a. The spot rate is the quoted price for immediate settlement of trade of an asset. It is the current market value at which the trade can be settled. It reflects the current demand and supply situation of the asset being traded
b. Thus far we have seen the computation of present value using a single discount rate. But the right way to value the cash flows of a bond is to use multiple discount rates, i.e. valuing the cash flows of a bond by using different discount rates that are unique to the time period in which the cash flow would be received.
c. The present value of the bond should thus be calculated using the formula:
 |
Here, Z1, Z2, ……, Zn are the spot rates or zero rates during each period. That is, the yield-to-maturity on the zero-coupon bonds maturing at the date of each cash flow.
d. The present value arrived at, using these multiple spot rates is also called the bond’s ‘no-arbitrage value; since the investors cannot take advantage using an arbitrage opportunity by taking the opposite position on zero-coupon bonds to offset any future cash-flow to be received.
e. For example, consider a 5-year, $ 100, 7% bond. The spot rates during the 5-year period were:
|
Year |
Spot Rates |
|
1 |
7.50% |
|
2 |
8.30% |
|
3 |
8.75% |
|
4 |
9.50% |
|
5 |
10.15% |
In this case, the PV of the cash flows would be:
|
Year |
Cash Flow (in $) |
PV (in $) |
|
1 |
7 |
6.51 |
|
2 |
7 |
5.97 |
|
3 |
7 |
5.44 |
|
4 |
7 |
4.87 |
|
5 |
107 |
65.99 |
|
Present Value |
88.78 |
We can also calculate the yield-to-maturity of the bond by considering the PV of $ 88.78 and the FV of $ 100. The YTM of this series of payments is 9.96 %.
f. Consider another example, of two 3% bonds of $ 100, having 4 years to maturity. The spot-rates of the bonds in the different years were:
|
Year |
Spot Rates (Bond A) |
Spot Rates (Bond B) |
|
1 |
0.39% |
4.08% |
|
2 |
1.40% |
4.01% |
|
3 |
2.50% |
3.70% |
|
4 |
3.60% |
3.50% |
The present value of the bonds would be calculated as follows:
|
Year |
Cash Flows ($) |
Spot Rates (Bond A) |
Spot Rates (Bond B) |
PVA ($) |
PVB ($) |
|
1 |
3 |
0.39% |
4.08% |
2.99 |
2.88 |
|
2 |
3 |
1.40% |
4.01% |
2.92 |
2.77 |
|
3 |
3 |
2.50% |
3.70% |
2.79 |
2.69 |
|
4 |
103 |
3.60% |
3.50% |
89.41 |
89.76 |
|
Present Value |
98.10 |
98.10 |
Thus, we can see that despite different spot rates in different years, the present value of the bond remains the same. Here, the spot rates for bond A are increasing for the longer maturities; whereas, the spot rates for bond B are decreasing for longer maturities.
Since the present value of the two bonds and the stream of cash flows for the two bonds are the same, its YTM would also be the same, i.e. 3.52%.