Spot and Forward Rate

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  • Topic: Zero-coupon bond, Bond, Bond duration
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  • Published : February 28, 2012
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Spot Rate and Forward Rate

Spot Rate:

It is nothing more than the YTM on bond. It is the rate of interest on bond maturing at any time in the future. It is also known as geometric average of 1 year forward rates in the future. Hence, YTM on bonds can be calculated as:

When forward rates are given then;
oS1 = oF1
oS2 = [(1+of1)(1+1F2)](1/2) -1
oS3 = [(1+of1)(1+1F2)(1+2F3)](1/3) -1 and so on.

Forward Rates:

It is the interest rate established today , that will be paid on money to be borrowed at some specific date in the future, and to be repaid at a specific but even more distant date in the future.. Forward rate is the interest that links the current spot rate over holding period to the current spot rate over a longer holding period. In particular, it is written as ‘ tFt+n’. Based on the spot Rates of bonds, we do calculate the forward rates as:

tFt+n = { [(1+oSt+n)t+n/(1+oSt)t]1/n} – 1
Where,
T= time when the bond will be issued.
N= maturity period of the bond.

PROBLEMS
Q.1Assume 4- year bonds are currently yielding 7 percent and 3 – years bonds are yielding 6 percent. What is the implied yield for 1-year bonds starting 3 years from now ? Show your work.

Q.2Use the following data:
BondMaturity, yearsYTM
W
X
Y
Z1
2
3
48.0%
9.0
10.5
12.0
Calculate the implied 1- year forward rate starting in year 2.
Calculate the implied 1-year forward rate starting in year 3.
Calculate the implied rate for a 3- year bond starting in year 2. Q.3If a 15-year T-bond is yielding YTM =12% and a 5- year T- bond is yielding YTM =8%,what is the expected return on a 10 – year bond starting at the end of Year 5 ? Bond’s Duration

The formula use to calculate the bonds basic duration is the Macaulay duration, which was created by Fredrick Macaulay in 1938.it, has been commonly used since 1960s. It is also known as Macaulay’s duration. Bond duration is the average amount of time required by the security to receive the interest and the principle. Therefore duration is a weighted average of time that interest payments and the final returns of principle received. The weights are present value of payments, using the bonds yields to maturity as the discount t rate.

The duration, therefore, calculated the weighted average of the cash flows (interest and principal payments) of the bond, discounted to the present time.

Duration is stated in terms of years. The duration measure will predict by how much a bonds price should change given a 1% change in interest rate. Thus, a bond with duration of 4 years will decrease by 4 % in price if the yields rise by 1%.

Duration helps an investor to identify the percentage change in the price of a bond. For example, if a bond has duration of 8 years and interest rate falls 6% to 4% ( a drop of a 2% point), the bonds price expected to rise by 16% (8 × 2).

The duration of bonds can be computed by the following formula:

Or, D = (W x T)

PV (Ct)= PRESENT VALUE OF the cash flows to be received at time t P0=current market price of bond
T= bond remaining life
D = Macaulay s duration

Alternatively,
1+ y _ (1+y ) + T (c –y)
D = y c [(1+ y)n – 1] + y

Where,
C= the annual coupon rate (as a percentage)
T = the number of years to maturity
Y= the yield to maturity

Illustration 1

Consider a Rs. 1000 bond with three years to maturity and a 9% coupon rate. Currently, the rate of interest on comparable bonds is 12%. Calculate the duration of bond.

Solution
Given
Years to maturity (n) = 3 years
Par value of bond (M)=Rs.1000
Coupon rate ©=9% annually
Macaulay’s duration (D)=?

Calculation table

yearCash flowsPVIF@12%PVweightproduct
(1)(2)(3)(4)=(2)×(3)(5)(6)=(5)×(1)
1Rs.900.8929Rs.80.360.08660.0866
2900.797271.750.07730.1546
310900.7118775.860.83622.5086
Rs.927.971.00002.7498

Therefore Macaulay s duration is 2.7498 = 2.75 years....
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