5  Varying Rates

5.1 Varying Compound Interest Rates

Definition: Let’s use the following notation for an interest rate that is constant from time \(s\) to time \(t\):

\[ i_{s,t}= \text{Effective interest rate }\textbf{per unit of time} \text{that applies from time $s$ to time $t$} \]

Proposition: The accumulated value is the present value times a product of accumulation factors:

\[ AV_{t_n} = PV_0 (1 + i_{0, t_1})^{t_1} (1 + i_{t_1, t_2})^{t_2-t_1} ... (1 + i_{t_{n-1}, t_n})^{t_n-t_{n-1}} \]

5.2 Varying Compound Discount Rates

Definition: Let’s use the following notation for a discount rate that is constant from time \(s\) to time \(t\):

\[ d_{s,t}= \text{Effective discount rate \textbf{per unit of time} that applies from time $s$ to time $t$} \]

Proposition: The present value is the accumulated value times a product of discount factors: \[ PV_0 = AV_{t_n} (1 - d_{t_{n-1}, t_n})^{t_n-t_{n-1}} ... (1 - d_{t_1, t_2})^{t_2-t_{1}} (1 - d_{0, t_{1}})^{t_1} \]

Example: In year 1, the discount rate convertible semiannually is \(6\%\). In year 2, the discount rate convertible quarterly is \(8\%\). For the first 6 months of year 3, the discount rate convertible monthly is \(12\%\). For the last 6 months of year 3, the annual effective discount rate is \(7\%\). Calculate the accumulated value at the end of 3 years of \(\$350\) deposited at the beginning of the first year.

Solution: We have: \((1 - d_{0,1}) = (1 - \frac{6\%}{2})^2\), \((1 - d_{1,2}) = (1 - \frac{8\%}{4})^4\), \((1 - d_{2,2.5}) = (1 - \frac{12\%}{12})^{12}\), \((1 - d_{2.5,3}) = (1 - 7\%)\)

By the proposition, we can formulate the equation: \[\begin{align} AV_3 &= PV_0 (1-d_{0,1})^{-1} (1-d_{1,2})^{-(2-1)}(1-d_{2,2.5})^{-(2.5-2)}(1-d_{2.5,3})^{-(3-2.5)} \\ &= 350 (1-0.03)^{2\times-1} (1-0.02)^{4\times-1} (1-0.01)^{12\times-0.5} (1-0.07)^{0.5} \\ &= 444.19\end{align}\] ## Varying Force of Interest Definition: For discretely varying force of interest, we use the following notation for a continuously compounded interest rate that is constant from time \(s\) to time \(t\): \[ \delta_{s,t} = \text{Force of interest }\textbf{per unit of time} \text{ that applies from time $s$ to time $t$} \]

For continuously varying force of interest, we denote the force of interest at time \(t\): \[ \delta_t = \text{Force of interest at time $t$} = \frac{\frac{d}{dt}(CV_t)}{CV_t} = \frac{d(\ln CV_t)}{dt} \]

Proposition: The accumulated value is the present value times a product of accumulation factors:

\[ AV_{t_n} = PV_0 e^{\delta_{0,t_1} \times t_1} e^{\delta_{t_1,t_2} \times (t_2 - t_1)} ... e^{\delta_{t_{n-1},t_n} \times (t_n-t_{n-1})} \]

For general force of interest, we can integrate \(t=0\) to \(n\) to obtain the relationship between force of interest and current value:

\[ \int_{t_1}^{t_2} \delta_s ds = \ln(CV_{t_2}) - \ln(CV_{t_1}) = \ln\bigg(\frac{CV_{t_2}}{CV_{t_1}}\bigg) \iff \begin{cases} AV_{t_2} &= PV_{t_1} \times e^{\int_{t_1}^{t_2} \delta_s ds} \\ PV_{t_1} &= AV_{t_2} \times e^{-\int_{t_1}^{t_2} \delta_s ds} \end{cases} \]

Example: Given \(\delta_t = .08+.005t\), calculate the accummulated value over five years of an investment of 1000 made at each of the following times:

• Time 0
• Time 2.

