Question

Can anyone solve this?

You are given the following information:
• The 1-year interest rate is 4.5% (with continuous compounding).
• The 2-year swap rate is 5.11% (with simple compounding).
• A 3-year, 7% coupon bond is priced at $104.30.
• A 4-year, 3% coupon bond is priced at $90.80.
Assume that all swaps and bonds have annual payment frequency. Bond prices are given as $ and ¢ per $100 par (not in “32nds” as customary in US bond markets).
(a)What are the spot rates (with continuous compounding) for 2, 3, and 4 years’ maturity? What are the implied discount factors?
(b) What are the implied forward rates (continuous compounding) for “1-into- 2”, “2-into-3”, and “3-into-4” years? What do these forward rates tell us about market participants’ expectations about future spot rates?
Consider a 4-year swap, in which one party will pay fixed annual interest at a rate of 6%, whilst receiving floating interest (also annually), both on a principal of $1 million.
(c) Given the rates you have calculated in parts a) and b) above, what (to the nearest $1) is the “fair” value of the swap (to the payer of fixed)? (Hint: don’t forget to convert the forward rates from part b) into simple compounding!)
(d) For the swap considered in part c), what is the “swap rate”? Confirm your result by re-valuing the swap, using the swap rate for the “fixed leg”. Why is the swap rate higher / lower than that of the 2-year swap?

Answer 1

Solution:

Assumption:

1. all swaps and bonds have annual payment frequency
2. Bond prices are given as $ and ¢ per $100 par (not in “32nds” as customary in US bond markets).
3.           =1 (i.e. T₁ =1 and T₂ =2 ......)
4. B_0,1 , B_0,2 , B_0,3 , B_0,4 are the discount factors for maturities of 1,2,3 and 4 years respectively.
5. r_0,n (n=1,.....4) is the continuously compounded spot rates for k years' maturity

Part (a):

Spot rates (with continuous compounding) for 2, 3, and 4 years’ maturity and the implied discount factors are given as follows:

1. From the information given, we know that 1-year rate i.e. r_0,1=4.5%. This implies a discount factor of

B_0,1 =exp(-r_0,1)=exp(-0.045) = 0.9560

2. Using            =1, the two year swap rate is given by R₂ = (1- B_0,1 )/( B_0,1 + B_0,2 ). From the information given, R₂ = 5.11% and from above B_0,1 = 0.9560. Therefore, to find the 2-year discounting factor, we solve:

0.0511 = (1- B_0,2 ) / (0.9560 + B_0,2)

Re-arranging the above, we get:

B_0,2= (1-0.0511*0.9560)/(1+0.0511)

= 0.9049

Which means 0.9049 = exp(-r_0,2 *2) hence r_0,2 = - IN(0.9049)/2 = 5 (rounded). Thus, we can conclude that the 2-year rate (Continuously compounded) is approximately 5%

3. The price of the 3-year bond must be equal to the present value of its future cash flows. As the bond pays 7% annually, the cash flows (per $100 par) are $7 in the next two years, and $ 107 in three years' time. hence:

$104.30 = (B_0,1 *7$) + (B_0,2 *7$) + (B_0,3 *107$).

As we know B_0,1 = 0.956 and B_0,2= 0.9049. Substituting both, we get:

$91.27= B_0,3 * $107.

i.e. B_0,3 = $91.27 / $107

i.e. B_0,3 = 0.8530

Now, 0.8530 = exp (-r_0,3 *3) = 5.3 % rounded.

Thus, we can conclude that the 3-year rate (Continuously compounded) is approximately 5.3%

4. The 4-year bond is priced at $90.80 and pays 3% annual coupons. Hence:

$90.80 = (B_0,1 *$3) + (B_0,2 *$3) + (B_0,3 *$3)+ (B_0,4 *$103)

Using the above discounted factors, this is brought down to $82.66 = B_0,4 *$103.

Thus, B_0,4 = 0.8025. Like before, we solve:

0.8025 = exp (-r_0,4 *4) to get (r_0,4 ) = 5.5 % rounded.

Thus, we can conclude that the 4-year rate (Continuously compounded) is approximately 5.5%

Part (b):

Forward rate from time T_n to T_n+1 (with countinous compunding) is f_{n,n+1 =} (r_0,n+1* T_n+1) - (r_0,n*T_n)

thus, the 0-into-1 year forward rate is equal to the year 1 spot rate i.e. 4.5%. We also know the spot rate for year 2 which is 5%. Together, this enables us to conclude the 1-into 2 year forward rate:

f_1,2 = (5.0% *2) - (4.5% *1) = 5.5%.

Similarly, using the spot rates derived in part (a), we get 5.9% and 6.1% for the 2-into-3 and 3-into-4           forward rates, respectively.

Part (c)

The "payer of fixed" will pay 6% fixed interest. We can value this stream of cash flows by replacing all payments at floating rates at the corresponding forward rates and then discounting the resulting cash flows. Hence, the value of the swap is given as follows:

0.9560 *(4.60% -6%) *$ 1,00,000

+0.9049 *(5.65% -6%) *$ 1,00,000

+0.8530 *(6.08% -6%) *$ 1,00,000

+0.8025 *(6.29 -6%) *$ 1,00,000 = -$13,542 (rounded to nearest 1)

Part (d):

We use the equation for the swap rate:

R₄ = (1-B_0,4)/ (B_0,1 + B_0,2 + B_0,3 + B_0,4 )

From part (a) and part (b), R₄ = 5.6165%

To check, we re-value this swap, this time using 5.6165% on the fixed side.

0.9560 *(4.60% -5.6165%) *$ 1,00,000

+0.9049 *(5.65% -5.6165%) *$ 1,00,000

+0.8530 *(6.08% -5.6165%) *$ 1,00,000

+0.8025 *(6.29 -5.6165%) *$ 1,00,000 = -$56

The value is not exactly zero due to rounding error , but given the notional of $1 million, the value of $56 is negligible. In a nutshell, the swap rate must be a weighted average of the forward rates.