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Good afternoon,
It’s not clear to me what they did in the part:
“Macaulay duration based on fixed annual repayments of interest and capital”Why and how did they get $22.17 per year?
If one bond value is 100$ and pays 3.57$ interest per year, if I need to repay capital and interest in 5 years, I would split the capital through the 5-year period and adding interest.
Therefore I would use 23.57 every year.Clearly I am wrong, but no idea the rational behind their calculation as per below:
“Annuity factor: (3·57%, 5 years) = (1 – 1·0357–5)/0·0357 = 4·51 approximately
Annual payments of capital and interest required to pay back new bond issue = $100/4·51 = $22·17 per $100 bond approximately”Any help?
Thanks
Giuseppe
The reason that you are wrong is that as the capital is repaid, the interest each year will reduce. The interest each year is not $3.57, but is 3.57% of the amount owing at the start of each year.
It is a fact that however a loan is repaid, the present value of the interest and repayments when discounted at the cost of borrowing will always equal the amount of the loan. (Try it yourself if you want with made up figures, and however the loan is repaid the PV will always be the amount borrowed.)
Here, the repayment and interest are of equal amounts each year.
Therefore is the repayment is X per year, then the PV of X per year for 5 years when discounted at 3.57% will be equal to $100.Therefore X x 5 year annuity factor at 3.57% = 100
Therefore X = 100 / (the 5 year annuity factor at 3.57%).You can easily prove that it ‘works’. The interest in the first year will be 3.57 and after a repayment of 22.17 they will be owing 81.40 at the end of the first year.
So the interest in the second year will be 2.91 (3.57% x 81.40) and so they will be owing 81.40 + 2.91 – 22.17 = 62.14 at the end of the second year.If you carry on each year in the same way, you will find that at the end of the 5th year the amount owing will be zero (not exactly zero but that is simply because of roundings).
Good morning,
thanks for your quick and helpful answer.
It makes sense to me.Thanks again and thank you for the great job done with your lectures.
Giuseppe
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