# How to calculate YTM from spot rates

In the correction of Fixed income - Akron, they said “using the spot rates, calculates the YTM for the bond using your calculator”.

I just realize I don’t know how to do that.

Its the geometric mean of sport rate

Spot rates are :

Y1: 5,5%

Y2: 6,25%

Y3 : 7%

Y4 : 8,25%

When I do ((1,055)*(1.0625)*(1.07)*(1.0825))^(1/4) = 6,75% but the YTM in the excercise is 8,14%.

You must discount each payment by its spot rate. Then you get a value which you enter as PV in your calculator.

Along with coupon as PMT, FV, N, you compute I/Y=>8,14%

i could be wrong, but i believe you set it up like the swap term structures…where u find PV/discount factors for each year. in this case 0.9479, 0.8858, 0.8163, 0.7283. then use the formula 1- PV(4) / PV1 + PV2 + PV3 + PV4 …ie. just like finding the fixed rate leg on a swap. and the answer i got was 8.04%

to be honest, i’ve never been asked to calculate the YTM without knowing the bond price. if you know the bond price, you simply plug in the inputs into your calculator to solve for YTM.

You can calculate the bond price with the spot rates, then use th CF function to calculate the IRR.

When I calculate the bond price with the spot rates I have 107.364

When I do:

PV = 107.364

PMT = -4

FV = -100

N = 4

I got YTM = 2.06. What I am doing wrong?

Not even close.

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Step 1)Discount the Cash flows with the respective Spot rates & you will get the present value of the cash flows.

1. Then plug the data in the Calulator to find the YTM…

Hope this will help you…

Can u please post the example here.

^^ what if there are no cash flows. ie. and what i was getting at in my answer above, what if simply the spot rates are given and nothing else. you can find the spot rates from the par curve (yield curve) and you can find foward rates from those two curves, but can you not find the par curve from the spot curve as well? i believe you can, and it’s what i was trying to figure in my answer. just not sure if that’s correct in how you do it, b/c in CFAI text/curriculum they never explicitly ask us to find the par curve (yield curve) from spot rates, i don’t believe.

Your PV is wrong. This is how you do it for a 4% coupon, 4 year bond (assume a \$100 par value for calculation purposes). Your cash flows will be (coupon * par) each year, so \$4. FV will be \$100 + the final \$4 coupon. Therefore:

Y1: 5,5%

Y2: 6,25%

Y3 : 7%

Y4 : 8,25%

PV = [CF1 / (1+ Spot1)] + [CF2 / (1 + Spot2)^2] + [CF3 / (1+Spot3)^3] + [CF4 / (1+Spot4)^4]

----note that you are raising each (1+spot) to the “n” power of whatever # spot rate it is. I.E. if its the 3rd spot you are raising it to the 3rd power

PV = ( 4 / 1.055 ) + ( 4 / 1.0625^2) + ( 4 / 1.07^3 ) + ( 104 / 1.0825^4) (quite laborious on a calc)

PV = 86.34

Now take your PV and input that into the TVM function on your calculator like this:

PV = -86.34

PMT = 4

N = 4

FV = 100 (do not include the final coupon payment here as you did in the calculation above, the calculator can figure that out itself)

then hit [CPT] + [I/Y] and you should arrive at ~8.13%

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^ Pocketsquare!!! Way to nail your first analyst forum participation point!!!

I’m not entirely sure what you’re trying to do here but I suspect you are making this more complicated then it really is.

When there are cash flows

Simply take this discounted cash flows at each spot rate (which you may or may not have to derive from forward rates that are given).

Once you’ve got the discounted cash flows, simply plug them into your calculator’s CF function. You do this by making CF0 equal the sum of your cash flows multiplied by negative one (i.e., just make the sum negatvie). Then you just plug in each period’s discounted cash flow into the respective period. Finally, solve for IRR.

When there are no cash flows

Someone please correct me if I’m wrong on this part but I believe this is even easier (well quicker to solve at least). Since there are no cash flows, the interim spot rates don’t matter because you don’t have to invest the coupons. The YTM will be equivalent to the spot rate. You can test this by discounting the par value of the zero coupon bond using the appropriate maturity length and spot rate. Use that PV figure as you negative CF0 and then plug in 0 for all of the interim cash flow periods until your’re maturity date. Then calculate IRR (i.e., YTM).

For example, assume you have a 10 year zero coupon bond at 100 par value and the 10 year spot rate is 7.5%. What’s the YTM? Well the 100 par value discounted back 10 years (100/(1.075^10)) will give you a present value of ~\$48.52. So plug in the following into your calculator (assuming you’re using a BA 2 +, I’ve never used the HP):

CF0 = -48.52

C01 = 0

F01 = 9

C02 = 100

F02 = 1

Calculate IRR (i.e., YTM) -> 7.5%

Discount Factors

All a discount factor really tells you is what \$1 discounted at the spot rate will give you. Therefore, if you’re given discount factors, you can use algebra to derive the spot rate. You can then use spot rates to derive forward rates.

I hope that helps with some of the problems you’re having.