Interest, discount and the force of interest: questions

Discount and interest on a promissory note

Paper A style · Interest rates · Straightforward · 6 marks

A finance company buys a promissory note that will pay £25,000 in exactly one year. It pays £23,850 for the note today.

(i)
Calculate the effective annual rate of discount earned by the company, as a percentage to 4 decimal places.
1 mark
Your answer
(ii)
Calculate the effective annual rate of interest earned, as a percentage to 4 decimal places.
1 mark
Your answer
(iii)
Explain by general reasoning why $d$ $= iv$ , where $v$ $= 1/(1+i)$ , and verify that your answers to (i) and (ii) satisfy this relationship.
2 marks
(iv)
The company's alternative is to lend at an effective rate of interest of 4.9% a year. Calculate the maximum price it should pay for the note, to the nearest penny, and comment on whether the purchase at £23,850 is attractive.
2 marks
Your answer

Equivalent rates from a monthly nominal rate

Paper A style · Interest rates · Straightforward · 7 marks

A building society quotes a nominal rate of interest of 5.4% a year convertible monthly on its savings account.

(i)
Calculate the equivalent effective annual rate of interest, as a percentage to 4 decimal places.
2 marks
Your answer
(ii)
Calculate the equivalent effective annual rate of discount, as a percentage to 4 decimal places.
1 mark
Your answer
(iii)
Calculate the equivalent nominal rate of discount convertible quarterly, $d^{(4)}$ , as a percentage to 4 decimal places.
2 marks
Your answer
(iv)
Calculate the equivalent force of interest, as a percentage to 4 decimal places.
2 marks
Your answer

Simple and compound interest compared

Paper A style · Interest rates · Straightforward · 8 marks

(i)
£3,000 is invested for 5 years and 6 months at simple interest of 4.5% a year. Calculate the accumulated amount, to the nearest penny.
1 mark
Your answer
(ii)
Calculate the accumulated amount if instead compound interest is earned at an effective rate of 4.5% a year, to the nearest penny.
2 marks
Your answer
(iii)
A 91-day government bill with nominal value £100 is priced using a simple rate of discount of 4.2% a year, with a 365-day year. Calculate the price, to 4 decimal places.
2 marks
Your answer
(iv)
Calculate the effective annual rate of interest earned by an investor who buys the bill at this price and holds it to maturity, as a percentage to 4 decimal places.
2 marks
Your answer
(v)
Explain why simple interest is rarely used for transactions lasting more than a year.
1 mark

Constant force of interest on a deposit

Paper A style · Interest rates · Standard · 8 marks

An investor deposits £5,000 in an account that credits interest at a constant force of interest of 4.2% a year. No further deposits or withdrawals are made.

(i)
Calculate the equivalent nominal rate of interest convertible half-yearly, as a percentage to 4 decimal places.
2 marks
Your answer
(ii)
Calculate the amount in the account after 7 years and 3 months, to the nearest penny.
2 marks
Your answer
(iii)
Calculate how long, in years to 2 decimal places, it takes for the deposit to grow to £8,000.
2 marks
Your answer
(iv)
Explain why the force of interest can be described as a nominal rate of interest convertible continuously.
2 marks

Accumulation under a time-varying force

Paper A style · Interest rates · Standard · 9 marks

The force of interest at time $t$ (in years) is $\delta(t)$ $= 0.02 + 0.004t$ for $0$ $\le t$ $< 5$ and $\delta(t)$ $= 0.04$ for $t$ $\ge 5$ .

(i)
Calculate the accumulated value at time 8 of £1,000 invested at time 0, to the nearest penny.
3 marks
Your answer
(ii)
Calculate the constant effective annual rate of interest that would give the same accumulation over the 8 years, as a percentage to 4 decimal places.
2 marks
Your answer
(iii)
Calculate the present value at time 2 of £2,000 due at time 10, to the nearest penny.
3 marks
Your answer
(iv)
Comment on why your answer to (ii) is below the long-term level of $e^{0.04} - 1$ $= 4.0811\%$ .
1 mark

Real returns on a savings account

Paper A style · Interest rates · Standard · 8 marks

An investor deposits £10,000 in a fixed-rate account paying an effective rate of interest of 5.8% a year. Inflation is expected to be 2.6% a year throughout.

