# Annuity problem

1. Jun 24, 2017

### issacnewton

1. The problem statement, all variables and given/known data
The annually compounded discount rate is 5.5%. You are asked to calculate the present
value of a 12-year annuity with payments of $50,000 per year. Calculate PV for each of the following cases. a. The annuity payments arrive at one-year intervals. The first payment arrives one year from now. b. The first payment arrives in six months. Following payments arrive at one-year intervals (i.e., at 18 months, 30 months, etc.). 2. Relevant equations Annuity PV formula 3. The attempt at a solution I have done the part a. I need help for part b. Let $r = 0.055$ and $C = 50000$. The payments arrive at one-year intervals after the first payment which arrives in six months. So 11 payments will arrive at one-year intervals after the first payment which arrives in six months. PV of these payments at 6 month is given by the Annuity formula $$\mbox{PV } = 50000+\frac{C}{r} \left[ 1 - \frac{1}{(1+r)^{11}} \right]$$ So $\mbox{PV } = 454626.8$. Now this is PV at 6 month. We want to convert this to today's value. $r$ here is annual rate. I want to convert this into equivalent monthly rate. For this, I did the following. Let $r_m$ be the equivalent monthly rate. Now $$P(1+r_m)^{12} = P(1+r)$$, where $P$ could be initial principal. So solving this for $r_m$, we get, $r_m = (1+r)^{1/12}-1$. So $r_m = 0.00447169$. So discounting, the present value would be $454626.8/ (1+r_m)^6 = 442617.70$ But the answer is$442,603.98. So where have I gone wrong ?

2. Jun 24, 2017

### Buzz Bloom

Hi issacnewton:

You seem to be using the wrong equation for the present value of an annuity.

Regards,
Buzz

3. Jun 24, 2017

### issacnewton

Buzz Bloom, I think I am using the correct one. I am just adding the payment done at 6 month to the PV of the 11 future payments. And this total would be the PV at 6 month

4. Jun 24, 2017

### Buzz Bloom

Hi issacnewton:

I was thinking a bit differently. If I am understanding the problem statement correctly, the PV you calculated for (a) is the same as a similar annuity started at the time six months earlier for the (b) annuity adjusted for the six month start date difference. That is the difference between the (a) and (b) PVs would be a six month interest at the same annual rate on the (a) PV.

Hope this helps.

Regards,
Buzz

5. Jun 24, 2017

### issacnewton

Buzz, I am not exactly following what you are trying to say here.

6. Jun 24, 2017

### Buzz Bloom

Hi issacnewton:

Sorry for my lack of clarity. I am suggesting that the difference between the PV for (a) and the PV for (b) is six months of interest on the PV of (a). The thought behind this suggestion is that the payments for an (a) annuity bought six months earlier are the same as the (b) annuity payments.

Regards,
Buzz

7. Jun 24, 2017

### issacnewton

So where I am going wrong in my calculations ? I think probably my equivalent monthly rate calculation is wrong.

8. Jun 24, 2017

### Buzz Bloom

Hi issacnewton:

I think that your formula is correct, except for the unlikely possibility that the fraction rate for a year is calculated in terms of days rather than months. A simpler way to calculate the rate for 1/2 a year would be
(1+r6mos) = (1+r)1/2.​

I confess I could not follow the details of what you did. I am guessing the problem is in the formula you used for PV.

Regards,
Buzz

9. Jun 24, 2017

### issacnewton

Even with your formula for semi annual interest, I calculated the PV and its same as mine, which is 442617.70. This is not the answer.

10. Jun 25, 2017

### issacnewton

Hello Buzz, it seems my solution is correct one. I was looking at the wrong answer sheet. This problem is also solved in MIT link alo.mit.edu/wp-content/uploads/2015/06/PS_Part1.pdf In this document, its problem 34 on page 13 of the pdf document. Its solution is given on page 47 of the pdf document. Their answer is $442,617.74$ , which matches with my answer. I can trust the MIT document anytime.

11. Jun 25, 2017

### Buzz Bloom

Congrtulation issac. Well done.