Annuities less frequenty payable than interest is convertible

[esmeralda1]

Good morning everyone,

I have trouble with these questions:

1)The payments you received from a 20 year annuity−immediate paying 500 every 6 months have been left to accumulate in a fund and are now worth 40,000. If i(12) = 0.06 is the rate earned by your fund, calculate how long it is since the last annuity payment was made.

The answer is 10.15 months.

2)At a nominal rate of interest i, convertible semiannually, the present value of a series of payments of 1 at the end of every 2 years forever, is 5.89. Calculate i.
The answer is 8%, but I can't exactly get that figure.

[killjacker]

(1) Well, first we need to find the value of your annuity-immediate at the end of 20 years. i(12) = .06, so you six month rate = i = [(1+i(12)/12))^(6) -1] = 1.030377509. Then the accumulated value of an annuity-immediate w/ 500 payments over 40 periods is:

X = 500[(1.0304)^40 - 1]/.0304 = 38,024.91583

40,000/38,024.91583 = (1.005)^x, implies x = 10.15 months

(2) This is actually a very basic problem. First, just find the 2 year interest rate from the given information.

PV of a perpituity = 5.89 = 1/i , which implies i = 1/5.89
That's your 2 year interest rate i, not the i we are looking for. To convert, you can just follow your formulas.

i(2) = 2[(1+i)^.25] = 8%

[esmeralda1]

Thank you Killjacker. I see I have trouble interpreting the wording. Can you please help me with another question I haven't been able to solve?

Show that the present value at time 0 of 1 payable at times 7, 11, 15, 19, 23, and 27, where the effective rate per annum is i, is given by

(a angle 28 − a angle 4) / (s angle 3 + a angle 1).

I'm ok with the upper part, but I don't understand the denominator. I found in a study guide this explanation:

s3 + a1 is the present value — just after the third payment — of a 4-payment
annuity of 1 for which the first payment was 1 unit ago. Hence a single payment of 1 at time 3 is equivalent to payments of 1
s3 + a1 at times 1, 2, 3, 4.