Simple vs Compounded Interest

[Atin]

Hi David,

For some instruments like - FRA, Eurodollar Futures, IRS - I noticed that simple interest has been used even when the questions say - interest rate is quoted with semi annual compounding (e.g. Hull's example 7.2 for IRS valuations in terms of FRAs). I am quite confused now as to when to use what! Are there certain conditions/instrument types where we need to use simple interest directly? Can you please help?

 

[Atin]

Hi David,
I am certain that you might be quite caught up at this time with lots of updates/additions but can you please take a look at this when you get a chance?

Thanks much!

 

[David Harper CFA FRM]

Hi @Atin,

Thanks for your patience. It's a good question, but there is no hardwired conditions, as far as I know.

First, in practice (I refer just to the FRM) we do not tend to employ simple interest rates because, by definition, simple rates do not account for the time value of money, TVM. (maybe the nearest we get is with respect to short term money market usage of so-called discount rates where, in Hull's example [Chapter 6] the rate of 8.0% is applied to the face value without compounding.) Our mostly likely, if not only usage, of simple rates is for short or very short-term instruments where TVM is not especially consequential.

Okay, so our usage primarily concerns discrete versus continuous compounding. Here is my general view:, please don't interpret it strictly (you will see why):

• We want to distinguish between interest rates that are features of an (interest rate) product or instrument;, as opposed to:
• Discount rates used to present value

The first rate frequency tends to, unsurprisingly, follow its instrument; for example, Eurodollar futures quote with quarterly (discrete) compounding because they are three-month contracts. Most of our bond rates quote semi-annually because the coupons pay semi-annually.

The second rate, discount rate, can vary and GARP is fully aware that practice vary such that the FRM will make it clear to the candidate. In fact, the FRM employs various authors with various defaults.

Hull defaults to continuous, which explains the IRS as FRAs approach:

• The swap coupons (future cash flows) as features of the instrument do not pay cash continuously (how would that work?), they settle semi-annually. So the FRA-as-equivalent is naturally quoted with a semi-annual rate. This is way easier than the alternatives. By this, if the swap rate pays 8% semiannual on $1,000 notional, we know the "coupon" is $40 every six months without any conversion fuss.
  
• What about discounting this stream of future cash flows? It can be done either with continuous or discrete; it must be defined or given. On the one hand, if the spot curve is flat at 8.0% then assuming 8.0% semiannual will ensure this "bond" prices exactly at par; on the other hand, Hull tends to discount at continuous due to its elegant properties.

Over the years, we've sincerely been diligent about submitting instances to GARP such that, at this point, we can be highly confident that compound frequency will be explicitly stated (although don't let that dull you into lack of awareness about certain interest rate products which have their own conventions, or obviously, interest rate futures which have concrete specifications). I hope that helps,

 

[Atin]

Thank you so much David for the detailed explanation! It is so amazing to see that you explain such tricky basics with so much ease!
To your point re Simple Interest, I can see it has been used, specifically for short term instruments. To be more precise, its just the interest calculated using the face value and given discount rates.