Introduction: Navigating the Taxman’s Share
Welcome! In our previous studies, we looked at how to value bonds based on their gross cashflows. However, in the real world, investors rarely get to keep every penny. Governments typically take a portion of the interest as income tax and a portion of the profit as capital gains tax (CGT).
For an actuary, calculating the "fair price" of a bond means looking at the net cashflows—the money that actually lands in the investor's pocket. Don't worry if this sounds a bit technical; once you understand the logic of "net of tax," the formulas become much easier to manage!
1. Understanding the Two Types of Tax
When an investor buys a fixed-interest bond, they usually face two types of taxation:
A. Income Tax
This is a tax on the coupons (the regular interest payments). If a bond pays a gross coupon of \(D\), and the income tax rate is \(t_1\), the investor only receives:
Net Coupon = \(D \times (1 - t_1)\)
B. Capital Gains Tax (CGT)
This is a tax on the "profit" made between the purchase price (\(P\)) and the redemption price (\(R\)). If you sell (or redeem) the bond for more than you paid for it, the government takes a cut of that profit.
Capital Gain = \(R - P\) (provided \(R > P\))
Tax Amount = \(t_2 \times (R - P)\), where \(t_2\) is the CGT rate.
Quick Review:
- Income Tax affects the annuity part of the bond price (the coupons).
- Capital Gains Tax affects the lump sum at the end (the redemption).
2. The Basic Bond Price Formula with Tax
To find the price (\(P\)) of a bond that provides a net yield of \(i\), we set up an equation of value. We assume the investor wants to find the present value of all future net cashflows.
The general formula looks like this:
\(P = g(1 - t_1) a_{\overline{n}|i} + R v^n - t_2(R - P) v^n\)
Where:
- \(P\) = Purchase price
- \(g\) = Gross coupon rate per annum (so \(g \times R\) is the annual coupon)
- \(t_1\) = Income tax rate
- \(t_2\) = Capital gains tax rate
- \(R\) = Redemption value (usually 100)
- \(n\) = Time to redemption
- \(i\) = The investor’s required net redemption yield
Why is \(P\) on both sides?
You might notice that \(P\) appears in the main formula and also inside the CGT term. This is because the tax depends on the price, but the price depends on the tax! In exams, you will often need to rearrange this equation to solve for \(P\).
Step-by-Step Rearranging for \(P\):
1. Expand the bracket: \(P = g(1 - t_1) a_{\overline{n}|} + R v^n - t_2 R v^n + t_2 P v^n\)
2. Move all \(P\) terms to one side: \(P - t_2 P v^n = g(1 - t_1) a_{\overline{n}|} + R v^n(1 - t_2)\)
3. Factorise \(P\): \(P(1 - t_2 v^n) = g(1 - t_1) a_{\overline{n}|} + R v^n(1 - t_2)\)
4. Solve: \(P = \frac{g(1 - t_1) a_{\overline{n}|} + R v^n(1 - t_2)}{1 - t_2 v^n}\)
3. The "Capital Gains Test": Do We Pay Tax?
This is a crucial point where many students lose marks! Capital Gains Tax is only paid if there is actually a gain.
If the purchase price (\(P\)) is higher than the redemption price (\(R\)), the investor has made a capital loss. In the CM1 syllabus, we generally assume that the investor cannot claim tax back on a loss in this specific context. Therefore:
If \(P \geq R\): The CGT term is simply zero.
If \(P < R\): The CGT term is included in the formula.
Did you know? This creates a "logic loop." To know if you need to include CGT, you need to know if \(P < R\). But you don't know \(P\) yet!
The Trick: Compare the net coupon rate with the net yield.
If \(g(1 - t_1) < i\), then the bond is being bought at a discount (\(P < R\)), and CGT will be payable.
If \(g(1 - t_1) > i\), then \(P > R\), and no CGT is payable.
4. Yields: Running Yield vs. Redemption Yield
While this chapter focuses on the equation of value, it's important to distinguish between these two terms:
- Net Running Yield: This only looks at the income. It is calculated as \(\frac{g(1 - t_1)R}{P}\).
- Net Redemption Yield: This is the overall interest rate \(i\) that satisfies the full equation of value (including the capital gain/loss and time value of money).
Note: For more on the basic definitions of these yields, see the chapter "Running yield, redemption yield and index-linked bonds".
5. Step-by-Step Example
Question: A bond pays coupons of 8% per annum annually in arrears and is redeemed at 100 in 10 years. An investor pays income tax at 25% and CGT at 20%. Calculate the price to provide a net yield of 6% per annum.
Step 1: The Capital Gains Test
Net coupon = \(8\% \times (1 - 0.25) = 6\%\).
Net yield required = \(6\%\).
Since net coupon = net yield, the price should be exactly par (\(P = 100\)). In this case, \(P = R\), so no CGT is payable.
Step 2: What if the yield was 7%?
Net coupon (6%) < Net yield (7%). Therefore, \(P < 100\) and CGT is payable.
Use the rearranged formula:
\(P = \frac{8(1 - 0.25) a_{\overline{10}|0.07} + 100 v^{10}(1 - 0.20)}{1 - 0.20 v^{10}}\)
Using \(i = 0.07\):
\(a_{\overline{10}|} = 7.0236\)
\(v^{10} = 0.5083\)
\(P = \frac{(6 \times 7.0236) + (100 \times 0.5083 \times 0.80)}{1 - (0.20 \times 0.5083)} = \frac{42.1416 + 40.664}{0.89834} = 92.17\)
6. Common Pitfalls to Avoid
- Confusing \(t_1\) and \(t_2\): Always double-check which rate applies to coupons and which to capital gains. They are often different!
- Ignoring the frequency of coupons: If coupons are paid semi-annually, remember to use \(a_{\overline{n}|}^{(2)}\) or adjust your interest rate \(i\) accordingly.
- Forgetting the CGT test: Always perform the quick check (\(g(1-t_1)\) vs \(i\)) before starting your calculation to see if CGT applies.
- Maths errors in rearranging: It is often safer to write out the full equation of value first before trying to move terms around.
Key Takeaways
1. Net of Tax: Always work with net cashflows. Net Coupon = \(Gross \times (1 - t_1)\).
2. CGT Logic: Capital Gains Tax only applies if \(P < R\). If you buy for more than you sell, ignore CGT.
3. The Formula: The price \(P\) is the present value of net coupons plus the present value of the net redemption amount.
4. Rearranging: Be comfortable solving for \(P\) when it appears on both sides of the equation.