# Issuing bonds at discount - computing effective interest rate

Suppose Southwest Airlines issued \$100,000 of 9%, 5-year bonds when the market interest rate is 10%. The market price of the bonds drops, and Southwest receives \$96,149.

First of all, in my textbook only term market interest rate is used, but if I understood it correctly it is also called effective interest rate , right?

In this problem(stated above) the market interest rate is given. But many problems given to us by our teacher is without market interest rate - so what we have is - for example corporation A issued bonds at discount (nominal interest rate is less than market interest rate) - I am left with coupon interest rate, amount of money raised, and maturity date. So using the problem above I would only know that I have 5 years bonds, bonds \$100,000, money received \$96,149, coupon interest rate 9%.

So finally my question - how would I compute market interest rate from the information given?( I hope what I am asking is clear)

In this case the market interest rate is the discount rate that sets equal the market price (current value) of the bond to its present value.

To find the market interest rate which is also referred to as promised yield YTM you would have solve for the interest rate in the bond price formula

A market price of bond is the sum of discounted coupons and the terminal value of the bond.

Most spreadsheet programs and calculators have a RATE function that makes possible finding this market interest rate.

First see this for finding a coupon paying bond price

``````C . PVIFA(i%, n) + BT . PVIF(i%, n)

C is the periodic coupon
PVIFA(i%, n) = [ 1 - (1+i%)^-n) ] / i

BT is the par value of bond
PVIF(i%, n) = (1+i%)^-n
``````

The coupon payments are discounted so is the par value of the bond and sum of such discounts is the market price of the bond.

The TVM functions in Excel and calculators make this possible using the following equation

``````PV + PMT . PVIFA(RATE NPER) + FV . PVIF(RATE NPER) = 0

PV = market price
PMT = periodic coupon
NPER = number of coupons
FV = Par value of bond
RATE = Periodic market interest rate
``````

Let us take your data, 9% \$100,000 coupon with 5 years remaining to maturity with market interest rate of 10%. Bonds issued in the US mostly pay two coupons per year. Thus we are finding the present value of 10 coupons each worth \$4500 and par value of \$100,000. The semi-annual market interest rate is 10%/2 or 5%

``````PV = ?
PMT = 9% / 2 * 100,000 = \$4500
NPER = 5 * 2 = 10
FV = \$100,000
RATE = 10% / 2 = 5%

-96139.13
``````

Now solving for RATE is only possible using numerical methods and the RATE function is programmed using Newton-Raphson method to find one of the roots of the bond price equation. This rate will be the periodic rate in this case semi-annual rate which you have to multiply by 2 to get the annual rate. Do remember there is a difference between annual nominal rate and an annualized effective rate.

To find the market interest rate

``````PV = -96139.13
PMT = 9% / 2 * 100,000 = \$4500
NPER = 5 * 2 = 10
FV = \$100,000
RATE = ?

0.0499999206927984
4.99%

0.0999998413855968
9.99%
``````

If you don't have Excel or a financial calculator then you may opt to use my version of these financial functions in this JavaScript library tadJS

Yes, the "effective" and "market" rates are interchangeable.

The present value formula will help make it possible to determine the effective interest rate.

Since the bond's par value, duration, and par interest rate is known, the coupon payment can be extracted.

Now, knowing the price the bond sold in the market, the duration, and the coupon payment, the effective market interest rate can be extracted. This involves solving large polynomials.

A less accurate way of determining the interest rate is using a yield shorthand.

To extract the market interest rate with good precision and acceptable accuracy, the annual coupon derived can be divided by the market price of the bond.

• thank you for providing useful link, is there a already derived formula with extracted market interest rate? I consider myself to be quite good at math, but the algebraic manipulation here is little bit hard for me.
– cgnx
Feb 12, 2014 at 20:01

If the market rate and coupon were equal, the bond would be valued at face value, by definition. (Not 100% true, but this is an exercise, and that would be tangent to this discussion).

Since the market rate is higher than the coupon rate, the value I am willing to pay drops a bit, so my return is the same as the market rate.

This can be done by hand, a time value of money calculation for each payment. Discount by the years till received at the market rate to get the present value for each payment, and sum up the numbers. The other way is to use a finance calculator and solve for rate.

The final payment of \$10,000 (ignore final coupon just now) is \$10,000/(1.1^5). In other words, that single chunk of cash is worth 10% less if it's one year away, (1.1)^2 if 2 years away, etc. Draw a timetable with each payment and divide by 1.1 for each year it's away from present. If the 9% coupon is really 4.5% twice a year, it's \$450 in 6 month intervals, and each 6 mo interval is really 5% you discount.

Short durations like this can be done by hand, a 30 year bond with twice a year payments is a pain.

Welcome to Money.SE.

• Thank you,I need it for amortization of the discount. Can you please give me some explanation for doing it manually? For exams I am allowed to use normal calculator(not the finance one).
– cgnx
Feb 12, 2014 at 19:34
• Edited to add more details. Feb 12, 2014 at 19:42