###### Backtesting VaR

After completing this reading you should be able to: Describe backtesting and exceptions... **Read More**

**After completing this reading you should be able to: **

- Explain the role of interest rate expectations in determining the shape of the term structure.
- Apply a risk-neutral interest rate tree to assess the effect of volatility on the shape of the term structure.
- Estimate the convexity effect using Jensen’s inequality.
- Evaluate the impact of changes in maturity, yield, and volatility on the convexity of a security.
- Calculate the price and return of a zero-coupon bond incorporating a risk premium.

This chapter explains the role of interest rate expectation in determining the shape of the term structure. It shows how spot or forward rates are determined by expectations of future short-term rates, the volatility of short-term rates and an interest rate premium.

Expectations imply uncertainty. For example, an investor might expect the one-year rate next year to be, say, 10%, but they know very well that the actual rate might fall short or even be slightly higher.

Expectations among investors have a major bearing on the shape of the term structure of interest rates. The resulting yield curve can be flat, upward sloping, or downward sloping.

Suppose the 1-year interest rate is 6% and investors have estimated the future 1-year forward rates at 6% for the next two years. Given these interest rate expectations, the price (present values) of 1-, 2-, and 3-year zero-coupon bonds per $1 face value assuming annual compounding are calculated as follows:

$$ \begin{align*} \text{Price of 1-year zero} & =\cfrac {$1}{1.06}=$0.9434 \\ \text{Price of 2-year zero} & =\cfrac {$1}{(1.06)(1.06)}=$0.8900 \\ \text{Price of 3-year zero} & =\cfrac {$1}{(1.06)(1.06)(1.06)}=$0.8696 \\ \end{align*} $$

In summary, investors expect the 1-year spot rates for the next three years to be 6%. What’s the implication? The yield curve is **flat**. Investors would be willing to lock in interest rates for two or three years at 6% by purchasing, say, a three year bond with an annual coupon of 6%. The 1-year, 2-year, and 3-year spot rates are all **equal**.

Now suppose the 1-year spot rate remains at 6%, but investors expect the 1-year rate in a year’s time to be 8%, and the 1-year rate in two years to be 10%. In this case, the 2-year and 3-year spot rates will not be 10%. The 2-year spot rate, \(\hat r\)(2), will be 7%, calculated as follows:

$$ \text{Price of 2-year zero} =\cfrac {$1}{(1.06)(1.08)}=\cfrac {1}{(1+{\hat r}(2))^2 }; \hat r(2)= 0.07 $$

Similarly, The 3-year spot rate, \(\hat r\)(3), will be 7.99%, calculated as follows:

$$ \text{Price of 3-year zero} =\cfrac {$1}{(1.06)(1.08)(1.1)}=\cfrac {1}{(1+r ̂(3))^3 }; r ̂(3)= 0.0799 $$

In summary, investors expect the 1-year spot rate to be 6%, the 2-year spot rate to be 7%, and the 3-year spot rate to be 7.99%. What’s the implication? The yield curve is **upward sloping**. Would investors be willing to lock in an interest rate of 6% for two or three years? No

Now suppose the 1-year spot rate remains at 6%, but investors expect the 1-year rate in a year’s time to be 4%, and the 1-year rate in two years to be 2%. In this case, the 2-year and 3-year spot rates again will be different. The 2-year spot rate, \(\hat r\)(2), will be 5%, calculated as follows:

$$ \text{Price of 2-year zero} =\cfrac {$1}{(1.06)(1.04)}=\cfrac {1}{(1+{\hat r} (2))^2} ; {\hat r}(2)= 0.05 $$ Similarly, The 3-year spot rate, \({\hat r}\)(3), will be 3.99%, calculated as follows: $$ \text{Price of 3-year zero}=\cfrac {$1}{(1.06)(1.04)(1.2)}=\cfrac {1}{(1+r ̂(3))^3} ;{\hat r}(3)= 0.0399 $$

In summary, investors expect the 1-year spot rate to be 6%, the 2-year spot rate to be 5%, and the 3-year spot rate to be 3.99%. What’s the implication? The yield curve is **downward sloping**. And unlike in the upward sloping case, investors would have an incentive to lock in an interest rate of 6% for two or three years.

Interest rate expectations inform the shape and level of the term structure for short-term horizons. However, investors have **much less confidence** in their expectations about spot rates several periods into the future. Thus, expectations are unable to describe the shape of the term structure for long-term horizons. This is because future expectations must factor in **inflationary** tendencies.

Using a risk-neutral interest rate tree, it is possible to demonstrate that when there is uncertainty regarding expected rates, the volatility of expected rates causes the future spot rates to be **lower**.

