I am quite stuck because I do not know in what time unit to solve for in for this question. "A loan of $600,000 is repayable by an annuity payable half-yearly in arrear for 10 years. The half-yearly repayment is calculated at an effective rate of interest of 5% per annum and decreases by $100 each half-year. Which of the following is the amount of the first repayment?" If it increases by 100 each half year, does that mean it is 100*(Ia)20? or 200(Ia)10? are these both equivalent? What interest rate do you use? Please help it is driving me crazy!
Hi Gerry, Thanks for your post. It can take a while to get your head around these, I appreciate. First, just to be clear, as it's a decreasing annuity, you'll actually need to do a level annuity and subtract an increasing annuity, to essentially get a decreasing annuity. I think you're aware of this, and it's the increasing annuity part you're querying. Often there are multiple ways to do questions like this. But in this case, what I would do is to work in a time period of "half years". So you have (for the increasing annuity part) $100 in 6 months, $200 in 12 months, $300 in 18 months etc. Which is an increasing annuity which starts at $100, payable for 20 "half years" So that would be $100(Ia)<20>. Importantly, because you're working in half-years, you need to use the half-yearly rate of interest. So if i = 5% pa, the half yearly effective rate of interest would be (1.05^0.5) - 1 = 2.4695%. So you'd use $100(Ia)<20> at a rate of 2.4695%. Note that everything has been expressed in half yearly units, it's a payment every half year, starting at $100, for 20 half years, at a half-yearly effective rate of interest. It would be possible to work annually instead...but it would be harder because the increases are twice per year, which would make the maths more complicated. When you're working with increasing annuities, it tends to be easier - where possible - to do everything in the same unit of time as the frequency of the increases. Just for completeness - if you were instead valuing level (ie flat) annuities, there's often 2 or more ways you could do it. So if the question had instead been to value a level (flat) annuity of £100 paid every 6 months, payable in arrears for 20 years, then you could either use: $100 a<20>, ie $100 every half year, for 20 half years - in which case you'd be doing everything in "half years" and would use the half yearly interest rate of 2.4695%; or $200 a<10>(2), ie $200 paid every year, but in two instalments per year, for 10 years. You're then working in years, and would use the annual interest rate of 5%. But to calculate a<10>(2), you'd need to use the p-thly annuity formula which uses a v on the numerator (based on i) but an i(2) on the denominator, which you'd also have to calculate. You might want to try this both ways. You should get the same answer from the two methods. Hope that helps. All the best with your studies.
