RS Capital LLC

The downfall of Silicon Valley Bank (SVB) was a stark reminder of the destructive power of rising interest rates. It demonstrated how even seemingly robust financial institutions can unravel rapidly when rates climb unexpectedly. In this new environment, investors find themselves grappling with uncertainty, as changes in interest rates can significantly impact the valuation of assets in their portfolios. Understanding these potential impacts is crucial for effective risk management and portfolio strategy.

This research delves into a quantitative approach to addressing these challenges, focusing primarily on bonds and equity investments. We will explore how to measure bond price sensitivity through duration and convexity, examine the impact of interest rates on equity valuations via the Dividend Discount Model (DDM), and outline portfolio optimization techniques to manage interest rate risk.

Bonds: Duration and Convexity

Bonds are particularly sensitive to changes in interest rates due to their fixed-income nature. When interest rates rise, the present value of a bond's future cash flows declines, leading to a drop in its price. This inverse relationship means that bond investors need to be vigilant about interest rate movements. Two critical measures for understanding a bond's sensitivity to interest rate changes are duration and convexity.

Duration provides an estimate of how much a bond's price will change in response to a 1% change in interest rates. It represents the weighted average time it takes for an investor to receive the bond's cash flows, factoring in both the coupons and the principal repayment at maturity. Mathematically, Macaulay duration is defined as:

D = [∑(Ct/(1+r)^t)*t + M/(1+r)^N*N] / P,

where:

- Ct represents the cash flow (the coupon) in period t,

- r is the yield per period,

- M is the maturity value,

- N is the number of periods,

- P is the bond price.

Duration can also be thought of as the bond’s interest rate risk: the higher the duration, the more sensitive the bond is to interest rate changes. For example, a bond with a duration of 5 years will approximately decrease in price by 5% if interest rates rise by 1%. Conversely, it will increase in price by 5% if rates fall by 1%.

While duration provides a linear estimate of price sensitivity, it becomes less accurate for larger changes in interest rates because bond price movements are convex, not linear. This is where convexity comes into play. Convexity measures how the duration of a bond changes as interest rates change, providing a more accurate assessment of price sensitivity, especially for significant rate movements. The formula for convexity is given by:

C = [∑(Ct*(t*(t+1))/(1+r)^(t+2)) + M*(N*(N+1))/(1+r)^(N+2)] / P.

Bonds with higher convexity experience less price decline when interest rates rise and a larger price increase when rates fall. Therefore, in a rising interest rate environment, investors may prefer bonds with shorter durations and higher convexity, as these characteristics help cushion the impact of rate hikes on bond prices.

Equity Valuation: Dividend Discount Model

Interest rate changes also affect the valuation of equities, particularly through their impact on future cash flows. As interest rates rise, the required rate of return for equities increases, leading to a lower present value of expected future dividends. One of the most commonly used models to capture this effect is the Dividend Discount Model (DDM).

The Gordon Growth Model

A simplified version of the DDM is the Gordon Growth Model, which values a stock based on the present value of its future dividend stream:

P = D1 / (k - g),

where:

- D1 is the expected dividend in the next period,

- k is the required rate of return (a factor of the risk-free rate, which moves with interest rates),

- g is the dividend growth rate.

Under higher interest rates, k increases, hence lowering the present value P.

Coping with Rising Interest Rates: Portfolio Optimization

Given the sensitivity of both bonds and equities to interest rate changes, investors need to adopt strategies to manage risk in a rising rate environment. One approach is to use portfolio optimization techniques, such as the Markowitz Mean-Variance Optimization Model.

Markowitz Mean-Variance Optimization

This model aims to construct an optimal portfolio by maximizing expected return for a given level of risk or minimizing risk for a given level of expected return. The optimization problem can be formulated as:

min w'Σw - q*(w'R),

subject to w'1 = 1,

where:

- w is the vector of portfolio weights,

- R is the vector of asset expected returns,

- Σ is the covariance matrix of asset returns,

- q is a parameter that adjusts the risk-return tradeoff,

- 1 is a vector of ones.

By adjusting q, investors can model various interest rate scenarios and assess their portfolio’s sensitivity to rate changes. The resulting portfolio weights help identify which assets to overweight or underweight to navigate the changing interest rate environment effectively.

Conclusion

Rising interest rates pose a multifaceted challenge for investors, impacting both bond and equity markets. However, understanding and applying quantitative tools like bond duration and convexity, the Dividend Discount Model, and portfolio optimization techniques can provide valuable insights into managing these risks.

• Duration and Convexity: Enable investors to assess bond price sensitivity and adjust exposure to limit losses in a rising rate environment.

• Dividend Discount Model: Helps understand the impact of interest rate changes on equity valuations, guiding investors on which stocks may become more or less attractive.

• Portfolio Optimization: Provides a framework to construct resilient portfolios by modeling different interest rate scenarios and optimizing asset allocation accordingly.

  
  

It’s crucial to remember that these models depend heavily on the accuracy of their input variables, such as interest rate projections, future cash flows, and market volatility. While quantitative approaches offer a structured way to address rising rates, incorporating a diversified investment strategy can further mitigate risks in an uncertain market environment.