a single
interest rate (R-I) instead of discounting 80 bond payments made over
20 years.

However, note that in order to do this simplification you must
annualize your bond coupon payments.  Bond coupons are traditionally
paid 4 times per year in the U.S. and only twice per year in Japan and
Europe, so the annualization of those payments would provide slightly
different interest rates (I), depending on the country.

Best regards,

Omnivorous-GA

Request for Answer Clarification by thanksmate-ga on 02 Oct 2004 08:27 PDT
Thank you very much for the prompt reply!

I believe "r" in that equation is the required rate of return (yield
to maturity) and in my question this equals 15%. So we have:

Par Value (B) = $50,000
Coupon Rate (c) = $6,000 (annual coupon payment)
Years to Maturity (n) = 20
Yield to Maturity (r) = 15%
   
Now we need to calculate P as follows.

P= 6000 (1 + 0.15)^-1 + 6000 (1 + 0.15)^-2 + 6000 (1 + 0.15)^-3 ...
6000 (1 + 0.15)^-20 + 50,000 (1 + 0.15)^-20

= 5217.39 + 4536.86 + 3945.10 + 3430.52 + 2983.06 + 2593.97 + 2255.62
+ 1961.41 + 1705.57 + 1483.11 + 1289.66 + 1121.44 + 975.17 + 847.97 +
737.37 + 641.19 + 557.56 + 484.83 + 421.59 + 366.60

= $37,554.99

Can you please explain why this is wrong?

Thank you very much!

Clarification of Answer by omnivorous-ga on 02 Oct 2004 11:36 PDT
TM --

If the bond were paid once each year at $6,000 it would be correct. 
But you're getting paid $1,500 every three months -- which increases
the effective yield.  Your April $1,500 actually makes another $168.75
between April 1 and January 1; your June $1,500 payment makes another
$112.50 before the anniversary date; your October payment makes
$56.25.  This is increasing your bond yield from the 12% that you've
used in your calculations to 12.68%.

If you adjust the $6,000 to add those numbers in, your forumula will work.

Now it's $6,337.50 per year.  Run the 20 calculations that way: you'll
come out to $31,605.26

Best regards,

Omnivorous-GA

Request for Answer Clarification by thanksmate-ga on 15 Oct 2004 03:21 PDT
Hi,

I think your answer is wrong. Next week I will know for sure.
If it is wrong, I would like to request a refund.

If you would like to re-do your answer and it is correct by next week,
I will not request the refund.

Thanks

Clarification of Answer by omnivorous-ga on 15 Oct 2004 04:37 PDT
TM --

I stand by the answer. 

As for what you do with it -- that's up to you.

Best regards,

O.

Clarification of Answer by omnivorous-ga on 15 Oct 2004 04:41 PDT
You did use the $6,337.50, didn't you?

Thanksmate, I suspect that's where your misunderstanding of this
finance question is.

Request for Answer Clarification by thanksmate-ga on 15 Oct 2004 06:30 PDT
You mustn't understand my post if you think using P = $6,337.50 will
give the correct answer. For example, if using P = $6,000 results in
$37,554.99, it is not possible to have a larger P ($6,337.50) result
in a smaller amount  ($31,605.26) so there is no need to re-calculate.
Maybe that formula doesn't even relate to this type of question.

We will know for sure the correct answer soon enough and I'll post it here. 

Maybe your answer is even correct (for which there will be a nice tip)
but I doubt it and I will request a refund if it is wrong; it would
not be fair that I pay if your answer is wrong.

Clarification of Answer by omnivorous-ga on 15 Oct 2004 08:08 PDT
TM --

Be very, very careful here.  In looking at your calculations in
detail, you've ignored the NPV of the final bond payout, thereby
UNDERVALUING the bond's current price.

Here is the forumla from the Moneychimp pages, which replicated poorly
above due to the way that these pages handle text:
c(1 + r)^-1 + c(1 + r)^-2 + . . . + c(1 + r)^-n + B(1 + r)^-n = P

You have corrected calculated the present value of all of the coupons.
 Indeed using 15% elminates the need to calculate the effective annual
return the way that I did it.  So you don't have to worry about my
calculation of annual returns.

Also, note that I doubled-checked your calculations -- and they come within $1.

BUT you're missing the 21st element of that calculation:
B(1 + r)^-n 

You get all of the bond -- all $50,000 -- back at the end of the 20th
years, so it's NPV is:
$50,000(1.15)^-20 = $3,055.01

So, according to the Moneychimp formula, the price, P should be:
$37,553.99 + $3,055.01 = $40,610.00

Now, as to why this number is different from my original number?  I
have yet to figure that out (and I've been doing NPVs for 25+ years). 
But if I don't, you won't have to worry about paying for this question
because I'll withdraw it.

Both methods -- Moneychimp and my own NPV calculations -- appear
logical.  Yet the answers are too different to make them both
accurate.

HOWEVER, please note that you do have that one error -- missing the
final bond payout.

Best regards,

Omnivorous-GA

Clarification of Answer by omnivorous-ga on 15 Oct 2004 08:20 PDT
TM --

I hate to add one more complexity here but the $6,337.50 SHOULD be
used in these calculations.  Moneychimp assumes ANNUAL bond payments,
as is done in Japan, but you have quarterly payments -- advancing cash
flow.

Using $6,337.50 each year and Moneychimp, here are your numbers:
NPV of coupon payments (discounted at 15%): $39,668.51
NPV of bond (doesn't change): $3,055.01

PRICE = $42,723.52

As you'd predicted in your last note, the higher cash flow should
increase (not decrease) the bond price, which it does.

So, as they say on "Who Wants to Be a Millionaire?" -- that would be
my final answer.

Best regards,

O.
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