Question

You have just won a very special prize, a "consol" bond that pays $30,000 every six months (semi-annual payment) forever. The only caveat is that the first payments is given 5.5 years from now. You are not that enthusiastic about waiting that long. In fact, you really want to simply receive 13 quarterly payments with the first payment starting one quarter from now. If the APR is 12% compounded monthly, what is the quarterly payment amount you want to receive instead? For your answer round to the nearest dollar, do not enter the dollar ($) sign, and use commas for the thousands. For example, if you obtained $1,240.75 then enter 1,241

solve step by step

Answer 1

Effective APR for 6 months = [{1+(.12/12)}^6]-1
Effective APR for 6 months(half year)			=	[1+0.01]^6
			=	[(1.01)^6]-1
			=	1.06152-1
			=	0.06152 or 6.152%
Value of annuity of $30,000 to be received after 5 years					=	annuity/Effective APR for 6 months
Since the first payment is received 5.5 years(i.e 6 months after 5 years)					=	$30,000/0.06152
					=	$487,646.29
5 years= 10half years
Value of annuity ($487,646.29) today = amount of annuity after 10 half years*PVF(half yearly APR,no. of half years)
Value of annuity today			=	$487,646.3*PVF(6.152%,10)
			=	$487,646.3*0.5505
			=	$                   268,449.29
Now we want to receive this amount in 13 quaterly installments starting from next quarter
Present value of annuity			=	Payment [{1-(1+r)^-n}/r]
		where
		P=payment
		r=rate per period =quaterly=0.030301 or 3.0301% (3 months APR calculated same as 6 months APR)
		n=No.of periods=13 quarters
		268449.29		Payment{1-(1+0.030301)^-13}/0.030301
		268449.29		Payment{1-(1.030301)^-13}/0.030301
		268449.29		Payment{1-0.6784}/0.030301
		268449.29		Payment{0.32163/0.030301}
		268449.29		Payment*10.6145
		268449.29/10.6145		Payment
		25290.808	=	Payment
		Quaterly amount	=	$     25,290.808
	there may be slight difference due decimal places.Please do not downvote on that basis.
	Please upvote