Pricing Bonds Between Coupon Dates

[JAbdo9644]

Hi Everyone !!

Suppose that at time t = 0, I want to price a bond that pays annual coupon at t = 1, t = 2 and t = 3. Maturity is 3 years. Assume annual compounding for simplicity.

The pricing at time t = 0 should be trivial because I can discount each of the coming 3 cash flows at the 1 year, 2 year and 3 year spot rates.

However, imagine now that 6 months have elapsed, so my cash flows are now due in 0.5, 1.5 and 2.5 years. How should I price this bond now, given that at any point in time, I only have access to the 1 year, 2 year and 3 year spot curve? I cant use them as is because the cash flows are due in 0.5, 1.5 and 2.5 years NOT 1, 2 and 3 years.

Can I just discount the first cash flow at 1 / (1 + 1yr spot) ^ 0.5 because it will occur in 0.5 years?

Similarly, can I discount the second cash flow at 1 / (1 + 2yr spot) ^ 1.5 because it will occur in 1.5 years and not 2 years exactly?

and so on....


Thanks alot
Jamal

 

[David Harper CFA FRM]

Hi Jamal, That's actually a great question: exam-wise some never notice the realistic problem is retrieving full/flat settlement price between coupons because the problems tend to query t = 0 pricing. Yes, should absolutely can discount the cash flows per your "irregular" (but totally correct!) syntax; e.g., 1 / (1 + 1yr spot) ^ 0.5 because it will occur in 0.5 years. Discounting cash flows never lies! (you just have to be careful about the fraction vis à vis day counts). In fact, in my PQs I do exactly what you show as a means to backup (reconcile) my solution; i.e., I do this to check my work ...

... because this approach is too tedious if there is a long remaining maturity. And, to my knowledge, the calculator's TVM will not get you the result because it doesn't really work with non-integer N values. Such that: the more efficient thing to do is to price the bond, as usual, at the last coupon and then simply compound the price forward to settlement at the yield. The price at the last coupon will be, by definition, when the full (dirty) price equals the flat (quote) price, and then if you compound forward at the yield--e.g., Price[at last coupon]*(1+yield)^(fraction of year)--will be a cash (aka, full price) at the settlement. So that's my efficient solution, then I also perform the tedious discounting to make sure I'm correct (they must match). You can see the general rule is: you can always trust a proper discounted cash flow. I hope that's helpful,

 

[JAbdo9644]

Hi Jamal, That's actually a great question: exam-wise some never notice the realistic problem is retrieving full/flat settlement price between coupons because the problems tend to query t = 0 pricing. Yes, should absolutely can discount the cash flows per your "irregular" (but totally correct!) syntax; e.g., 1 / (1 + 1yr spot) ^ 0.5 because it will occur in 0.5 years. Discounting cash flows never lies! (you just have to be careful about the fraction vis à vis day counts). In fact, in my PQs I do exactly what you show as a means to backup (reconcile) my solution; i.e., I do this to check my work ...

... because this approach is too tedious if there is a long remaining maturity. And, to my knowledge, the calculator's TVM will not get you the result because it doesn't really work with non-integer N values. Such that: the more efficient thing to do is to price the bond, as usual, at the last coupon and then simply compound the price forward to settlement at the yield. The price at the last coupon will be, by definition, when the full (dirty) price equals the flat (quote) price, and then if you compound forward at the yield--e.g., Price[at last coupon]*(1+yield)^(fraction of year)--will be a cash (aka, full price) at the settlement. So that's my efficient solution, then I also perform the tedious discounting to make sure I'm correct (they must match). You can see the general rule is: you can always trust a proper discounted cash flow. I hope that's helpful,

Thank you so much for the help Prof. David, I'm assuming if the syntax for the first cash flow is correct, then I should also be able to the same for the second cash flow, i.e. 1 / (1 + 2yr spot rate) ^ 1.5 correct? Although the solution you suggest is clearly much more efficient

 

[David Harper CFA FRM]

Hi @JAbdo9644 Yes, you are correct. In fact, you can see a similar example via a question that I recently wrote here at https://forum.bionicturtle.com/threads/p1-t3-23-1-corporate-bond-issuance.24359/ i.e.,

23.1.2. A U.S. corporate bond that matures on April 1st, 2026, pays a semi-annual coupon of 9.0% per year. The coupon payment dates are Jan 1st and July 1st. Settlement is on April 7th, 2024. The year 2024 is a leap year. The bond's yield happens to be about 7.00%. If the cash (aka, dirty) price is $1,064.50, then which is nearest to the quoted (aka, clean) price? (Inspired by GARP EOC PQ 17.13)

a. $960.14
b. $974.50
c. $1,040.50
d. $1,088.50

The solution is captured in my XLS (file attached). See how I check (reconcile) the $1,064.50 with a longhand calculation? Thanks!