Future Value of Annuity Compounded Daily?

Question

(a) What is the future value of $4$ payments of $\$300$ made at the end of each year with interest rate being $11\%$ p.a. compounded daily?

I did $300 (1 + 0.11/365)^{365}\cdot 4 -1)/0.11/365 = $550092.45$ which is wrong.

The options are

a. 1200.00 b. 4062.90 c. 1425.96 d. 918.43

(b) Obtas offers a mobile phone plan that charges $35 per month for 7 years. If you subscribe to this plan, calculate the present value of this plan, assuming you could have invested this money into a bank account that pays 6% p.a. payable annually.

a. 77329.59 b. 3621.34 c. 578.97 d. 2408.40

I have no idea how to start here.

Thank you so much.

Answer 1

(a)

1. Apart from proper placement of brackets, since the compounding is daily whereas the payment is annual, you first need to find the effective annual rate (APY), $[{(1+\frac{0.11}{365}})^{365} - 1]$ which will obviously be higher than 0.11 (decimal fraction), say r

2. Then use $A = \dfrac{300(({1+r})^4 - 1)}{r}$

By commonsense, if you do things rightly, you should get 1425.96

(b)

You should be knowing the formula to find the present value of a series of payments (PMT)

$PV = \dfrac{PMT[1-(1+r)^{-n}]}{r}$

The computations here are straightforward

Answer 2

(a) Let $i^{(365)}=11\%$ the nominal interest rate compounded daily, so that the effective annual interest rate is $$i=\left(+1\frac{i^{(365)}}{365}\right)^{365}-1=11.63\%$$ and the future value $S$ of the periodic payment $P=300$ for $n=4$ years will be $$ S=P\,s_{\overline{n}|i}=P\,\frac{(1+i)^n-1}{i}=1425.96 $$

(b) The effective interest rate is $i=6\%$, so you have that the monthly interest rate is $$ j=\frac{i^{(12)}}{12}=(1+i)^{1/12}-1=0.4868\% $$ corrisponding to a nominal annual interest rate $i^{(12)}=5.84\%$ companndly monthly.

Thus the present value of the annuity immediate of payments $P=35$ for $n=12\times 7=84$ months is $$ PV=P\,a_{\overline{n}|j}=P\,\frac{1-(1+j)^{-n}}{j}=2,408.40 $$