Suppose you are observing the bond market and notice two seemingly identical 5-year corporate bonds issued by the same company. Both bonds offer an annual coupon rate of 6% but are currently trading at different prices:
Both bonds have a face value of $1,000 and will mature in 5 years.
Question: Analyzing Arbitrage Opportunity and Price–Return Relationship
(a) Calculate the exact yield to maturity (YTM) for Bond A and Bond B. The YTM for each bond is the rate \(r\) that satisfies the equation:
\[P = \sum_{t=1}^{n} \frac{C}{(1 + r)^t} + \frac{F}{(1 + r)^n}\]
where:
(b) Based on the YTM calculations, identify the bond that should be purchased and the bond that should be sold to take advantage of an arbitrage opportunity.
(c) Explain the relationship between the price and yield (return) of these bonds, illustrating the concept of inverse relationship.
(a) To find the YTM for each bond, we solve for \(r\) in the standard bond pricing equation. Expanding for \(n = 5\) years:
Bond A (priced at $980):
\[980 = \frac{60}{(1 + r)} + \frac{60}{(1 + r)^2} + \frac{60}{(1 + r)^3} + \frac{60}{(1 + r)^4} + \frac{60}{(1 + r)^5} + \frac{1000}{(1 + r)^5}\]
Bond B (priced at $1,050):
\[1050 = \frac{60}{(1 + r)} + \frac{60}{(1 + r)^2} + \frac{60}{(1 + r)^3} + \frac{60}{(1 + r)^4} + \frac{60}{(1 + r)^5} + \frac{1000}{(1 + r)^5}\]
Solving numerically for \(r\) in each case:
The exact YTMs demonstrate that Bond A offers a higher return than Bond B for the same risk, indicating it is priced more attractively.
(b) To exploit an arbitrage opportunity, an investor should buy Bond A (undervalued, yielding a higher return) and sell Bond B (overvalued, yielding a lower return). The assumption is that the market will eventually correct these discrepancies, and the prices of Bond A and Bond B will adjust such that their YTMs align, given their identical risk profiles and cash flows.
(c) The relationship between bond price and yield illustrates the fundamental inverse relationship: as the price of a bond decreases, its yield increases, and vice versa. This is because the fixed coupon payments and maturity value represent a larger return percentage on a lower purchase price, and a smaller return percentage on a higher purchase price.
In this scenario, Bond A's lower price relative to Bond B results in a higher yield, demonstrating that as bond prices fall, yields rise. This principle is a fundamental concept in bond valuation, illustrating how returns adjust to reflect changes in bond pricing.
Consider an extension of the Solow model where the economy experiences exogenous technological progress at a constant rate \(g\), with the production function given by:
\[Y_t = K_t^{0.5}(A_t L_t)^{0.5}\]
where \(A_t\) grows at rate \(g = 0.02\) per period.
1. Effective Capital per Effective Worker Evolution.
Express the equation for the evolution of effective capital per effective worker in the presence of technological growth, where effective capital per effective worker is defined as \(k_t = K_t/(A_t L)\) and effective output per effective worker is defined as \(y_t = Y_t/(A_t L)\).
2. Steady-State Analysis.
Discuss how the steady-state level of effective capital per effective worker,
\[k^* = \frac{K}{A_t L},\]
is affected by technological progress. Derive the steady-state value of \(k^*\).
3. Long-Run Growth Implications.
Explain the implications of continuous technological progress for long-term economic growth and per capita income growth.
1. Effective Capital per Effective Worker Evolution
Begin with the standard capital accumulation equation:
\[K_{t+1} = (1 - \delta) K_t + s Y_t\]
Define effective capital per effective worker as \(k_t = K_t/(A_t L)\), so \(K_t = A_t L k_t\). With labor-augmenting technology, rewrite the production function as:
\[Y_t = K_t^{0.5} (A_t L)^{0.5}\]
so that effective output per effective worker is:
\[y_t = \frac{Y_t}{A_t L} = \left(\frac{K_t}{A_t L}\right)^{0.5} = k_t^{0.5}\]
Dividing both sides of the capital accumulation equation by \(A_{t+1}L = (1 + g)A_t L\):
\[\frac{K_{t+1}}{A_{t+1}L} = \frac{(1 - \delta)K_t}{(1 + g)A_t L} + \frac{s Y_t}{(1 + g)A_t L}\]
The left-hand side is \(k_{t+1}\). Substituting \(k_t\) and \(y_t = k_t^{0.5}\) on the right-hand side yields the law of motion:
\[k_{t+1} = \frac{(1 - \delta) k_t + s k_t^{0.5}}{1 + g}\]
This equation shows that effective capital per effective worker next period depends on what survives depreciation plus new investment, both deflated by the growth factor \((1 + g)\) that continuously expands the effective labor denominator.
2. Steady-State Analysis
At the steady state, \(k_{t+1} = k_t = k^*\) by definition. Substituting into the law of motion:
\[k^* = \frac{(1 - \delta) k^* + s k^{*0.5}}{1 + g}\]
Multiply both sides by \((1 + g)\):
\[(1 + g) k^* = (1 - \delta) k^* + s k^{*0.5}\]
Collect the \(k^*\) terms on the left:
\[(1 + g) k^* - (1 - \delta) k^* = s k^{*0.5}\]
\[(g + \delta) k^* = s k^{*0.5}\]
Divide both sides by \(k^{*0.5}\) (valid since \(k^* \neq 0\)):
\[(g + \delta) k^{*0.5} = s\]
\[k^{*0.5} = \frac{s}{g + \delta}\]
Squaring both sides:
\[k^* = \left(\frac{s}{g + \delta}\right)^2\]
The steady-state level of effective capital per effective worker depends:
3. Long-Run Growth Implications
With continuous technological progress, the economy does not reach a steady state in terms of actual capital per worker \(K/L\) or output per worker \(Y/L\). Instead, it reaches a steady state only in effective units. The long-run implications are:
(a) Sustained per capita growth. Even in the steady state (where \(k^*\) is constant), output per capita \(Y/L\) grows continuously at the rate of technological progress \(g\), because \(A_t\) keeps rising.
(b) Total output growth. If the labor force grows at rate \(n\), total output grows at:
\[\text{Growth of } Y = g + n\]
reflecting both technological progress and labor force expansion.
(c) Role of technology. Unlike changes in \(s\) or \(\delta\), which only affect the level of the balanced growth path, technological progress \(g\) drives the rate of long-run per capita growth. This is the central message of the Solow model: capital accumulation alone cannot sustain growth; only technological progress can.