Calculate spot rate from par rate

Hi there,

Could you kindly help with the following question.

How do you calculate the 5th year spot rate, when given the following information?

1 year: Par rate = 2.5% Spot rate = 2.5%

2 year: Par rate = 2.99% Spot rate = 3.0%

3 year: Par rate = 3.48% Spot rate = 3.5%

4 year: Par rate = 3.95% Spot rate = 4%

5 year: Par rate = 4.37% Spot rate = X%

The answer is 4.45% - however, I don’t know how to calculate this. If you could kindly show me the quickest method on BA II plus, it would be most appreciated.

Many thanks,


We need to use bootstrapping, and it’s time consuming even with calculator.

par rate = coupon rate

assuming 100 par value, 4.37 being the 5-year par rate is the coupon rate

100 = 4.37/(1.025) + 4.37/(1.03)2 + 4.37/(1.035)3 + 4.37/(1.04)4 + 104.37/(1+x)5

(you can also use 1 as the par and 0.0437 as the coupon but the extra decimals makes me twitchy)

solving for x gives you 4.45%

Thank you Edbert. To solve for X, is there a quick way to do this? Many thanks.

yes? when asked for bond pricing using spot rates (either now or back in Level 1), how do you usually do it in the calculator? it’s almost the same process.

I use the STO and RCL function of my calculator most of the time (using TI BA2 plus) and that’s enough for this question.

I have actually never been shown how to do this.

Could you kindly write out the functions of the calculator that you would press, to get the answer for this question, so I can learn.

Many thanks,

do you at least know how to store and recall values using STO and RCL? If not, follow this page:

this is the formula that I wrote and we’re looking for x: 100 = 4.37/(1.025) + 4.37/(1.03)2 + 4.37/(1.035)3 + 4.37/(1.04)4 + 104.37/(1+x)5

now, you know the coupon is 4.37, here’s what I’d do:

  1. find 4.37/1.025, then store it

  2. find 4.37/1.032 then store it

  3. find 4.37/1.0353 then store it

  4. find 4.37/1.044 then store it

  5. press 100, and then subtract it by the sum of values from step 1-4

  6. using the value you got from step 5, press 1/x and then multiply it by 104.37

  7. using the value you got from step 6, press yx and enter 0.2

  8. subtract value from step 7 by 1 to arrive at the 5-year spot rate

if you need help or explanations on how/why to do some of these steps quickly, message me directly rather than replying in this thread. I can do all 8 steps in less than a minute. practice using your calculator to its utmost potential. I can do all these steps in less than a minute and so can you.

if you are not confident with storing and recalling values in your calculator, then you should just write down the values you get in each step.

Bootstrapping is a Level I burden. And it’s gross.

Don’t worry about it at Level II.

(If you feel that you need to worry about it, I cover it in detail in this article: It’s one of the free articles on my website, so I encourage you to read through it.)


can you please explain steps 5 & 6 and also plz explain the reason for multiplying 0.2 in step 7

You’re not multiplying by 0.2; you’re raising it to the 0.2 (= 1/5) power.