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In converting Continuous rate to Discrete, does it matter how often interest is paid?

Steve Jobs

Active Member
Thread starter #1
Assume the following:

An asset is quoted at 12% annually with continuous rate.
Interest is paid quarterly.

Is this correct for equivalent rate with monthly compounding?
r = 12 * [ e^(.12/12)) - 1] = 12.06%

Does it matter whether interest is paid quarterly, monthly or annually? What about doing the reverse convert from continuous to discrete?
 

ShaktiRathore

Well-Known Member
Subscriber
#2
hi,
let i deposit 1 unit today,
then from monthly compounding my deposit value after 1 yr is if r is the annual rate,
(1+r/12)^12
also from continuous compounding assuming continous rate rc is in annual terms after 1 yr is,
e^rc
pays from both should be equal,
(1+r/12)^12=1.e^rc
(1+r/12)=e^(rc/12)
(r/12)=[e^(rc/12)-1]
r=12*[e^(rc/12)-1]
for n periods compounding,
r=n*[e^(rc/n)-1]
for e.g. n=4(12/3) quarterly, r= 4* [ e^(.12/4)) - 1]
hence it matters whether interest is paid quarterly, monthly or annually
for reverse convert from continuous to discrete, following the same logic as above
(1+r/n)^n=e^rc
ln((1+r/n)^n)=rc
rc=n ln(1+r/n)

thanks
 

Steve Jobs

Active Member
Thread starter #3
Thanks Shakti,

When we say compounded monthly, are we assuming that the interest income is withdrawn each month too?

Otherwise there will be two periods, one period specifying how often the rate is compounded and the second one for how often the interest is actually withdrawn and paid. My confusion is regarding these two periods and whether I should be worry about this in the exam?
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
#4
That's interesting, Steve (to me, because I write so many questions).

With regard to "An asset is quoted at 12% annually with continuous rate. Interest is paid quarterly." Note three timeframes are invoked:
  1. Interest paid quarterly (4 per year)
  2. The rate curve used to compound or discount (FV or PV more likely) should always be expressed "per annum" which is independent of compound frequency; i.e., even if the "annually" were omitted, we would assume the 12.0% is per annum
  3. Compounding frequency is continuous
A modern version of the question is more likely (imo) to rephrase, in a manner typical of Hull, as follows (eg):
"An asset pays interest quarterly and the [spot | zero | discount | swap rate curve] is flat at 12.0% per annum with continuous compounding"
... Note in a carefully phrased question, how we can easily see that purpose of the 12% is to discount to price (or compound forward to an expected future price)

So, as phrased above, the continuous would not be converted;
For example, assume a bond with a 12% quarterly coupon when the spot rate curve is flat at 12.0% per annum with continuous compounding.
It's price is slightly less than par:
($100*12%/4)*exp(-0.25*12%); first quarterly coupon discounted continuously
+ ($100*12%/4)*exp(-0.50*12%); i.e. coupon paid quarterly but discounted continuously
....
+ $100+($100*12%/4)*exp(-n*12%)
= slightly less than $100

which is why it's more typical to assume, or better yet to explicitly set the coupon frequency = compound frequency (but to keep in mind they are different things!):
"an asset pays pays a quarterly 12.0% coupon when the [spot] rate curve is flat at 12.0%"
.... does not explicitly give us the compound frequency, so assume that it means:
"an asset pays pays a quarterly 12.0% coupon when the [spot] rate curve is flat at 12.0% with quarterly compounding"
... and that will discount a bond to par, as we would expect

my point is that it helps to separate (conceptually) the curve being used to discount from the coupon (interest payments) which has its own per annum r% and compound frequency, often the same. (sorry for length). You asked i think a great question:
"When we say compounded monthly, are we assuming that the interest income is withdrawn each month too?"
My answer is: no we do not make this assumption, unless we have no other information from which to assume.
On the other hand, while it is unnatural to infer from compound frequency --> coupon frequency, the reverse inference is quite natural:
"An asset pays interest monthly"
... while a good question explicates the compound frequency, in most cases, it is totally natural to infer from monthly coupon --> monthly compound frequency, as that describe the actual nature of reinvested cash flows.
 

Steve Jobs

Active Member
Thread starter #5
Thanks David for the clarification,

I understand that how often the interest is paid is:
a. not important when covering discrete to continuous and vice versa.
b. important when calculating the PV of the bond.

