I treat the each coupon payment period as one unit. So if the bond is paid semiannually and priced at par, it has two units for the payment. YTM = (1 + annual coupon rate/2) ^ 2 -1. Is YTM here, on annual basis or semi-annual basis? I think it is on annual basis. If I’m right, how to get the YTM on semi-annual basis? Would it be YTM = Coupon Rate? I feel i know it but I don’t completely get it.
And if we need to calculate the price of the semiannual paid bond, N = Maturity year * 2 = coupon payment periods, I/Y = YTM/2, FV = Par Value, PMT = (Coupon Rate * Par Value) / 2. How about YTM here? Is this YTM on semiannual basis?
Question 4 on Kaplan Book 5 page 64. Spot Rates for 0.5 years, 1 year, 1.5 year and 2 year are 4%, 4.4%, 5.0%, and 5.4%. Calculate the value of a 2 year, 4.5% coupon, semi-annual pay bond. I don’t understand why the discount rates for each coupon payment period are the spot rates divided by 2. In this case, the discount rates are 2%, 2.2%, 2.5% and 2.7%.
The basis of the resulting YTM will depend of the periodicity of the cash flows. If you say you will receive 100$ per month over 3 years for investing 1000$, the resulting YTM will be monthly-based; say 0.8%. If you want annualize it, just multiply by 12 (9.6%). The same for semi-annual payments, multiply by 2 to get the annual YTM.
As I said above, it would be in semi-annual basis because the payments are semi-annual.
In real life, bonds are not valued using a single rate, you use a curve, the spot curve is one of them. If the bond pays semi-annual coupon of 4.5%, then the coupon is 4.5% / 2 = 2.25% of face value. The spot rates are ALWAYS quoted in annual terms so you must divide by 2 to get the semi-annual rates, simple as that.
For example, the LIBOR rate (London Inter Bank Offering Rate) of 1-month loan is 0.15%. Which period is that? Yes, annual… so it would pay you 0.15% / 12 = 0.0125% interest for the 1 month loan.
The meaning of “on annual/semin-annual basis” is different from the periodicity of the cash flows, right? Like the YTM on semi-annual basis (YTM1) for the annual paid coupon is different from the YTM on annual basis (YTM2) for the same annual paid coupon. Cuz YTM servers as the discount rate to make the present value of the sum of the cash flows equals to the current market price. In this case, YTM2 = (1+YTM1 / 2)^2 -1. Am I right?
The spot rates are always quoted in annual terms. It seems all the rates (YTM, Coupon Rates, Forward Rates) are quoted in annual terms just on different time basis. Am I understanding correctly?
Nope, they are related. As I told in my previous post, the basis of the cash flows will determine the basis of the calculated YTM. If your cash flows are semi-annual, the calculated YTM will be on semi-annual basis. In order to get the annual YTM, just multiply by 2.
Nope. It is not correct to calculate compound interest because coupons are not compounded when paid, they are nominal. We can demonstrate this with a little example:
Suppose a bond with 1,000 face value that pays semi-annual with coupon of 5% and a life of 3 years. The bond is priced at 1,000 (par). We know that par bonds’ YTM is the same as their coupons. So this bond yields 5%.
Value the bond cash flows as:
(0) -1,000
(1) 25
(2) 25
(3) 25
(4) 25
(5) 25
(6) 1,025
Solving for the rate that equals the price and the cash flows we get exactly 2.50%. If we need the annual YTM, then just multiply by 2 and get 5.00%.
If we use compound formula, we will get another number: (1+0.025)^2 - 1 = 0.05063 = 5.06% which is incorrect. The YTM is 5%, not 5.06% because semi-annual cash flows does not generate compound interest.
Correct!, the convention is that all rates are on annual terms or annualized. We start from there when another time frame is needed.
Yes, thanks for pointing out that the YTM time basis is related with the payment periodicity.
I understand that YTM on semi-annual basis for the semi-annual paid bond is using the period discount rate to multiple 2. In your example, it would be 2.5% * 2 = 5%. But in my pervious post, I was thinking to calculate YTM on annual basis for the semi-annual paid bond. I think 5.06% in your example, should be the answer for the YTM on annual basis. Cuz if I need to convert YTM on annual basis to the semi-annual basis, I would do (1 + 5.06%)^1/2 - 1 = 2.5% * 2 = 5%.
This logic makes sense to me now. Am I following the correct lead?
Yes, thanks for pointing out that the YTM time basis is related with the payment periodicity.
I understand that YTM on semi-annual basis for the semi-annual paid bond is using the period discount rate to multiple 2. In your example, it would be 2.5% * 2 = 5%. But in my pervious post, I was thinking to calculate YTM on annual basis for the semi-annual paid bond. I think 5.06% in your example, should be the answer for the YTM on annual basis. Cuz if I need to convert YTM on annual basis to the semi-annual basis, I would do (1 + 5.06%)^1/2 - 1 = 2.5% * 2 = 5%.
This logic makes sense to me now. Am I following the correct lead?
