# A very different spot rate and forward rate question

Hey guys,

long time lurker here. Just came across this question and have confusion about it. Could you help:

Bond Principal    Maturity (yrs)    Coup Rate    Quoted Yield    Bond price  Zero Coupon(Spot) Rates   Last period implied forward rate

100                     0.25                  0.00%           1.6064              99.6000                      ?                                                ?

100                     0.50                  0.00%            2.0202             99.0050                      ?                                                ?

100                     1.00                  2.50%            2.1500             100.3557                    ?                                                ?

100                      1.50                 4.00%            2.2500             102.5857                    ?                                                ?

100                      2.00                 5.00%            2.4000              105.0762                    ?                                                ?

We have to solve for the question marks. How to go about it ? Which all formulas do we use ?

Thanks

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How often are the coupons paid?

Simplify the complicated side; don't complify the simplicated side.

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I would say as the maturities are in quarters jumps, so we can assume that the coupons are paid quarterly.

If we assume that, we cannot solve the problem, because we don’t have yields for 9 months, 15 months, and 21 months.

If they haven’t specified the frequency of coupon payments, I suspect that they want us to use semiannual payments.  I was simply hoping that they’d made that explicit.

Simplify the complicated side; don't complify the simplicated side.

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Perhaps, you are right.

How do we solve this if we use semi annual payments ?

The yield on the 3-month bond is compounded quarterly while the yield on the 6-month bond is compounded annually.

The data aren’t consistent.

Simplify the complicated side; don't complify the simplicated side.

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One approach as follows provides zeros of 2.1508% (for year 1), 2.2529% (year 1.5), and 2.4031% (year 2)

Step one. Determined coupon payment frequency for bonds 3 through 5 is semi annual by checking consistency between bond price and quoted yield.

Step two. Solve for 1 Year Zero.  First two bonds provide zero for 0.25, 0.5 years respectively. We will use the rate for 0.5 years to discount the first coupon and then solve for 1.0 year discount rate (which is the one year zero);

PV of One Year Bond = PMT1/(1+zero_half_year)0.5 + PMT2/(1+zero_year1)1

100.3557 = 1.25/(1.020202)0.5 +101.25/(1+r)  Solving for r = 2.1508% is the year 1 zero

Step three.  With the 0.5 year and one year zeros we can go ahead and solve for the 1.5 year zero;

102.5857=2/(1.020202)0.5 + 2/(1.021508) +102/(1+r(1.5years))1.5   Solving for r(1.5years) is 2.2529%

Step four, repeat for the 2 year zero.

One approach as follows provides zeros of 2.1508% (for year 1), 2.2529% (year 1.5), and 2.4031% (year 2)

Step one. Determined coupon payment frequency for bonds 3 through 5 is semi annual by checking consistency between bond price and quoted yield.

Step two. Solve for 1 Year Zero.  First two bonds provide zero for 0.25, 0.5 years respectively. We will use the rate for 0.5 years to discount the first coupon and then solve for 1.0 year discount rate (which is the one year zero);

PV of One Year Bond = PMT1/(1+zero_half_year)0.5 + PMT2/(1+zero_year1)1

100.3557 = 1.25/(1.020202)0.5 +101.25/(1+r)  Solving for r = 2.1508% is the year 1 zero

Step three.  With the 0.5 year and one year zeros we can go ahead and solve for the 1.5 year zero;

102.5857=2/(1.020202)0.5 + 2/(1.021508) +102/(1+r(1.5years))1.5   Solving for r(1.5years) is 2.2529%

Step four, repeat for the 2 year zero.

How did you get the spot rates for 0.25 and 0.5 period bonds ? What formula to use ? and then how to get the Last period
implied forward rate ?

It will be super helpful if i have more detail.

Freaky guy: The first two bonds (0.25, 0.5 years) have no coupon. Therefore I took their quoted yields as the zero rates for 0.25 and 0.5 years respectively.

I have not gotten around to solving the forward rate part of the question. However, one can solve it using the forward rate model equation.

Freaky guy: The first two bonds (0.25, 0.5 years) have no coupon. Therefore I took their quoted yields as the zero rates for 0.25 and 0.5 years respectively.

I have not gotten around to solving the forward rate part of the question. However, one can solve it using the forward rate model equation.

Got it.

I have the spot rates now

Bond 1 - 1.6161.

Bond 2 - 2.0202

Bond 3 - 2.1393

Bond 4 - 2.2402

Bond 5 - 2.3944

How do i get the forward rates ?

In general, if you have the m-period spot rate, sm, and the (m+1)-period spot rate, sm+1, then the 1-period forward rate starting m periods from today. 1fm is calculated by:

1 + 1fm = (1 + sm+1)m+1 / (1 + sm)m

However, there’s still the problem of how these rates are quoted, which is inconsistent.  You need effective rates to use in the formula I wrote, but the rates you’re given aren’t (necessarily) effective rates.

Simplify the complicated side; don't complify the simplicated side.

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Two comments: I agree with S2000 this question is incomplete - we need to know the compounding periods as the numbers are inconsistent.

That said, if the bonds are semi-annual then I think a correction to my earlier post is warranted. If bonds 3 to 5 are semi-annual, there are two periods per year. Two years = 4 periods.  Therefore the spot for each period is found sequentially by equating the bond PV to the PMTs discounted by the spot rates.  It is then annualized.

For instance, the spot for year one is found with:

100.3557 = 1.25/(1.010101)1 +101.25/(1+r(2))2

r(2) is the discount rate for the second period which is the second half year.  Multiply by 2 to obtain the annualized value for the spot rate for year one.