So I'm trying to work my way through a finance textbook through self study, and I can't figure out why a 5% interest rate for 30 periods is different than a 10% interest rates for 15 periods? I'm not talking about annuities here, but rather how much I would value $1 in the future (i.e. the present value of a future dollar). I'm calculating the price of a bond, and I'm trying to calculate the "present value of the principal payment." The bond matures in 15 years with a required rate of return of 10% annually. The book uses a discount factor based on 30 periods and 5%. Intuitively it seems to me, one should be able to use a discount factor based on 15 periods and 10% and arrive at the same value. However, when I try calculating it this way, I get a different price. Why is my intuition incorrect?
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Would you rather get $5 on June 30 and $5 on December 31 of every year or would you rather get $10 on December 31? If you are a savvy investor, the answer is that you would rather get $5 semi-annually than $10 annually because you get re-invest the $5 you get every year on June 30 and earn a return such that it is worth more than $5 come December 31 (such that $5 June 30 and $5 December 31 is worth more than $10 December 31). Therefore, if the bond pays a 5% coupon semi-annually (15 periods) you will get a different (re: higher) value for the bond than if it pays a 10% coupons annually (30 periods).
I'm not talking about the annuity portion of the bond, I'm only talking about the principal payment at maturity.
Aren't we talking about getting the face value of a bond at maturity, so in both cases wouldn't we be talking about getting $X 15 years from now? In which case, we ask, if I require a rate of return of 10% annually, how much am I willing to pay for the $X now? Why does it matter how we split up the periods between now and 15 years? We could think of it as one period, 15 period, 30 period, a period per day, etc (obviously we would have to adjust the required rate of return to account for the units, i.e., days, months, a half a year, a year, etc).
Also, 5% per semester means less price volatility (A former PM at a BB told me why, but i forgot)
5% for 30 periods = 1.05^30 = 432.1% 10% for 15 periods = 417.7%
if you break something into smaller periods and apply the same interest rate adjusted for the time split you still have to take into account compounding. x1.1 does not equal x1.05*1.05
That makes sense. My next question is, how do we know using 30 periods at 5% is the correct discount factor for the lump sum payment of the bond in question? Is it just because the bond pays an annuity for 30 periods, and there is a convention to keep the periods constant in both calculations (by both calculations, I mean the annuity part and the lump sum part)?
Does the book make a mistake then (or maybe just use an estimation without explicitly mentioning it)? Because the book says 10% required rate of return for 15 years, and then they just say that is the same as 5% required rate of return for 30 6-month periods.
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