# Converting the annual yield on a bond | to comaparable bonds that make quartely coupon payments

In page 417 Example 7:

A five year, 4.50% semiannual coupon government bond is priced at 98 per 100 of par value. Calculate the annual yield-to-maturity stated on a semiannual bond basis, rounded to the nearest basis point. Convert the annual yield to:

a) an annual rate that can be used for direct comparison with otherwise comparable bonds that make quarterly coupon payments.

The solution states the YTM on a semiannual bond basis is 4.96% (0.0248 x 2).

Thus, how would you convert 4.96% from a periodicity of two to a periodicity of four?

My calculation is wrong: (1+ 0.0496/4)^4 = 0.0505

Any help on this would be really appreciated You calculate the yield first. N=10, PV=-98, PMT= 2.25, FV=100, CPT I/Y=2.47, since it’s semi-annual, multiply by 2 to get the annualized value,I.e., 4.96%. To convert this to a quarterly compounding period yield, the formular is [1+(annualized yield ÷ 2)]^2 = (1 + quarterly yield)^4. That is [1 + (4.96÷2)]^2 = (1+ quarterly yield)^4. Doing a little maths, you should get the quarterly yield as 0.0123. Multiply by 4 to annualize it. You get 0.0493. Hope I didn’t lose you.

Thanks adekunle, do you mind showing the actual steps of how you solve to get APR4?

• Find the nominal annual rate on a semi-annum bond basis:

N=10, PMT=2.25 (semi-annual coupon), PV=-98, FV=100

Compute I/Y we get 2.478261%, mutiply it by 2 to get 4.956522% which is the nominal annual rate on a semi-annum bond basis.

• Change periodicity from 2 to 4:

(1+ 4.956522%/2 )^0.5 - 1= 1.2315%, the effective quarterly yield

We mutiply it by 4 to get the nominal annual rate of 4.93% ,now on a quarterly bond basis.

hope it helps This. By hand. Just cos I can. http://tinypic.com/r/w1qt1j/8

adekunle and Ernest, this is fantastic! thank you! Hello All,

Will anyone please give a guide solving the question below with a BA II Plus using ICONV? The quarterly compounding gives 5.05% rather than 4.93.

A five year, 4.50% semiannual coupon government bond is priced at 98 per 100 of par value. Calculate the annual yield-to-maturity stated on a semiannual bond basis, rounded to the nearest basis point. Convert the annual yield to:

a) an annual rate that can be used for direct comparison with otherwise comparable bonds that make_quarterly_ coupon payments

When using the TVM worksheet to solve this problem, ensure that P/Y=C/Y=2. This will produce the correct semi-annual rate of 4.95652%.

For ICONV, enter 4.95652 as NOM and 2 for C/Y. Calculate EFF (produces 5.01794). Reset C/Y to 4 and recalculate NOM (produces 4.92619)

Absolute genius! Thank you.

But could you please explain the reset to 4 after the initial setting of C/Y to 2 to get the 5.02& to rather than outright setting to 4 and getting the 4.93% straightaway?

I use ICONV to solve for the effective annual interest rate first. The effective rate then acts as a reference point from which I can calculate any nominal rate for any value of C/Y.

In the TVM worksheet, you could also set P/Y=2 and C/Y=4 to get the nominal rate compounded quarterly.

Gracias

Today I was faced with same issue. Ex. 5,81 % YTM semiannual for quarterly compounding. I solved in this manner:

1. (1+0,0581 /2) for semiannual yield = 1,02905

2. convert to quarterly yield 1,02905^1/2 = 1,01442

3. convert to annual yield based on quarterly compounding (1.01442-1)*4 = 5,768 %

For monthly, weekly etc. root is 1/m and in last step should be multiply times n.

Make sure you know what the question wants. Here, you have created a _ nominal _, annual rate (compounded quarterly); if the question asked for an _ effective _ annual rate, this calculation would be incorrect.

Thanks for this tip. Could you correct my steps to correct calcualtion of EAR? Appreciate your help. I am also trying to understand formulas beside using my BA II Plus.

I have been using this formula for EAR so far (1+r/m)^n. m is frequency (2 for half of year, 4 for quarter etc…) and n is number of compounding periods. Fixed Income chapter is a bit confusing to me.

My friend, any help on this via BA II Plus?

A \$1000, 7% 10 year annual pay bond has a yield of 7.8%. If the yield remains unchanged, how much will the bond value increase over the next 4 years?

Calculate the value of the bond today, n = 10, i = 7.8%, PMT = 70, FV = 0, solve for PV.

Calculate the value of the bond four years from today, n = 6, i = 7.8%, PMT = 70, FV = 0, solve for PV.

The difference is the increase.

The answer should be less than \$20.

I think FV=1000 should do the trick! 