Solution:

• If the investment is made at time 0, then \(AV_5 = PV_0 \times \exp\bigg[\int_0^5 (.08+.005t)dt\bigg] = 1000 \times e^{.4645} \approx1588.04\)
• If the investment is made at time 2, then \(AV_7 = PV_2 \times \exp\bigg[\int_2^7 (.08+.005t)dt\bigg] = 1000 \times e^{.5125} \approx 1669.46\)

5.2.1 Force of Interest for Simple Interest

Theorem:For simple interest \(i\) per annum, the force of interest is \[\delta_t = \frac{i}{1+it}\]

We see that as \(t\) increases, the force of interest has on accumulated value decreases. One should be aware that simple interest usually means that the linear function starts all over again from the date of each deposit or withdrawal.

Example: Suppose we want to determine the accummulated value at time 7 of $1.00 invested at time 2, with annual simple interest rate of 5%. If we follow the methodology above, the answer would be: \[AV_7 = AV_2 \times \exp \bigg[\int_2^7 \delta_t dt\bigg] = 1\times \exp \bigg[\int_2^7 \frac{.05}{1+.05t} dt\bigg] = 1.22727\] Where if we follow the notion that the amount of interest earned is proportional to the amount of time of accumulation in simple interest, we will obtain the accummulated value of \(1\times(0.05)(7-2) = 1.25\).

Thus we should be very careful when handling simple interest.

5.2.2 Mix of Varying Rates

Not only can the rates vary over time, but the type of the rate provided can vary over time as well.

Example: In the first year, the annual effective interest rate is \(4\%\). In the second year, the discount rate convertible semiannually is \(5\%\). In the third year, the force of interest is \(6\%\). In the fourth year, the force of interest is \(\delta_t = 0.02t\), for \(3<t<4\). \(\$700\) is lent today. What is the accumulated value of the loan at the end of the fourth year?

Solution: To calculate the accumulated value, we use a mixture of rates:

\[\begin{align}AV_4 &= 700 (1.04)\bigg(1-\frac{0.05}{2}\bigg)^{-2} e^{0.06} e^{\int_3^4 (0.02t) dt}\\ &= 872.1284\end{align} \]

5.3 Average Annual Rate of Return

The Average Annual Rate of Return is the interest rate is the average annual return (profit) from an investment. It can be determined from:

\[(1 + i_{avg})^{t_n} = (1+i_{0, t_1})^{t_1} (1+i_{t_1, t_2})^{t_2-t_1} ... (1+i_{t_{n-1}, t_n})^{t_n-t_{n-1}}\] Example: The excerpts below are taken from the 2016 year-end report of National Bank Global Equity Fund, a fund managed by a Canadian mutual fund company. The excerpts below focus on the performance of the fund and the Dow Jones Industrial Average during the five year period ending December 31, 2016. The Dow Jones Industrial Average is a price-weighted average of stocks traded on major American stock exchanges.

Annual Rate of Return	2016	2015	2014	2013	2012
NB Global Equity	0.00%	18.08%	13.38%	33.57%	14.14%
Dow Jones Ind. Avg.	13.42%	-2.23%	7.52%	26.50%	7.26%

For the five year period ending December 31, 2016, the total compound growth in the Global Equity Fund can be found by compounding the annual rates of return for the 5 years.

\[(1+.00)(1+.1808)(1+.1338)(1+.3357)(1+.1414)=2.0411\]

This would be the value on December 31, 2016 of an investment of 1 made into the fund on January 1, 2012. This five year growth can be described by means of an average annual return per year for the five-year period. In practice the phrase “average annual return” refers to an annual compound rate of interest for the period of years being considered. This would be \(i\) that satisfies \((1+i)^5 = 2.0411 \implies i = .1534\).

Similarly, for the Dow Jones average, an investment of 1 made in January 1, 2012 would have a value on December 31, 2016 of

\[(1+.1342)(1-.0223)(1+.0752)(1+.2650)(1+.0726) = 1.6178\] resulting in an average rate of return of \(i=.1010\)