You buy an 8-year bond at exactly three quarters of the way through its fifth year. Its price is 110 (i.e 10% over par) plus the interest accrued since the last coupon was paid. The coupon is 8% per year, payable every six months. What is the yield to maturity?

The YTM is the discount rate which makes the PV of all your cash flows net to zero.

First, get a good handle on all your cash flows, both as to their amounts and their exact timing. At time t = 0 you have a cash outflow equal to the 110 puchase price, plus the accrued interest. Subsequently, you have cash inflows of $4 on each of the remaining coupon dates, and then the final maturity payoff of $104 (including the final coupon).

Having scheduled out your cash flows, you'll then go through the usual PV exercise of discounting all the cash inflows by some discount rate. The objective is to identify a discount rate such that the sum of the inflows' PV is exactly equal to your total immediate outflow (which is equivalent to saying that the sum of all the cash flows' PV is zero, since the t = 0 outflow has a negative PV equal to its actual amount). Once identified, that particular discount rate is your bond's Yield To Maturity.

Very roughly, if you have more than two CFs (as you do here), it amounts to trial-and-error. If you're familiar with the algebraic rendering of the PV of a cash-flow series, you'll see that it's really just a polynomial. With two cash flows, you've got a quadratic, whose roots are easily determined by your favorite weapon of choice. Three or four cash flows, and you're dealing with a cubic or quartic--roots are 'findable' but difficult. Five or more CFs, and it's off to trial-and-error land.

That's the bad news. The good news is that many tools, such as Excel, can blaze through the trial-and-error iterations for you in a blink. Having laid your bond's cash flow amounts and timing, try hitting them with Excel's Goal Seek or Solver tools, or its IRR function.

Go through the foregoing procedure and you'll have the bond's YTM in no time. Best of luck!

...it was early and I was full of no coffee...

The YTM is the discount rate which makes the PV of all your cash flows net to zero.

First, get a good handle on all your cash flows, both as to their amounts and their exact timing. At time t = 0 you have a cash outflow equal to the 110 puchase price, plus the accrued interest. Subsequently, you have cash inflows of $4 on each of the remaining coupon dates, and then the final maturity payoff of $104 (including the final coupon).

Having scheduled out your cash flows, you'll then go through the usual PV exercise of discounting all the cash inflows by some discount rate. The objective is to identify a discount rate such that the sum of the inflows' PV is exactly equal to your total immediate outflow (which is equivalent to saying that the sum of all the cash flows' PV is zero, since the t = 0 outflow has a negative PV equal to its actual amount). Once identified, that particular discount rate is your bond's Yield To Maturity.

Very roughly, if you have more than two CFs (as you do here), it amounts to trial-and-error. If you're familiar with the algebraic rendering of the PV of a cash-flow series, you'll see that it's really just a polynomial. With two cash flows, you've got a quadratic, whose roots are easily determined by your favorite weapon of choice. Three or four cash flows, and you're dealing with a cubic or quartic--roots are 'findable' but difficult. Five or more CFs, and it's off to trial-and-error land.

That's the bad news. The good news is that many tools, such as Excel, can blaze through the trial-and-error iterations for you in a blink. Having laid your bond's cash flow amounts and timing, try hitting them with Excel's Goal Seek or Solver tools, or its IRR function.

Go through the foregoing procedure and you'll have the bond's YTM in no time. Best of luck!

...it was early and I was full of no coffee...

Thank you very much

This is what I have done! Is this oK?


Date Day Count % of Period
Previous Coupon 30/06/Y5
Today 30/09/Y5 90 50%
Next Coupon 30/12/Y5 90 50%
Total Days(30 day Convention) 180

Nominal IRR 8%
Periodic IRR 4%

Period Date PMT PV

0 30/09/Y5 112,00 -100,00
0,5 30/12/Y5 4,00 3,92
1,5 30/06/Y6 4,00 3,77
2,5 30/12/Y6 4,00 3,63
3,5 30/06/Y7 4,00 3,49
4,5 30/12/Y7 4,00 3,35
5,5 30/06/Y8 4,00 3,22
6,5 30/12/Y8 104,00 80,60

Yep, looking good so far. You've set up your PV model correctly, to allow for the fractional periods involved.

What you've done is to determine the bond's price IF the proper discount rate is 8% per year (or 4% per coupon period). When you add up the results you have there--that is, the PVs of the individual cash flows--you see that you've got $101.98, which is the expected price of $102 after allowing for a bit of rounding. (The expected price of the bond, when discounted by its coupon rate, would be the par value plus the partial-period accrued interest when purchased, or 100 + 2 = 102).

Now you just need to take the final step. Your question was to determine the bond's yield, or IRR. So you just need to determine, through trial-and-error, the discount rate which makes all those cash flow PVs add up to your acquisition cost of $112, instead of the $101.98 they add up to now. When you determine the discount rate that does the job, you've found the bond's yield, for a purchaser who buys it and holds it under the scenario you originally described.

(Hint: It'll be less than 4%, since it'll take a lower discount rate to make the PV of the cash flows greater than $102. That's why bond prices move inversely to market rates.)

Best of luck!

...it was early and I was full of no coffee...