Assume that the following tree gives the true process for the one-year rate:

$$ \begin{array} {} & {} & {} & {\scriptsize { 1 }/{ 2 } } & 10\% \\ {} & {} & 8\% & {\Huge \begin{matrix} \diagup \\ \diagdown \end{matrix} } & {} \\ 6\% & {\begin{matrix} \scriptsize { 1 }/{ 2 } \\ \begin{matrix} \begin{matrix} \quad \quad \quad \Huge \diagup \\ \end{matrix} \\ \quad \quad \quad \Huge \diagdown \end{matrix} \\ \scriptsize { 1 }/{ 2 } \end{matrix} } & {} & {\scriptsize \begin{matrix} \begin{matrix} { 1 }/{ 2 } \\\scriptsize \begin{matrix} \\ \end{matrix} \end{matrix} \\ \begin{matrix} \\ \end{matrix} \\ { 1 }/{ 2 } \end{matrix} }& 6\% \\ {} & {} & 4\% & {\Huge \begin{matrix} \diagup \\ \diagdown \end{matrix} } & {} \\ {} & {} & {} & {\scriptsize { 1 }/{ 2 }} & 2\% \\ \end{array} $$ $$ \begin{array} \\ \text{Year 0} & {} & {} & {} & \text{Year 1} & {} & { } & {} & \text{Year 2} \\ \end{array} $$

The expected interest rate on year 1 is .5 × 8% + .5 × 4% or 6% and that the expected rate on year 2 is .25 × 10% + .5 × 6% + .25 × 2% or 6%.

The price of a one-year zero is, by definition, 1/1.06 or .9434, implying a one-year spot rate of 6%.

Under the assumption of risk-neutrality, the price of a two-year zero may be calculated by discounting the terminal cash flow using the preceding interest rate tree:

$$ \begin{array} {} & {} & {} & {\scriptsize { 1 }/{ 2 } } & 1 \\ {} & {} & 0.9259 & {\Huge \begin{matrix} \diagup \\ \diagdown \end{matrix} } & {} \\ 0.8903 & {\begin{matrix} \scriptsize { 1 }/{ 2 } \\ \begin{matrix} \begin{matrix} \quad \quad \quad \Huge \diagup \\ \end{matrix} \\ \quad \quad \quad \Huge \diagdown \end{matrix} \\ \scriptsize { 1 }/{ 2 } \end{matrix} } & {} & {\scriptsize \begin{matrix} \begin{matrix} { 1 }/{ 2 } \\\scriptsize \begin{matrix} \\ \end{matrix} \end{matrix} \\ \begin{matrix} \\ \end{matrix} \\ { 1 }/{ 2 } \end{matrix} }& 1 \\ {} & {} & 0.9615 & {\Huge \begin{matrix} \diagup \\ \diagdown \end{matrix} } & {} \\ {} & {} & {} & {\scriptsize { 1 }/{ 2 }} & 1 \\ \end{array} $$ $$ \begin{array} \\ \text{Year 0} & {} & {} & {} & {} & \text{Year 1} & {} & { } & { } & {} & \text{Year 2} \\ \end{array} $$

$$ \left[0.5 x \left( \frac {$0.9259}{1.06} \right) \right] + \left[0.5 x \left( \frac {$0.9615}{1.06} \right) \right] = $0.8903 $$

Hence, the two-year spot rate, \(\hat r\)(2) is 5.9819%, calculated as follows:

$$ \begin{align*} 0.8903 & = \cfrac {1}{\left(1+\hat r(2) \right)^2} \\ 0.8903(1+\hat r(2))^2 & =1 \\ (1+\hat r(2))^2 & =1.1232 \\ \hat r (2) & =1.059819-1 =0.059819 \\ \end{align*} $$

We can see that even though the one-year rate is 6% and the expected one-year rate in one year is 6%, the two-year spot rate is 5.9819%, demonstrating the fact that when there is uncertainty regarding expected rates, the volatility of expected rates causes the future spot rates to be **lower**.

The 1.8-basis point difference between the spot rate that would obtain in the absence of uncertainty, 6%, and the spot rate in the presence of volatility, 5.9819%, is the **effect of convexity on that spot rate**.

