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From: koch@chopin.DEC (Kevin Koch LTN1-2/B17 DTN229-6274)
Newsgroups: net.invest,net.taxes
Subject: Calculating yield to maturity for zero coupon instruments
Message-ID: <1663@decwrl.UUCP>
Date: Sun, 14-Apr-85 09:57:26 EST
Article-I.D.: decwrl.1663
Posted: Sun Apr 14 09:57:26 1985
Date-Received: Tue, 16-Apr-85 04:10:34 EST
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Xref: linus net.invest:572 net.taxes:680

> > There are five different bonds which each cost $2,000.00.
> > Maturity of the five bonds is as follows:
> > 
> > 	 7 years	$  4,000.00
> > 	10 years	$  6,000.00
> > 	15 years	$ 10,000.00
> > 	21 years	$ 20,000.00
> > 	29 years	$ 50,000.00
> > 
> > Is there any way to figure out the actual interest rate that
> > each bond is receiving. I am sending for more information today,
> > but until I receive it, I though I would solicit information from
> > the net.
> 
>      Figuring out the actual interest rate is relatively easy.  In the
> following formulas, these variables mean:
> 	N : number of years to maturity
> 	V : value of bond at maturity
> 	C : initial cost of bond
> 	i : interest rate
> 	e : 2.71828... (the natural number) (not a variable, actually.)
> 
>     From the definition of compound interest:
> 
>     C * (1 + i)^N = V
> 
>     Solving this equation for i:
> 
>     i = e^(1/N * ln(V/C)) - 1
> 
>     That wasn't so hard, now, was it?

     The quick and dirty method is to use the rule-of-72.  
Conveniently, the 7 year bond matures at exactly double the purchase 
price, so the yield is 72/7 = 10.29%.

     Its also obvious that the yield curve is back to normal -- long 
term investments paying higher yields.  If the yield on the 21 year 
bond were also 10.29%, the bond would mature at $16000 (the value
doubles every seven years).  

     You can also see the yield going up on the longer term bonds by 
noting that the doubling time goes from 7 years (short term) to 6 
years (15 years to 21 years).

     Note that neither the rule of 72 or the accurate formula stated 
above works when there is a coupon.  The general yield-to-maturity 
calculation can only be done numerically.