Is the above correct?
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
#6
Hi Steve, no, i don't think i said that (?). We need to know how often interest is paid. I mean to suggest that inferring compound frequency given coupon frequency is more natural than vice versa. My point is compound frequency may or may not coincide with coupon payment frequency; for example, Tuckman's almost entire text is an example of where semi-annual-pay coupons are discounted semi-annually (but he has exceptions)

Just back to your original: "An asset is quoted at 12% annually with continuous rate. Interest is paid quarterly."

How should we price this asset if it is, for example, a two-year bond? (is why I find this an interesting question, ultimately imprecise IMO). I see three (3) ways to price it.
  1. First, to ensure bond prices to par, translate coupons: 12% continuous = [exp(12%/4)-1]*4 = 12.182% per annum = quarterly payout = $3.045. Because we have converted coupons to continuous, this bond discounts (PV) to par of $100.
  2. Second, also to ensure bond prices to par, convert 12% continuous to LN(1+12%/4)*4 = 11.82%. Now, $3 coupons (quarterly) discounted at 11.82% (quarterly) will also price to par. Question: why isn't this exactly the same as the first?
  3. Third, discount $3 coupons (quarterly) at 12% continuous, which will price below par (< 100)
 

Steve Jobs

Active Member
Thread starter #8
Hi again! I reviewed more practice questions and now I've even more questions!

Assume the following deposits which are similar to BT practice question no. 157.3:

Deposit no. 1: The bank quotes the interest rate on a $1,000 as 9.0% per annum with continuous compounding and the interest is paid annually.
Deposit no. 2: The bank quotes the interest rate on a $1,000 as 9.0% per annum with continuous compounding but the interest is paid monthly.

For each deposit, calculate the equivalent discrete rate per annum with monthly compounding.

I think in the above cases:
a. the equivalent rates will be the same for both deposits
b. but there could be differences in the amount of interest received by the customer for one year, dependent on whether the the interest that is paid monthly is being re-invested again or withdrawn by the customer. So the difference is not in the equivalent rates but in deposit terms/conditions and customer decision.
 

Steve Jobs

Active Member
Thread starter #9
The question was asking for the amount of interest, I changed it to equivalent rate according to the point wanted to raise. Actually, the only similarity now is the interest rate.
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
#10
I can see that I did write a question just like Hull's
BT 157.3 Suppose a lender quotes the interest rate on a $1,000 loan as 9.0% per annum with continuous compounding but the interest is actually paid monthly. What are the monthly interest payments?

Hull 4.10 A deposit account pays 12% per annum with continuous compounding, but interest is actually paid quarterly. How much interest will be paid each quarter on a $10,000 deposit?
If the question seems awkward (i.e., we could skip the continuous rate and quote the corresponding monthly or annual rate), that's only because the whole point of the question is to test the equivalence-translation from one rate to another.

So, the meaning of 9% per annum with continuous compounding is: regardless of how frequently the interest is paid, the terminal value at the end of one year will be $1,000*exp(9%) = $1,094.17; it is also same as saying, regardless of the interest payout frequency, the effective annual return = exp(9%)-1 = 9.417%.

Any conversion-to-discrete will assume reinvestment (we generally do assume reinvestment). As below, i input your questions, the first scenario is quarterly interest (first two columns), which implies $22.76 interest paid per quarter , but those interest payments are reinvested, so they accumulate to the same $94.17 at the end of the year. If the interest pays annual (k =1), then the $94.17 is just paid once, at the end of the year. I didn't detail the monthly, but each $7.5282 will get reinvested and at the end of the year, this will also equal $94.17. So the profile of cash flow varies, but the very design of the question means to treat 9.0% continuous as the rate earned regardless of interest payout frequency. Regardless of (k), the terminal value of $1,094.17 is unchanged, is how i look at this. thanks,

 

Steve Jobs

Active Member
Thread starter #11
Thanks David,

So in the exam, for the equivalent rate questions, I should ignore any details provided regarding the frequency of interest payout because it's assumed that the interest is reinvested. In reality I guess such details should be documented.
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
#12
It's safe to assume reinvestment (which is different than assuming we know the reinvestment rate: yield assumes it knows. Reinvestment risk refers to the reality we don't know. But the existence of some re-investment assumption is tantamount, I think, to assuming time value of money, and we always assume TVM).

Yes, details will be documented. If there is a #1 topic upon which GARP has received feedback with respect to specificity, i think it probably is careful specification of compound frequency assumptions. It's one of the more seasoned elements such that they know to give details, thanks,
 
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