(i)
Calculate the real effective annual rate of interest earned, as a percentage to 4 decimal places.
2 marks
Your answer
(ii)
Calculate the value of the account after 6 years, expressed in today's money, to the nearest penny.
2 marks
Your answer
(iii)
The investor wants a real return of 2.5% a year. Calculate the effective annual money rate of interest that would provide this, as a percentage to 4 decimal places.
2 marks
Your answer
(iv)
The investor estimates the real rate as $5.8\% - 2.6\%$ $= 3.2\%$ . Comment on this approximation.
2 marks

Choosing between three quoted rates

Paper A style · Interest rates · Standard · 8 marks

A saver is comparing three accounts. Account A pays a nominal rate of interest of 4.8% a year convertible quarterly. Account B quotes a nominal rate of discount of 4.7% a year convertible monthly. Account C pays a constant force of interest of 4.75% a year. All three are otherwise identical.

(i)
Calculate the effective annual rate of interest for Account A, as a percentage to 4 decimal places.
2 marks
Your answer
(ii)
Calculate the effective annual rate of interest for Account B, as a percentage to 4 decimal places.
2 marks
Your answer
(iii)
Calculate the effective annual rate of interest for Account C, as a percentage to 4 decimal places.
2 marks
Your answer
(iv)
State which account gives the best return and explain why comparing the quoted figures directly is misleading.
2 marks

Accumulation under changing interest regimes

Paper A style · Interest rates · Standard · 10 marks

A fund earns interest as follows: an effective rate of 3.2% a year for the first 2 years; a nominal rate of 4% a year convertible half-yearly for the next 3 years; and a constant force of interest of 4.5% a year thereafter. Time is measured in years from the start.

(i)
Calculate the accumulated value at time 8 of £12,000 invested at time 0, to the nearest penny.
3 marks
Your answer
(ii)
Calculate the present value at time 1 of £20,000 payable at time 7, to the nearest penny.
3 marks
Your answer
(iii)
Calculate the constant effective annual rate of interest that gives the same accumulation over the 8 years, as a percentage to 4 decimal places.
2 marks
Your answer
(iv)
Explain why the answer to (iii) is not the simple average of the three effective annual rates weighted by the length of each period.
2 marks

Money and real returns from an index

Paper A style · Interest rates · Standard · 9 marks

An investor bought a holding of shares for £8,400 and sold it exactly 4 years later for £10,950. No dividends were received and there were no expenses or taxes. A consumer price index stood at 108.6 when the shares were bought and at 127.3 when they were sold.

(i)
Calculate the effective annual money rate of return, as a percentage to 4 decimal places.
2 marks
Your answer
(ii)
Calculate the average annual rate of inflation over the 4 years, as a percentage to 4 decimal places.
2 marks
Your answer
(iii)
Calculate the effective annual real rate of return, as a percentage to 4 decimal places.
3 marks
Your answer
(iv)
Comment on your answer to (iii) as a measure of how much better off the investor is.
2 marks

A force of interest that declines

Paper A style · Interest rates · Hard · 10 marks

The force of interest at time $t$ (in years) is $\delta(t)$ $= \dfrac{0.1}{1 + 0.04t}$ for $t$ $\ge 0$ .

(i)
Show that the accumulation factor from time 0 to time $t$ is $A(0,t)$ $= (1 + 0.04t)^{2.5}$ .
2 marks
(ii)
Calculate the accumulated value at time 10 of £2,000 invested at time 3, to the nearest penny.
2 marks
Your answer
(iii)
Calculate the constant effective annual rate of interest equivalent to this force over the period from time 0 to time 10, as a percentage to 4 decimal places.
2 marks
Your answer
(iv)
Calculate the time, in years to 2 decimal places, at which an investment made at time 0 has doubled.
2 marks
Your answer
(v)
Comment on how the effective rate of interest earned in a single year changes over time under this force, and what this means for an investor choosing when to invest for a fixed 5-year term.
2 marks

Finding when an average rate is reached

Paper A style · Interest rates · Hard · 10 marks

The force of interest at time $t$ (in years) is $\delta(t)$ $= 0.03 + 0.002t$ for $0$ $\le t$ $\le 15$ .

(i)
Show that the accumulation factor from time 0 to time $t$ is $\exp(0.03t + 0.001t^2)$ for $0$ $\le t$ $\le 15$ .
2 marks
(ii)
Find the term $T$ (in years, to 2 decimal places) such that the constant effective annual rate of interest equivalent to this force over $[0, T]$ is exactly 4%.
3 marks
Your answer
(iii)
Calculate the nominal rate of interest convertible half-yearly that is equivalent to the accumulation from time 5 to time 6, as a percentage to 4 decimal places.
3 marks
Your answer
(iv)
Calculate the present value at time 0 of £5,000 payable at time 12, to the nearest penny.
2 marks
Your answer