Spot rates

CFA Level I
#1

Can someone pls explain this for me? I can’t wrap my head around this!!

analyst collects the following info on spot rates:

1-yr rate: 4%

2-yr rate: 5%

3-yr rate: 6%

4-yr rate: 7%

utilizing the pure expectation theory of the term structure of interest rates, the expected annualized 2-yr interest rate two years from today is?

a) 7%

b) 8.03%

c) 9.04%

#2

a?

#3

(4 * 7) - (2 * 5) = 18 /2 = 9

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Not 100% positive on this one. This is the shortcut/cheat way to get it, but I couldn’t remeber if this was it.

#4

Someone confirm the ans

#5

4yr bond/ 2 yr bond: (1.07^4)/(1.05)^2 =1.188930621 This give us the future value of a 2 yr bond at time 4 (from time 2 to time 4) Annualize it: sqrt(1.188930621)=1.090380952 You know what this means: 1.090380952 -1 = 0.090380952 Thus, the answer is c

#6

Think it through like this:-

  1. Short the 2-yr zero today. You receive 100/1.05^2 = $90.7029. The 2-yr zero will mature at $100 in 2-yrs.

  2. Invest the proceeds of $90.7029 in the 4-yr zero today. The 4-yr zero will mature at 90.7029 * 1.07^4 = $118.8931 in 4-yrs.

  3. So you’ve locked-in an investment of $100 in 2 years which matures at $118.8931 in 4 years.

  4. Then it’s clear the 2-yr rate 2 years from today is sqrt(1.07^4 / 1.05^2) as pointed out by EddieChen

#7

The question wants you to calculate the Forward Rate (2f2) from the given spot rates.

If you use Schweser 2012, there is an example like this one on page 121 of book 5 Fixed income.

The Answer is C and calculation as @EddieChen did above.

In the exam you may asked to calculate 3f1, 3f2… prepare for them :-s

#8

Bond maturity=1000rs. Time= 3 years. Coupn= 7%. YTM of treasury strips are Yields to Maturity Maturity 6.5% 1 yr 7% 2 year 8% 3 year Calculate value of bond using spot rates

#9

Bond maturity=1000rs. Time= 3 years. Coupn= 7%. YTM of treasury strips are Yields to Maturity Maturity 6.5% 1 yr 7% 2 year 8% 3 year Calculate value of bond using spot rates

#10

7*[1/1.08^3 + 1/1.07^2+1/1.065] + 100/1.08^3 97.626

so 976.36 for a 1000$ par bond.

#11

Yah man

But why we did not use bootstrapping to calculate spot rates for 2nd and 3rd year

#12

Great explanation here: http://www.youtube.com/watch?v=UPuqjYz0KwA

#13

Thanks buddy. Its great video

So YTM of treasury= Spot rates :slight_smile:

Can u provid link for video on bootstrapping. , arbitrage free bond valuation

#14

3 year spot rate= 4% 5 year spot rate= 5% 4 year forward rate ,three years from now- 6% 3 year forward rate, 7 years from today- 7% Calculate 2 year forward rate, 5 years from today a) Cant be calculated b) 5.48% c) 6.3%

#15

3 year spot rate= 4% 5 year spot rate= 5% 4 year forward rate ,three years from now- 6% 3 year forward rate, 7 years from today- 7% Calculate 2 year forward rate, 5 years from today a) Cant be calculated b) 5.48% c) 6.3%

#16

  1. Short the 5-yr zero today. You receive 100/1.05^5 = 78.3526. This will mature at par (100) in 5 yrs.

  2. With the proceeds, long the 3-yr zero. This will mature at 78.3526 * 1.04^3 = 88.1360 3 yrs from today.

  3. Roll the proceeds of (2) above into the 4-yr forward 3-yrs from today. This will mature in 7-yrs from today at 88.1360 * 1.06^4 = 111.2697.

You’ve effectively invested $100 5-yrs from today which matures at $111.2697 7-yrs from today. The annualised rate is sqrt(111.2697/100)-1 = 5.48%

If you’re clear on the steps, the short-cut is sqrt (1.04^3 * 1.06^4 / 1.05^5 ) - 1.

#17

Rambus this looks a lil timeconsuminng any shortcuts for such questions

#18

Short cut is the last line, i.e. sqrt (1.04^3 * 1.06^4 / 1.05^5 ) - 1

#19

Hey all,

for ppl that are having trouble with this question and forward rates in particular.

Lets think of this question in terms of an investment account where I have 1 dollar to invest today.

I have options on how long I want to invest this dollar (1, 2, 3 years etc.)

I also can see the interest rates at various maturities. As per the question above:

1-yr rate: 4%

2-yr rate: 5%

3-yr rate: 6%

4-yr rate: 7%

As an investor, I should be indifferent between investing my money at 4 years @ 7%, or say, investing my money for 2 years @ 5% and rolling over the principal ($1) and interest for another 2 years. The forward rate for a 2 years investment, starting in 2 years, is the rate that will make me indifferent between these two strategies.

Here are the calc’s

If I invest $1 for four years, my investment account will grow to 1*(1.07)^4 = $1.3108. Hence, on maturity I will receive $1.31 of proceeds for my four year investment.

Now, If instead I choose to invest for 2 years and roll over principle and interest for another two years:

For the first two years my investment account will grow to 1*(1.05)^2 = 1.1025

I will then roll over this 1.1025 for another two years and if I am indifferent between the two strategies, my 1.1025 should grow at a rate that in two years from now will mature with a value equal to my four year investment - $1.3108

1.1025 * ( 1 + x ) ^ 2 = 1.3108

now solve for X

(1.3108 / 1.1025) ^ (1/2) - 1= x

x = 9.38% ~ 9.4%

#20

If r[0,7] is the 7 year spot rate, then: (1+r[0,7])^7 = (1.04)^3 * (1.06)^4

r[0,7] = 5.1382%

Let the 2 year forward rate, 5 years from today be r[5,7].

(1 + 0.051382]^7 = (1.05)^5 * (1 + r[5,7])^2

r[5,7] = 0.0548 = 5.48%