Que 12.7

[sfischer]

In Que 12.7, we are told that the woman's salary increases by 6% each year and the interest on the fund is 8% each year. I would have assumed that the accumulation after 15 years is simply 1500+1500*1.06*1.08+.... However, the answer discounts back with the rate of 1.06/1.08 (as if 8% was a kind of inflation rate and we were asked for a real rate of return) and then accumulates at 8%. I am missing the logic here - thanks.

 

[Manasvee Kumar]

In Que 12.7, we are told that the woman's salary increases by 6% each year and the interest on the fund is 8% each year. I would have assumed that the accumulation after 15 years is simply 1500+1500*1.06*1.08+.... However, the answer discounts back with the rate of 1.06/1.08 (as if 8% was a kind of inflation rate and we were asked for a real rate of return) and then accumulates at 8%. I am missing the logic here - thanks.

ACTUALLY we generally calculate the net present value of the investment (or anything else) and then find the accumulated value if required.( and the accumulation rate is generaaly the rate of interest on lending)

 

[Calum]

Your accumulation is not quite right - I think it's 1500 + 1500 *1.06^14*1.0 +1500*1.06^13*1.08^2 +...

 

[Lloydie1990]

I haven't looked at the question but I remember it from last session!

The undiscounted payments are 1500 + 1500(1.06) + 1500(1.06^2)+... because of the 6% salary increase.

If we now discount these payments at the interest rate (as we normally would do in this type of question), we get

PV = 1500 + 1500(1.06)/(1.08) + 1500(1.06^2)/(1.08^2) + ...

This can be thought of as using v = 1.06/1.08, i.e we combine the salary increases and interest rate (discount rate) into one discount rate, for non-increasing payments of 1500 every year. This then gives i = (1/v) - 1 = 1.08/1.06 - 1.

I think the bit that confused you was the fact that it says interest rate. Remember that we're trying to find the PV of the cashflows, so we use this interest rate as our discount rate.

 

[Lloydie1990]

Didn't see the bit about accumulating! The easiest way to do this is once you've found the PV of the cashflows, add on the interest for the correct number of years to the PV to arrive at the accumulated value.

So for this question I think you need to do Acc. Value = PV x 1.08^15.

We just use the 8% for this bit because we only use the interest/discount rate for accumulating/discounting!

 

[sfischer]

Thanks for all your input. I think it's discounting in general that I haven't quite got. I get the concept of purchasing power - so if I have $100 in 5 years and inflation is expected to be 4% over that time, I need to discount it back to see what its worth in today's terms. But what to use as a discount rate is eluding me a bit - sometime we use loan interest rate (a negative cash flow) and sometimes we use bank interest (a positive cash flow) as in this case. In fact here we have 2 positive rates - bank interest and salary increase but we have chosen to accumulate by salary increase and discount by bank interest. Can anyone help me understand that concept - thanks.

 

[asmkdas]

Thanks for all your input. I think it's discounting in general that I haven't quite got. I get the concept of purchasing power - so if I have $100 in 5 years and inflation is expected to be 4% over that time, I need to discount it back to see what its worth in today's terms. But what to use as a discount rate is eluding me a bit - sometime we use loan interest rate (a negative cash flow) and sometimes we use bank interest (a positive cash flow) as in this case. In fact here we have 2 positive rates - bank interest and salary increase but we have chosen to accumulate by salary increase and discount by bank interest. Can anyone help me understand that concept - thanks.

Question 12.7's data somehow is not matching with your elaborated information. However, here we need to increase the money value by using the inflation rate for the future, and then need to discount it back by using the cost of capital (which is Bank interest rate/ Simply interest rate) to get the NPV. To use a combined interest rate we use \(i'=\frac{1+e}{1+i}-1=\frac{e-i}{1+i}\), where i=rate of interest pa and e=inflation rate pa.
We just need to use the cost of capital (whether it's loan capital interest rate or bank interest rate) to get the NPV, the idea is elaborated in Example 12.11.
Lloydie1990 has elaborated the the proper concept here.

 

[sfischer]

You're right - it was Que 12.17 not 12.7 - sorry about that. So the 5% is the increase in the cash flow (being the increase in the amount she is putting into the fund) and the 8% interest is the capital increase (the capital being the value of the fund itself). The increase in cash flows has nothing to do with yields or PVs - it just happens to be a % increase. It's the capital that accumulates or is discounted. Does that sound right?
The other bit I was confused about was why we discounted and then accumulated but a few have pointed this out. If I understand it, its because we had to construct an equation of value which equates to a PV.

 

[asmkdas]

You're right - it was Que 12.17 not 12.7 - sorry about that. So the 5% is the increase in the cash flow (being the increase in the amount she is putting into the fund) and the 8% interest is the capital increase (the capital being the value of the fund itself). The increase in cash flows has nothing to do with yields or PVs - it just happens to be a % increase. It's the capital that accumulates or is discounted. Does that sound right?
The other bit I was confused about was why we discounted and then accumulated but a few have pointed this out. If I understand it, its because we had to construct an equation of value which equates to a PV.

Your concept is correct now. To get the value of any cashflow or to compare it with any other cashflow we always need to find the present value rather than accumulated value.
All the best.

 

[sfischer]

So to add to this one then (because I thought I had it but now I'm not so sure), in Q&A 2.16(ii), for project B, in order the calculate the accumulated value, I first calculated the NPV=197,750v^8-7000 \( \require{enclose}a_{\enclose{actuarial}{8}} \) @7%. Then I accumulated the total by 1.07^8. Now as it turns out, this was the right answer but I had forgotten to take out 7000 loans each year and accumulate the interest for that which if I did would have decreased the profit and the answer would have been wrong. So what is the difference in the approach to this question vs Que 12.17?

Thanks.

 

[John Lee]

So to add to this one then (because I thought I had it but now I'm not so sure), in Q&A 2.16(ii), for project B, in order the calculate the accumulated value, I first calculated the NPV=197,750v^8-7000 \( \require{enclose}a_{\enclose{actuarial}{8}} \) @7%. Then I accumulated the total by 1.07^8. Now as it turns out, this was the right answer but I had forgotten to take out 7000 loans each year and accumulate the interest for that which if I did would have decreased the profit and the answer would have been wrong. So what is the difference in the approach to this question vs Que 12.17?

Thanks.

In this question the interest rate changes when you go into surplus. So you have to find the break-even point first - and accumulate to there at the borrowing rate and then accumulate at the surplus rate from there onwards.

 

[sfischer]

Looking at it again, the 7,000 annuity I included is the extra loans since for project B, there are no loan repayments, just extra loans.