The convexity effect arises from a special case of Jensen’s Inequality:

$$ E \left[ \cfrac {1}{[1+r]} \right]>\cfrac {1}{E[1+r]} $$

Assume that next year the 1-year spot rate will be 8% or 4% with equal (0.5) probability. Demonstrate Jensen’s inequality for a 2-year zero-coupon bond with a face value of $1 assuming the previous interest rate expectations apply (shown below for convenience)

$$ \begin{array} {} & {} & {} & {\scriptsize { 1 }/{ 2 } } & 10\% \\ {} & {} & 8\% & {\Huge \begin{matrix} \diagup \\ \diagdown \end{matrix} } & {} \\ 6\% & {\begin{matrix} \scriptsize { 1 }/{ 2 } \\ \begin{matrix} \begin{matrix} \quad \quad \quad \Huge \diagup \\ \end{matrix} \\ \quad \quad \quad \Huge \diagdown \end{matrix} \\ \scriptsize { 1 }/{ 2 } \end{matrix} } & {} & {\scriptsize \begin{matrix} \begin{matrix} { 1 }/{ 2 } \\\scriptsize \begin{matrix} \\ \end{matrix} \end{matrix} \\ \begin{matrix} \\ \end{matrix} \\ { 1 }/{ 2 } \end{matrix} }& 6\% \\ {} & {} & 4\% & {\Huge \begin{matrix} \diagup \\ \diagdown \end{matrix} } & {} \\ {} & {} & {} & {\scriptsize { 1 }/{ 2 }} & 2\% \\ \end{array} $$ $$ \begin{array} \\ \text{Year 0} & {} & {} & {} & \text{Year 1} & {} & { } & {} & \text{Year 2} \\ \end{array} $$

The left-hand side of Jensen’s inequality is the expected price in one year using the 1-year spot rates of 8% and 4%.

$$ E \left[\cfrac {$1}{(1+r)}\right]=0.5×\cfrac {$1}{1.08}+0.5×\cfrac {$1}{1.04}=$0.9437 $$ The expected price in one year using an expected rate of 6 % computes the right-hand side of the inequality as: $$ \cfrac {1}{E[1+r]} =\cfrac {$1}{0.5×1.08+0.5×1.04}=\cfrac {$1}{1.06}=0.9434 $$

Thus, the left-hand side is greater than the right-hand side, $0.9437 > $0.9434.

If the current 1-year rate is 6%, the price of a 2-year zero-coupon bond is found by dividing each side of the equation by 1.06. Thus, the price of the 2-year zero-coupon bond on the left-hand side of Jensen’s inequality equals $0.8903 (calculated as $0.9437/1.06). The right-hand side is calculated as the price of a 2-year zero-coupon bond discounted for two years at the expected rate of 6%, which yields $0.8900 (calculated as $1/1.06^{2}).

This effectively demonstrates that the price of the 2-year zero-coupon bond is greater than the price obtained by discounting the $1 face amount by 6% over the first period and by 6% over the second period. As a result, we know that since the 2-year zero-coupon price is higher than the price achieved through discounting, its implied rate must be lower than 6%.

The value of convexity is measured by the distance between the rates assuming no volatility and the rates assuming volatility.

It can be shown that all else held equal, the value of convexity **increases as maturity** increases. Thus, the value of convexity for a 1-year bond would be less than the value of convexity for a 30-year bond. The implication is that as the maturity of a bond increases, the price-yield relationship becomes more convex.

In addition, the value of convexity **increases with volatility**. This means that as the difference between interest rates in the upper and lower nodes increases, convexity increases.

Risk-averse investors (who are not risk-neutral) will require compensation for the volatility.

Suppose the following interest rate tree exists:

$$ \begin{array} {} & {} & {} & {\scriptsize { 1 }/{ 2 } } & 10\% \\ {} & {} & 8\% & {\Huge \begin{matrix} \diagup \\ \diagdown \end{matrix} } & {} \\ 6\% & {\begin{matrix} \scriptsize { 1 }/{ 2 } \\ \begin{matrix} \begin{matrix} \quad \quad \quad \Huge \diagup \\ \end{matrix} \\ \quad \quad \quad \Huge \diagdown \end{matrix} \\ \scriptsize { 1 }/{ 2 } \end{matrix} } & {} & {\scriptsize \begin{matrix} \begin{matrix} { 1 }/{ 2 } \\\scriptsize \begin{matrix} \\ \end{matrix} \end{matrix} \\ \begin{matrix} \\ \end{matrix} \\ { 1 }/{ 2 } \end{matrix} }& 6\% \\ {} & {} & 4\% & {\Huge \begin{matrix} \diagup \\ \diagdown \end{matrix} } & {} \\ {} & {} & {} & {\scriptsize { 1 }/{ 2 }} & 2\% \\ \end{array} $$

A risk-neutral investor would value a 2-year zero coupon bond at $0.99437 per $1 face value, calculated as follows:

$$ \cfrac { \left( \frac {$1}{1.08} + \frac {$1}{1.04} \right)0.5 }{1.06}=$0.9437 $$

The price of $0.9437 implies a 1-year expected return of 6%. However, this is only the average return. The actual return will be either 4% or 8%. As a result, a risk-averse investor will demand a risk premium for bearing this interest rate risk, and demand a return greater than 6% for buying a 2-year zero-coupon bond and holding it for the next year.

