1

How do I go about solving this please? I solve a lot of interest questions but this looks different. I'm just trying to solve as many as questions as possible. Am I ought to use the compound interest formula directly?

If an annual discount rate of $\left(\frac {39}8\right)\%$ is quoted for $3$ months treasury bills, what would it cost to buy a tranch of this bills with redemption value of $£100,000$? What would be the equivalent rate of return on the sum paid for them?

Narasimham
  • 40,495
john scott
  • 95

1 Answers1

0

As a preliminary, calculate the effective quarterly discount rate. If the annual discount rate quoted is a nominal rate (and not an effective one), then the effective quarterly discount rate is $$\frac{39}{8}\% \div 4 = \frac{39}{32}\%.$$

When a loan is phrased in terms of a discount rate, the interest on the loan is payed in advance. In effect, the borrower receives the principal $P$ less the interest $I$. If the principal is 100,000 and interest is paid in advance at a rate of $\frac{39}{32}\%$ of the principal, then the borrower receives 100,000 less $\frac{39}{32}\%$ of 100,000, or $$\begin{align*}100{,}000 - (0.0121875) 100{,}000 & = 100{,}000 (1 - 0.0121875) \\ & = 100{,}000(0.9878125) \\ & = 98{,}781.25.\end{align*}$$ If the tranche of T-bills is priced fairly, then the cost is 98,781.25.

The second question explores the relationship between discount rates and interest rates. If $i$ is the equivalent rate of return, i.e., the effective quarterly interest rate, then $i$ satisfies $$98{,}781.25(1 + i) = 100{,}000.$$ Solving for $i$, $$i = \frac{100{,}000}{98{,}781.25} - 1 = \frac{39}{3161}.$$ In general, if $d$ is the effective discount rate and $i$ the effective interest rate, then $i = \frac{d}{1 - d}$.

To express the effective quarterly interest rate as a nominal annual interest rate, simply multiply the effective quarterly interest rate by $4$: $$4i = 4 \left(\frac{39}{3161}\right) = \frac{156}{3161}.$$ To express the effective quarterly interest rate as an effective annual interest rate, solve the equation $$(1 + j) = {(1 + i)}^4,$$ where $j$ is the effective annual interest rate and $i$ is the effective quarterly interest rate. In this case, $$j = {\left(1 + \frac{39}{3161}\right)}^4 - 1 \approx 5\%.$$

Andrew
  • 459
  • 3
  • 8
  • If the annual rate is $\frac {39}8$, the quarterly rate should be one quarter of this, or $\frac {39}{32}$, not four times. If I could turn 80,500 into 100,000 in a quarter I would love it. – Ross Millikan Jan 28 '16 at 20:46
  • @RossMillikan, I edited my answer with your correction. Thank you. – Andrew Jan 28 '16 at 21:03
  • @Andrew thank you very much sir. Please how did you get the 80,500? – john scott Jan 29 '16 at 01:48
  • @Andrew I think it should be replaced with 98, 781.25. Also sir, what if I want to calculate the annual equivalent rate of return on the sum paid for them and not the quarterly equivalent rate of this same question, how do I go about that sir? – john scott Jan 29 '16 at 02:21
  • @Kolaoloke, it would appear my previous edit in response to Ross Millikan's correction was not as thorough as I had hoped. I have edited my answer once more: i) correcting all numerical quantities and ii) including a paragraph on converting the quarterly rate to annual rates. I hope that helps. – Andrew Jan 29 '16 at 02:52
  • @Andrew thanks again sir. You really helped. I really get it now. Can you please suggest any text for further reading on this topic, sir? – john scott Jan 29 '16 at 03:05