Yield to Maturity - Reinvestment Assumption

Hi all,

I'm slightly confused about the reinvestment assumption involved in calculating the YTM. I've always thought of the YTM as being the annualised yield earned by an investor who holds their bond to maturity, taking into account the initial price they paid for it, and assuming that all coupons are reinvested at the YTM rate. (It also represents the rate needed to discount all future cash flows of a bond into its present value (i.e. price)).

However, I am then confused at how a bond (say with a coupon rate of 4%), when bought at par, has a coupon rate that is equal to its YTM? I get that if a bond is bought at par, there will be no capital appreciation (since principal and price will be the same), but surely the reinvestment of the coupons would make the YTM greater than 4%, due to compounded interest? Explanations I see online only seem to be talking about the capital appreciation element, when comparing coupon rate with YTM, with no mention of the effect of reinvestments.

I guess this then brings into question whether the reinvestment assumption is even needed in calculating the YTM, as per this article?

https://www.economics-finance.org/jefe/issues/For…

Many thanks for your help.

 

Econ is right, I got a value of 102.31 per 100 of face value. When you're talking formulas interest rates are 99% of the time divided by 100. for example, 5%=.05 and so on. If you have a financial calculator (ba II plus pro):

PMT: 2.875 (coupon times face value/2) fv=100 N=4 i/y=2.2625

Remember you always need to adjust for semi-annual coupons and the easiest way (imo) is to just double the period and such as I did above. Hope this helps.

 
Hubristic:
Econ is right, I got a value of 102.31 per 100 of face value. When you're talking formulas interest rates are 99% of the time divided by 100. for example, 5%=.05 and so on. If you have a financial calculator (ba II plus pro):

PMT: 2.875 (coupon times face value/2) fv=100 N=4 i/y=2.2625

Remember you always need to adjust for semi-annual coupons and the easiest way (imo) is to just double the period and such as I did above. Hope this helps.

And just in case you want it in actual formula format:

=2.875(1/0.022625-((1/(0.022625(1+0.022625)^4))))+100/(1+0.022625)^4

 
Best Response

You can approximate quite closely in your head for bonds trading around par (give or take a few pts) by decomposing YTM into its current yield and capital appreciation components.

Eg a 5yr bond trading at 90 with a 10% coupon will have a yield of ~13pct. 10%/90 = 11.1% current yield + 10pts/5yrs = 2pts/year = 13.1%. Actual answer is 12.83% but that's about as good as you can do without a calculator.

 

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