Suppose the risk-averse investor demands a 30-basis point premium. To come up with the price of the 2-year zero-coupon bond would decrease to $0.8878, calculated as follows:

$$ \cfrac { \left( \frac {$1}{1.083} + \frac {$1}{1.043} \right)0.5 }{1.06}=$0.8878 $$

It follows that a risk-averse investor would be willing to pay much less for every $1 face value of the bond.

## Question 1

Investors value the current one-year interest rate at 8.30%. If they also forecast that for the following year, the one-year interest rate will be 9.43%, then the two-year spot rate, \(\rho \left( 2 \right) \), is closest to:

- 8.86%
- 9.43%
- 18.51%
- 9.26%
The correct answer is

A.The two- year spot rate \(\rho \left( 2 \right) \) is such that:

$$ \begin{align*}{ P }^{ 2 }&=\frac { 1 }{ \left( 1.0830 \right) \left( 1.0943 \right) }=\frac { 1 }{ { \left( 1+\rho \left( 2 \right) \right) }^{ 2 } } \\ &\Rightarrow { \left( 1+\rho \left( 2 \right) \right) }^{ 2 }=\left( 1.0830 \right) \left( 1.0943 \right)\\ \rho \left( 2 \right)& =0.886=8.86\% \end{align*}$$

## Question 2

Assume that the following tree gives the true process for the one-year rate.

$$ \begin{array} \hline {} & {} & {} & {\scriptsize 0.5 } & 22\% \\ {} & {} & 18.5\% & {\Huge \begin{matrix} \diagup \\ \diagdown \end{matrix} } & {} \\ 15\% & {\begin{matrix} \scriptsize 0.5 \\ \begin{matrix} \begin{matrix} \quad \quad \quad \Huge \diagup \\ \end{matrix} \\ \quad \quad \quad \Huge \diagdown \end{matrix} \\ \scriptsize 0.5 \end{matrix} } & {} & \scriptsize 0.5 & 15\% \\ {} & {} & 11.5\% & {\Huge \begin{matrix} \diagup \\ \diagdown \end{matrix} } & {} \\ {} & {} & {} & {\scriptsize 0.5} & 8\% \\ \end{array} $$

Compute the expected interest rate for dates 1 and 2, respectively.

- Both are 15%
- 12% and 19%
- 18.5% and 11.5%
- 8% and 22%
The correct answer is

A.The expected interest rate on date 1 is:

$$ 0.5\times 11.5+0.5\times 18.5=15\% $$

The expected interest rate for date 2 is:

$$ 0.25\times 22+0.5\times 15+0.25\times 8=15\% $$

## Question 3

Using the data provided in the following tree, apply the Jensen’s inequality for estimation of connectivity to show why the 2-year spot rate is less than 15%.

$$ \begin{array} \hline {} & {} & {} & {\scriptsize 0.5 } & 22\% \\ {} & {} & 18.5\% & {\Huge \begin{matrix} \diagup \\ \diagdown \end{matrix} } & {} \\ 15\% & {\begin{matrix} \scriptsize 0.5 \\ \begin{matrix} \begin{matrix} \quad \quad \quad \Huge \diagup \\ \end{matrix} \\ \quad \quad \quad \Huge \diagdown \end{matrix} \\ \scriptsize 0.5 \end{matrix} } & {} & \scriptsize 0.5 & 15\% \\ {} & {} & 11.5\% & {\Huge \begin{matrix} \diagup \\ \diagdown \end{matrix} } & {} \\ {} & {} & {} & {\scriptsize 0.5} & 8\% \\ \end{array} $$

- 0.853 > 0.657
- 0.798 > 0.713
- 0.758 > 0.756
- 0.658 > 0.456
The correct answer is

C.According to the Jensen’s inequality:

$$\begin{align*} E\left[ \frac { 1 }{ 1+r } \right] &>\frac { 1 }{ E\left[ 1+r \right] }=\frac { 1 }{ 1+E\left[ r \right] }\\ \Rightarrow 0.5\times \frac { 1 }{ 1.185 } +0.5\times \frac { 1 }{ 1.115 } &>\frac { 1 }{ 0.5\times 1.185+0.5\times 1.115 } =\frac { 1 }{ 1.15 } \end{align*}$$

Dividing both sides by 1.15:

$$ \begin {align*}\Rightarrow \frac { 1 }{ 1.15 } \left[ 0.5\times \frac { 1 }{ 1.185 } +0.5\times \frac { 1 }{ 1.115 } \right]& >\frac { 1 }{ { 1.15 }^{ 2 } }\\ \Rightarrow 0.758 &> 0.756\end{align*} $$