3.3 Spot Rates
The reinvestment rate for the MIRR is exogenous or external; so far for us, in other words, it was a “given.” The analyst must project the project’s reinvestment rate, based on economic and financial matters about which he is aware within the corporation’s business environment. Absent such information, s/he must, obviously, look elsewhere.
Some useful clues may be garnered from the bond yield curve (see The Yield Curve, the Spot Curve, and Pure Expectations section above), which embeds critical information about future short-term interest rates. The “spot curve,” which we delineate below, will provide not merely a single future reinvestment rate, but multiple and varying rates over the term of the project. The provision of varying reinvestment rates will satisfy the claim that future rates cannot be monolithic.
Example I
Choice 1:
1.5-Year Treasury Bond
Price = 99.45 ( Par = 100)
Coupon = .085
Choice 2:
6-Month Zero YTM = .08
1-Year Zero YTM = .083
- What is theoretical 1.5-year Spot Rate (1R3)?
- The two alternatives should be equal due to the Law of One Price.
- Note that the Zero-coupon Yields are the same as the Spot Rates for the six-moth and one-year periods. Spot Rates involve no reinvestment of coupons.
Solution I
Coupon Payments = ($100) (.085) (.5) = $4.25
[latex]99.45 = \frac{4.25}{(1.04)^{1}} + \frac{4.25}{(1.0415)^{2}}+\frac{104.25}{(1+y_{3})^{3}}[/latex]
Where y1 = (.080) (.5)
Where y2 = (.083) (.5)
Where y3 = ?
[latex]99.45= \frac{4.25}{(1.04)^{1}}+\frac{4.25}{(1.0415)^{2}}+\frac{104.25}{(1+y_{3})^{3}}[/latex]
y3 = .04465 AND BEY3 = (2) (.04465) = .0893
Where BEY = Bond Equivalent Yield
Example II
Price = 99.64 (Par = 100)
Coupon = .090
Spot Rates for y1 – y3 as in Example #1 above
- What is theoretical 2.0-year Spot Rate (y4)?
Solution II
[latex]99.64 = \frac{4.50}{(1.04)^{1}} + \frac{4.50}{(1.0415)^{2}}+\frac{4.50}{(1.04465)^{3}}+\frac{104.50}{(1+y_{4})^{4}}[/latex]
y4 = “1R4” = .046235
BEYy4 = (2) (.046235) = .09247
Summary (Examples I and II):
| Maturity | Coupon | Dollar Price | Y-T-M | Spot Rates |
|---|---|---|---|---|
| 6-months | 0.00% | 96.15 | 0.800 | 0.8000 |
| 1-year | 0.000 | 92.19 | .0830 | .08300 |
| 1.5 years | 0.085 | 99.45 | .0890 | .08930 (As calculated) |
| 2.0 years | 0.090 | 99.64 | .0920 | .09247 (As calculated) |
- Can you graph this?
- Show the terms-to-maturity on the horizontal.
- Show the yields for the two curves on the vertical.
- Note that, in this case, the curves diverge after one-year.
