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Question: Spot interest rate curve: $$\begin{array}{c|c} t & r_t \\ \hline 0.25 & 1.50\% \\ \hline 0.5 & 1.65\% \\ \hline 0.75 & 1.79\% \\ \hline 1 & 1.92\% \end{array}$$

I have a one year loan for $1,000,000$ with principal paid at the end of the one-year period. The loan has a variable interest rate that resets at the beginning of each three month period. The interest rate will be the spot interest rate at the beginning of each three month period. I enter an interest rate swap as the payer where the characteristics of the loan are matched exactly. What quarterly swap rate will I pay?

Attempt: I know that the present value of my swap payments must be equal to the present value of the variable swap payments that the other guy pays, so:

$$1000000R \left(\cfrac{1}{1.015} + \cfrac{1}{1.015 \cdot 1.0165} + \cfrac{1}{1.015 \cdot 1.0165 \cdot 1.0179} + \cfrac{1}{1.015 \cdot 1.0165 \cdot 1.0179 \cdot 1.0192}\right) \\ = 1000000\left(\cfrac{1.015}{1.015} + \cfrac{1.0165}{1.015 \cdot 1.0165} + \cfrac{1.0179}{1.015 \cdot 1.0165 \cdot 1.0179} + \cfrac{1.0192}{1.015 \cdot 1.0165 \cdot 1.0179 \cdot 1.0192}\right)$$

This equation does not bear the correct value for $R$, my quarterly rate. In fact I'm off by a little over a factor of $2$. My thought processes: these denominators discount the payments back to the present. Since I have a fixed rate $R$, all my payments should be $1000000R$. Since the counterparty pays the variable rate according to each spot rate, each of his payments should be $1000000r_t$.

Solution: $$R \left(\cfrac{1}{1.015^{0.25}} + \cfrac{1}{1.0165^{0.5}} + \cfrac{1}{1.0179^{0.75}} + \cfrac{1}{1.0192}\right) = \left(\cfrac{.003729}{1.015^{0.25}} + \cfrac{0.00477}{1.0165^{0.5}} + \cfrac{0.005137}{1.0179^{0.75}} + \cfrac{0.005728}{1.0192}\right)$$

This produces $R = .0048$. I don't understand why the payments are discounted in this way, and I also don't understand the numerators for the variable payments. I am told that these are $f^*_{[t_i,t_{i+1}]}$, but I don't see why my original answer isn't right. Thanks.

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The given spot interest rates $r_{t_i}$ ($i=1,\ldots,4)$ are annualized. The actual accrual over the time period $t_i$ is $(1 + r_{t_i})^{t_i}$, i.e., the amount that a principal of $\$1$ invested at the spot rates grows to at maturity. (There are other conventions for computing interest but this one is consistent with your given answer.)

Hence, the appropriate discount factor for a cash flow payed at time $t_i$ is $1/(1 + r_{t_i})^{t_i}$.

The forward interest rate $f_{t_i,t_{i+1}}$ for the period from $t_i$ to $t_{i+1}$ is defined for $i > 0$ by

$$(1+r_{t_i})^{t_i}(1 + f_{t_i,t_{i+1}})= (1+r_{t_{i+1}})^{t_{i+1}}$$

This is the future spot rate at which the investor is indifferent between investing at the spot rate $r_{t_{i+1}}$ out to time $t_{i+1}$ versus investing at the spot rate $r_{t_{i}}$ out to time $t_{i}$ and then reinvesting the proceeds at that future rate for the period for $t_i$ to $t_{i+1}$. The convention for the first period is $(1+f_{t_0,t_1}) = (1+ r_{t_1})^{t_1}$.

The par swap rate $R$ is the fixed rate that equalizes the present value of the fixed-rate and floating-rate cash flows. With notional amount $N$, we have

$$NR\sum_{i=1}^4\frac{1}{(1+r_{t_i})^{t_i}} = N\sum_{i=1}^4\frac{f_{t_{i-1},t_{i}}}{(1+r_{t_i})^{t_i}}$$

The given answer is obtained by substituting $N = 1000000$, the appropriate values for $t_, r_{t_i}$ and using the forward rates

$$f_{t_0,t_1} = (1+ r_{t_1})^{t_1}-1 = (1+ 0.015)^{0.25} -1 = 0.00379, \\f_{t_1,t_2} = \frac{(1+ r_{t_2})^{t_2}}{(1+ r_{t_1})^{t_1}}-1 = \frac{(1+ 0.0165)^{0.5}}{(1+ 0.015)^{0.25}} -1 = 0.00477, \\ \text{etc.}$$

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  • $\begingroup$ Thank you for your answer. How come the discount factor for a cash flow payed at time $t_i$ is $1/(1+r_{t_i})^{t_i}$ and not $1 / \prod_i (1+r_{t_i})^{t_i}$? I would think I have to discount over each period with that period's given spot rate $\endgroup$
    – jeremy909
    Commented May 25, 2020 at 20:54
  • $\begingroup$ @jeremy909: Because "spot" rate $r_{t_i}$ means the interest rate that applies to a deposit that matures at time $t_i$. For $t_2 = 0.5$ think of $r_{t_2}$ as the annualized rate that you earn on a six month certificate of deposit. $\endgroup$
    – RRL
    Commented May 25, 2020 at 20:56
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    $\begingroup$ At time $t_0 = 0$ you can earn $r_{0.25}$ for a deposit that matures in $3$ months and $r_{0.5}$ for a deposit that matures in $6$ months. We don't know what the 3-month rate will be $3$ months from now. The $3$-month rate $3$-months forward -- $f_{0.25,0.5}$ is the expectation of that rate. $\endgroup$
    – RRL
    Commented May 25, 2020 at 21:03

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